High-Order Accurate, Low Numerical Diffusion Methods for Aerodynamics

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Progress in Aerospace Sciences 41 (2005) 192–300 www.elsevier.com/locate/paerosci

High-order accurate, lownumerical diffusion methods for aerodynamics John A. Ekaterinaris

Foundation for Research and Technology FORTH, Institute of Applied and Computational Mathematics IACM, FORTH/IACM, 71110 Heraklion, Crete, Greece

Abstract

In recent years numerical methods have been widely used to effectively resolve complex flow features of aerodynamics flows with meshes that are reasonable for today’s computers. High-order numerical methods were used mainly in direct numerical simulations and aeroacoustics. For many aeronautical applications, accurate computation of vortex- dominated flows is important because the vorticity in the flow field and the wake of swept wings at an incidence and rotor blades largely determines the distribution of loading. The main deficiency of widely available, second-order accurate methods for the accurate computation of these flows is the numerical diffusion of vorticity to unacceptable levels. Application of high-order accurate, low-diffusion numerical methods can significantly alleviate this deficiency of traditional second order methods. Furthermore, higher-order space discretizations have the potential to improve detached eddy simulation predictions of separated flows with significant unsteadiness. Recently developed high-order accurate finite-difference, finite-volume, and finite-element methods are reviewed. These methods can be used as an attractive alternative of traditional low-order central and upwind computational fluid dynamics methods for improved predictions of vortical and other complex, separated, unsteady flows. The main features of these methods are summarized, from a practical user’s point of view, their applicability and relative strength is indicated, and examples from recent applications are presented to illustrate their performance on selected problems. r 2005 Elsevier Ltd. All rights reserved.

Contents

1. Introduction ...... 194 1.1. Prior work on high-order numerical methods ...... 196 2. Governing equations ...... 198 2.1. Conservative form of the NS equations ...... 198 2.2. Entropy splitting ...... 199 2.3. Time integration of the Euler and NS equations ...... 200 2.3.1. Explicit temporal schemes...... 201 2.3.2. Implicit schemes...... 202 2.3.3. Time accurate solutions with multigrid...... 202

E-mail address: [email protected].

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 193

3. High-order finite-difference schemes ...... 203 3.1. Centered compact schemes ...... 203 3.2. Boundary closures for high-order schemes ...... 204 3.3. Simultaneous evaluation of the first and second derivative with compact schemes ...... 205 3.4. Modified high-order finite-difference schemes ...... 206 3.4.1. Upwind high-order schemes ...... 206 3.4.2. Dispersion-relation-preserving (DPR) scheme ...... 207 3.5. Spectral-type filters...... 207 3.6. Characteristic-based (ACM) filters ...... 208 3.6.1. ENO and WENO ACM filters ...... 210 3.7. Finite-volume compact difference-based schemes...... 211 3.8. Results with ACM filters ...... 212 3.9. Results with compact schemes and filters ...... 218 4. ENO and WENO schemes ...... 227 4.1. High-order reconstruction ...... 228 4.2. Approximation in one dimension...... 228 4.3. One-dimensional conservative approximation of the derivative ...... 229 4.4. ENO reconstruction ...... 230 4.5. 1D finite-volume ENO scheme ...... 231 4.6. 1D finite-difference ENO scheme ...... 232 4.7. WENO approximation ...... 232 4.8. Multidimensional ENO and WENO reconstruction...... 234 4.8.1. Finite-volume reconstruction for Cartesian mesh...... 234 4.8.2. Two-dimensional reconstruction for triangles ...... 235 4.8.3. Multidimensional finite-difference ENO ...... 236 4.9. Optimization of WENO schemes ...... 236 4.10. Compact WENO approximation ...... 237 4.11. Hybrid compact-WENO scheme for the Euler equations ...... 239 4.12. Applications of ENO and WENO ...... 240 5. The discontinuous Galerkin (DG) method ...... 256 5.1. DG space discretization ...... 257 5.2. Element bases ...... 258 5.3. Arbitrary high-order DG schemes ...... 260 5.4. Analysis of the DG method for wave propagation ...... 260 5.5. Dissipative and dispersive behavior of high-order DG method ...... 261 5.6. Limiting of DG expansions ...... 261 5.6.1. Rectangular elements ...... 262 5.6.2. Triangular elements ...... 262 5.6.3. Component-wise limiters...... 263 5.7. DG stabilization operator ...... 264 5.8. DG space discretization of the NS equations ...... 264 5.9. The DG variational multiscale (VMS) method ...... 265 5.10. Implicit time marching of DG discretizations ...... 269 5.11. p-type multigrid for DG ...... 270 5.12. Results with the DG method...... 270 6. The spectral volume (SV) method ...... 276 6.1. Fundamentals of the SV method ...... 277 6.2. SV method in one dimension...... 279 6.2.1. Reconstruction for SV in one dimension ...... 279 6.3. Polynomial reconstruction in triangular SVs ...... 281 6.4. Linear, quadratic, and cubic SV reconstructions ...... 283 6.5. Spectral volume partitions...... 284 6.6. Multidimensional limiters ...... 284 6.7. The SV method for two-dimensional systems ...... 285 6.8. Multidimensional TVD and TVB limiters ...... 286 6.9. The SV method for the Navier–Stokes equations ...... 287 ARTICLE IN PRESS

194 J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300

6.9.1. SV formulation for 2D convection–diffusion equation ...... 287 6.9.2. Extension of the SV method to the NS equations ...... 287 6.10. The SV method in three dimensions...... 288 6.11. Results with the SV method ...... 288 7. Conclusions...... 292 . Appendix ...... 293 References ...... 293

1. Introduction imation is proportional to hn. If we cut the grid spacing in half, the error in the second-order scheme reduces by a In this paper, we explore applications of high-order factor of 4 and the error in the fourth-order scheme methods in aerodynamics. High-order methods typically reduces by a factor of 16. Provided that the computa- have at least third-order spatial accuracy. Traditionally, tional requirements for the second- and fourth-order second-order accurate numerical methods are often schemes are similar, the fourth-order scheme is clearly preferred in practical aerodynamic calculations because more efficient. Efficiency improvements are even greater of their simplicity and robustness. High-order accurate for higher-order approximations. A demonstration of methods are often perceived as less robust, costly to run, the superior performance of high-order, low-dissipative, and complicated to understand and code. As a result, centered schemes compared to lower-order more dis- there are very fewworkingcomputational fluid dy- sipative TVD schemes in preserving vorticity [1] is shown namics (CFD) codes that use higher-order accurate in Fig. 1. The conversion of a two-dimensional isentropic schemes for production numerical simulations of com- vortex is carried out for long time with the numerical pressible flow. This is especially true for codes designed solution of the inviscid compressible flowequations on a to compute steady flows. We will attempt to dispel this Cartesian mesh. At the absence of physical viscosity, the negative impression about high-order methods. loss in vortex strength due to the numerical diffusion of Before presenting the essential details of high-order the scheme is evident. For three-dimensional computa- methods, we point out that in many practical aero- tions, grid refinement in all three directions, which is dynamic problems, the solution structures are so necessary for better resolution of vortical structures that complicated and their time evolution is so long, that it are isotropic, becomes very expensive computationally. is impossible to obtain an acceptable solution with Therefore, application of high-order schemes is expected today’s computing speeds using high-grid density and to significantly decrease computing cost. low-order methods. These problems often involve To better demonstrate the potential of higher-order regions of complicated but smooth flowstructures, such methods make the following assumptions: (1) The error as vortices interacting with each other or with shear in the solution is OðhpÞ where h is the mesh length and p layers, and regions that contain both shocks and is the order of accuracy. (2) The number of intervals Ni complex smooth structures. Some examples of these in the grid, or elements for a finite-element method, is Àd flows are briefly discussed in the following paragraphs. related to the cell size by Ni ¼ Oðh Þ, where d is the Vortical flowfields are especially challenging for the spatial dimension of the problem. (3) Higher-order of low-order numerical methods that are typically found in accuracy is achieved by increasing the stencil, or the d current Euler and Navier–Stokes flowsolvers. The main number of unknowns per element, Ns ¼ Oðp Þ. Thus the cause for this deficiency is that these vortical flow total number of operations or unknowns, N, scales as d features deform and dissipate prematurely due to N ¼ NiNs ¼ Oððp=hÞ Þ. (4) The operation count, W, excessive numerical diffusion in the solution algorithms. required to solve the discrete problem scales as It is well recognized that low-order CFD algorithms W ¼ OðNwÞ, where w is the complexity of the discretiza- require extremely fine grid resolution to accurately tion method. (5) The total time required for the convect and preserve the strength of vortical flowfields. numerical solution is T ¼ W=F, where 1=F is the time This fine grid resolution requirement leads to extremely for a single operation, which depends on the processor large computational problems that cannot be solved on speed. These assumptions lead to the conclusion that the even the largest parallel computer architectures. computing time, T, required to achieve a specific error It is straightforward to demonstrate the power of tolerance E scales as high-order numerical methods by examining the numer- 1=p wd ical approximation of a smooth function on a regular T ¼ Oððp=E Þ =FÞ grid. For a second-order accurate numerical scheme, the or taking the logarithm error in the functional approximation is proportional to 2 1 h , where h the mesh size Dx. For an nth-order accurate log T wd À log E þ log p À log F. numerical scheme, the error in the functional approx- p ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 195

t = 50 t = 100 1.2 1.2

1.0 1.0

0.8 0.8 Exact Exact TVD22 TVD22 TVD44 TVD44 0.6 TVD66 0.6 TVD66

Density cut at y=0 0.4 0.4 (a) 0246810(b) 0246810 1.2 1.2

1.0 1.0

0.8 0.8 Exact Exact ACM22 ACM22 ACM44 ACM44 0.6 ACM66 0.6 ACM66

Density cut at y=0 0.4 0.4 0246810 0246810 (c) xx(d)

Fig. 1. Stationary vortex: comparison of the various orders of TVD and ACM methods with the exact solution, illustrated by density proﬁles at the centerline y ¼ 0, at t ¼ 50 and 100 for a 401 Â 81 grid (k ¼ 0:05).

For stringent accuracy requirements (E51), it is wall, where the grid density is small, is responsible for expected that the term log E dominates the log p terms. inaccuracies of the computed flowfield. Therefore, the computing time will depend exponentially Attempts to overcome these deficiencies of RANS on the order of accuracy, p, the complexity of the methods to accurately compute turbulent separated flow method, w, and the grid resolution d. The above have led to the development of large eddy simulation reasoning demonstrates that, from a practical point of (LES) [2–7], and more recently the detached eddy view, it pays off to improve the order of accuracy simulation (DES) approaches [8–10]. Both of these provided that the operation count does increase techniques, fully resolve the three-dimensional vortical dramatically. Furthermore, from the first equation it is structures and turbulent motions in detached flow evident that small changes of the ratio w=p can reduce regions. In addition, LES methods also resolve certain computing time, which scales only inversely by the range of the small-scale turbulent flowstructures in the processor speed F. attached or separated flowboundary layers. DES The most widely used flow solvers for aerospace methods, on the other hand, model the attached flow applications are based on the solution of the Reynolds- regions with RANS techniques. As a result, the averaged Navier–Stokes (RANS) equations. These resolution requirements of DES are significantly lower RANS methods have recently benefited from improved than LES for the attached boundary layers, where the performance of workstations, supercomputers, and RANS turbulence model is used. In the separated flow parallel processors. Hardware improvements combined field, however, the resolution requirements for DES are with advances in areas such as grid generation and more comparable to those for LES calculations. Like the computationally efficient solution algorithms made RANS, LES and DES can also benefit from the RANS a tool that often complements wind tunnel. application of high-order numerical algorithms to better RANS flowsolvers have proven useful for airloads resolve separated flowregions. Most of the high-order prediction, examination of detailed and gross flow numerical methods that we will discuss in this paper are features, as well as engineering design. The primary therefore expected to have applications to RANS, LES limitation of RANS is the inability to predict turbulent, and DES flowsolvers. separated flow. Turbulence models used in RANS, In particular, high-resolution requirements that are model the entire spectrum of turbulence and they are encountered in typical LES of complex flows can be met unable to accurately predict phenomena dominated by with the application of high-order numerical methods. turbulent eddies of massively separated flows. The For LES of flows in simple domains, the resolution deficiency of RANS to predict turbulent separated flow problem is solved with the use of the highly accurate combined with the unphysical diffusion of vorticity by spectral methods [11,12]. Spectral methods are not, the numerical scheme in flowregions awayfrom the however, easy to apply for complex domains and ARTICLE IN PRESS

196 J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 compressible flows with discontinuities. For flows with complex geometries that preclude the use of spectral methods, the use of high-order numerical methods is a necessity in order to minimize the overall computational cost. The continued increase of computing power and developments of parallel computing are expected to make possible DES or even LES of flows with increased complexity. For these applications, use of high-order accurate CFD methods is necessary. While the use of high-order accurate numerical methods is relatively common for direct numerical simulations (DNS) and LES methods, high-order numerical methods have not been commonly used for DES. As a result, DES of complex flows often exhibit significant grid dependence. It is expected that use of Fig. 2. Computed surface pressure distribution with different high-order spatially accurate numerical schemes com- grid densities and order of accuracy over a double delta wing at 6 bined with improvements of DES models [13] will M ¼ 0:22, a ¼ 19 ,Re¼ 4 Â 10 . eliminate the grid dependencies in DES calculations. We hope that the resulting high-order RANS/DES hybrid models will yield reliable predictions of realistic, separated, unsteady flows in a computationally efficient manner.

1.1. Prior work on high-order numerical methods

The favorable effects of high-order accurate methods was early recognized even in several RANS simulations. For example, simulations of delta wing vortical flows [14], wind tip vortices [15], and helicopter rotor [16]. The sensitivity of the numerical solution on grid resolution [14,16], and the order of accuracy of the numerical scheme [14,15] on the resolution of vortices was demonstrated. Delta wing flows are dominated by the leading edge vortices and accurate capturing of the correct strength of the leading edge vortex strength is essential for the prediction of loads and the vortex breakdown location. Fig. 2 shows the effects of increased grid resolution and high-order of accuracy for the computation of the leading edge vortices over a double delta wing [14]. It appears that the increase in order of accuracy can yield better resolution of the vortices even with lower grid resolution. Similar conclusions are reported for the numerical prediction of the wing-tip vortex [15]. In this study, increased grid density and higher-order numerical schemes significantly improved the accuracy of the final results. The comparisons of the predicted peak velocity in the vortex core [15] shown in Fig. 3 demonstrate that in addition to Fig. 3. Comparison of third- and fifth-order accurate differencing on the solution for a wake-case grid having 371,000 points turbulence model the order of the scheme plays a very ð35 Â 103 Â 103Þ. significant role. Furthermore, computations of rotorcraft flows [16,17] demonstrated that the tip vortex is severely diffused by the first passage even in computations performed with grid resolution for the tip vortex. This problem was 10 million grid points. These computations used curvi- addressed by the tetrahedral unstructured flowsolver linear structured meshes and suffered from an inap- approach [17]. It was concluded, however, that adaptive propriate placement of the grid points and insufficient tetrahedral mesh approaches could have only limited ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 197 success for even the simplest hovering rotor cases centered schemes were found particularly useful for because the adapted grid is anisotropic with computa- prediction of noise sources. These high-order centered tional cells of very large aspect ratio. The overset grid schemes, which can be extended into finite-volume approach [18] appears to resolve some of the problems context, and the spectral-type or characteristic-based associated with grid topology but it still requires very filters, which are necessary to stabilize centered schemes high grid resolution (greater than 60 million grid points) and suppress spurious modes, are summarized in for accurate capturing of the tip vortex. On the other Section 3. hand, calculations performed with high-order accurate For high-speed flows with shocks, numerical solutions schemes in [19–21] and more recently in [22] demon- that are uniformly high-order accurate up to the strated that significant improvements in the accuracy of discontinuity can be obtained with ENO [32] and rotor performance predictions can be obtained with just weighted ENO (WENO) [33,34] schemes. These meth- a small increase in computing cost. ods are presented in detail in Section 4. On the other For the computation of flows with shocks, methods hand, it was shown that explicit filters [30] could also be designed to regularize the numerical solution have been used with high-order centered schemes to obtain shock studied since the early attempts of von Neumann and capturing. Application of explicit filters [30] yielded Richtmayer [23] who used finite-difference techniques improved computational efficiency of flows with dis- combined with the so-called artificial viscosity or continuities compared to ENO [32] or WENO methods numerical dissipation. Use of numerical dissipation in [33,34]. Explicit filters can be easily implemented into the finite-difference and finite-volume context has found existing codes because the filter step is essentially widespread application in solution methods for com- independent of the basic differencing scheme and is pressible aerodynamic flows. The main difficulty in the applied as post processing. In the same spirit, Yee et al. application of these methods in DNS, DES, and LES of [1] showed that the dissipative part of a shock-capturing compressible flows is the control of numerical dissipa- scheme could be applied after each time step to tion necessary to capture discontinuities that occur in regularize the numerical solution and acts like a filter. such flows. Too much numerical dissipation smears out Moreover, to meet the requirement of a local application important flowfield features. Too little numerical of the numerical dissipation, the amplitude of the dissipation yields unstable solutions. It was recently dissipation is evaluated with a sensor derived from the demonstrated (see [24] and references therein) that the artificial compression method (ACM) of Harten [35]. inherent dissipation of nonlinear high-resolution meth- The filters of Yee et al. [1] are referred to as ods can be exploited to successfully compute certain characteristic-based filters and they are summarized in types of turbulent flows [25] without need to resort to an Section 3. The numerical test in Yee et al. [1] used total explicit turbulence model. variation diminishing (TVD) schemes to construct the In previous numerical investigations [26,27],itwas characteristic-based, non-linear filter. The possibility of shown that even the reduced numerical dissipation of using high-order non-linear filters based on essentially high-order, shock-capturing schemes can lead to sig- non-oscillatory (ENO) reconstruction has been demon- nificant damping of turbulence fluctuations and mask strated in [36]. the effects of the subgrid-scale (SGS) models. For these In parallel with the finite-difference [37], finite-volume cases, a local application of the shock-capturing scheme methods [38], and high-resolution methods [39], which was found absolutely necessary in order to minimize found widespread application in CFD and turbulence numerical dissipation. In the study of [26], for example, simulation, finite-element methods [40–44] were also this requirement was achieved by means of the applica- used for convection-dominated problems. Application tion of an essentially non-oscillatory (ENO) scheme only of the finite-element method is far from trivial for non- in the shock-normal direction and over a fewmesh linear convective problems, such as compressible flow points around the mean shock position. Unfortunate- with discontinuities. For such cases, the finite-element ly, in most cases, the shock position is unknown and numerical solution must capture the physically relevant one needs to introduce a sensor to detect possible discontinuities without introducing spurious oscilla- discontinuities. tions. This issue is addressed successfully by the In recent years, efforts were made to alleviate the discontinuous Galerkin (DG) method [45]. The DG effects of numerical diffusion introduced by shock method assumes discontinuous approximate solutions. capturing schemes and upwind methods. It was shown Subsequently, it treats discontinuities in a manner [28–31] that for compressible flows without disconti- analogous to high-resolution finite-difference and fi- nuities, high-order centered finite-difference schemes are nite-volume methods for nonlinear hyperbolic systems sufficiently stable and accurate for the computation of by incorporating suitably defined numerical fluxes and convecting vortical structures and aeroacoustic distur- slope limiters into the finite-element framework. Due to bances when they are combined with explicit, spectral- its local character, the DG finite-element method is very type filtering for numerical stability [28]. In addition, suitable for local grid refinement and becomes highly ARTICLE IN PRESS

198 J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 parallelizable. This method is also presented and accurate schemes are also summarized in Section 2. The analyzed in Section 5. presentation of high-order accurate spatial discretization Recently, another type of high-order accurate, con- methods is given in Sections 3–6. In Section 3, high- servative and computationally efficient scheme was order accurate discretization with finite-difference and introduced for the solution of conservation laws in finite-volume centered methods is presented. In Section unstructured grids. This scheme is known as the spectral 4, ENO and WENO reconstruction is explained and volume (SV) method [46]. The concept of a ‘‘spectral WENO and ENO high-order schemes are presented. volume’’ was introduced to achieve high-order accuracy The discontinuous Galerkin method is presented and in an efficient manner similar to spectral methods and at analyzed in Section 5. In Section 6, the recently the same time retain the benefits of the finite-volume developed spectral volume method is presented. At the formulations for problems with discontinuities. In the end of each section selected examples from the applica- SV method, each spectral volume, which is the same as tion of the high-order methods are shown. Finally, in the traditional triangular or tetrahedral finite volume, is Section 7, some comparisons and general remarks are further subdivided into volumes called control volumes. given. Cell-averaged data from these control volumes are used to reconstruct a high-order approximation in the spectral volume, while Riemann solvers are used to 2. Governing equations compute the fluxes at the spectral volume boundaries. The main difference of the SV method and DG method The full Navier–Stokes (NS) equations govern non- is that for the SV method, the cell-averaged variables linear fluid dynamics and aeroacoustics over complex in the control volumes are updated independently. configurations. For the majority of compressible flow Similar to the DG method, the SV method uses TVD simulations, these equations are cast in the strong or total variation bounded (TVB) limiters to eliminate/ conservation form [37]. For numerical solutions based reduce spurious oscillations near discontinuities. A very on the Galerkin/least-squares method, different sets of desirable feature of the SV method is that the variables may be used [54]. For all sets of variables reconstruction is carried out analytically, and does not tested with the Galerkin/least-squares method [54], involve large stencils in contrast to the computationally global conservation and correct shock structure was intensive reconstruction in high-order finite-volume achieved for any set of variables. This is not, however, methods. true for finite-difference and finite-volume numerical The DG and SC methods are both suitable for high- methods traditionally used in aerodynamics. These order discretization of complex domains using unstruc- methods use the conservative flowvariables to ensure tured meshes. In addition, they are fully conservative conservation and discontinuity capturing. The primitive due to the use of Riemann fluxes across element variable formulation was used rarely in aerodynamics boundaries. Another high-order conservative scheme either with shock fitting schemes [55], or for the for unstructured quadrilateral grids is the multidomain computation of subsonic compressible flows without spectral method on a staggered grid recently developed discontinuities [56]. The transformation from conserva- by Kopriva and Kolias [12,47–49]. The multidomain tive to primitive variables or other sets of variables is spectral method is similar to the spectral element obtained by multiplying the conservative variables method by Patera [50]. The spectral element method is vector with the appropriate flux Jacobian. For example, high-order accurate and more flexible compared to primitive variables are obtained by multiplying with spectral methods for discretizations of complex do- M ¼ qU=qV, U ¼½r; ru; rv; E , V ¼½r; u; v; p see [57] mains. It is however not conservative. Although very T for more details. high order of accuracy can be achieved with both the multidomain spectral and the spectral element method, these methods are difficult to extend to other cell types 2.1. Conservative form of the NS equations such as triangles or tetrahedral cells [51]. These spectral methods and other recent, less widely used, high-order The conservative variable formulation of the N–S methods [52,53] will not be presented here. Further equations in divergence form is information about these methods can be found in the qU 1 original references. A detailed presentation of the theory þrÁFi ¼ rÁFv, (2.1) and implementation of the spectral element method can qt Re also be found in the recent book by Karniadakis and where U is the conservative variable vector, Fi Sherwin [43]. is the inviscid flux vector, and Fv is the viscous flux This paper is organized as follows: The governing vector. equations in differential and integral form are presented Numerical solutions with finite-volume (FV) methods in Section 2. Time integration methods for high-order are obtained with the integral form of the NS equations ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 199 for a control volume O with boundary qO neighborhood of discontinuities. For the resolution of Z I smooth but complex flowfeatures and for wave q U dx þ ½FðU; nÞÀF vðU; rU; nÞ dS ¼ 0, propagation, the order of accuracy of the numerical qt O qO solution increases (p-refinement) and a relatively coarse, (2.2) canonical mesh is used. Finite-difference methods on structured type grids are where U ¼½r; rV; rET; FðU; nÞ¼V Á nU; F ðU; rU; nÞ v often used for the numerical solution of the governing ¼½0; t; t Á V À q Á nT, with t and q representing the stress equations because of their efficiency. The major and heat flux vectors, V ¼ðu; v; wÞT is the velocity difficulty with finite-difference methods is structured- vector, and n is the outward unit normal vector to the type grid generation. The task of generating structured boundary qO. For flows with discontinuities the integral grids over complex configurations is still a serious form holds and any discrete scheme must obey both challenge even with the multiblock structured grid local and global conservation in order to capture approach. The powerful approach of overlapping grids, correctly weak solutions [58]. Finite-volume (FV) or Chimera-type grids [67], where structured grids methods gained popularity in aerodynamics because generated about different simple components are they make possible numerical solutions in complex allowed to overlap, further facilitates volume grid domains with unstructured or mixed type grids [59]. generation. Numerical solutions in complex domains During the 1980s upwind mechanisms were introduced with finite-difference methods on body-fitted deformed into FV algorithms leading to increased robustness of meshes [68] are obtained by expressing, Eq. (2.1) (see FV for applications with strong shocks and provided [37]) in terms of a generalized non-orthogonal curvi- better resolution of viscous layers due to the decrease of linear coordinate system ðx; Z; zÞ using general trans- numerical dissipation compared to FV methods that formations x ¼ xðx; y; zÞ, Z ¼ Zðx; y; zÞ, z ¼ zðx; y; zÞ employed artificial dissipation [60,61]. as follows: Finite-element (FE) methods, which use the weak "# form of Eq. (2.1), gained popularity in aerodynamics q U qFb qGb qHb 1 qFc qGc qHc þ þ þ ¼ v þ v þ v . with the development of the discontinuous Galerkin qt J qx qZ qz Re qx qZ qz method [45] and the stabilized finite-element methods [62–64]. The weak form is obtained by multiplying the (2.4) t strong form of Eq. (2.1) by a test function W For turbulent flowcalculations, the molecular viscos- and integrating over the domain. The weak form of ity m is replaced by the turbulent eddy viscosity m þ mt. Eq. (2.1) is The turbulent eddy viscosity mt is obtained from Z I q turbulence models developed during the last decades. W tU dx þ W t½FðU; nÞÀF ðU; rU; nÞ Á n dS These turbulence models range from simple algebraic qt v OZ qO models to one- and two-equation turbulence models, or t À rW ½FðU; nÞÀF vðU; rU; nÞ dx ¼ 0. ð2:3Þ more sophisticated Reynolds stress models [69]. O The integral form, Eq. (2.2), is the weak form of 2.2. Entropy splitting Eq. (2.3) for unity weight function. The stabilized finite- element methods augment the weak form of the The unique properties of the compressible Euler governing equations with stabilization terms [63–65]. equations for a perfect gas allows splitting of the flux The real power of unstructured grid methods with FV vectors [70]. A special splitting of the flux derivative can or FE discretization is their ability to adapt not only to be obtained using a convex entropy function and certain complex geometries but also to solution features without homogeneous properties. This splitting yields a sum of being constrained by considerations such as grid a conservative portion and a non-conservative portion structure, orthogonality or topology. However, aniso- of the flux derivative referred to by Yee et al. [71] as tropic grid adaptation of three-dimensional, high the ‘‘entropy splitting’’. Yee et al. [71] investigated the Reynolds number flows, which contain both disconti- choice of the arbitrary parameter that determines the nuities and smooth but complex flowfeatures, still amount of splitting and its dependence on the type of remains a challenge. Consider for example high-lift physics of interest to CFD. In addition, the manner the systems where shock waves, confluent boundary layers, splitting affects the nonlinear stability of central schemes and wake roll-ups may be present in the flowfield. Grid for long time integration of unsteady flows, such as in adaption for these flows, which are common in nonlinear aeroacoustics and turbulence dynamics, was aerodynamic applications, combined with high-order assessed. methods offers the additional possibility of hp-type The first step for the derivation of the entropy refinement [43,66]. hp refinement uses low-order accu- splitting for the compressible Euler equations for a rate solution and high grid density (h-refinement) at the perfect gas ðp ¼ rRTÞ is to introduce a symmetry ARTICLE IN PRESS

200 J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 transformation form the vector of the conservative T at t = 160, Un−split T at t = 160, α= −3 variables U to a newvector of symmetry variables W, referred to as the ‘‘entropy variables’’. The transformation is chosen so that the flux Jacobian matrix with 40 40 respect to W, F W ¼ qF=qW is symmetric and positive definite. A family of symmetry transformations based on a scalar convex function Z, referred to as ‘‘entropy function’’ have been derived for the Euler equations for a perfect gas by Harten [72] and Gerritsen and Olsson 20 20 [73,74]. The entropy function Z has the form Z ¼ rxðsÞ, (2.5) where the function xðsÞ is an arbitrary but differentiable 0 0 function of a dimensionless physical entropy s ¼ log ðprÀgÞ. The entropy variables W are then given by W ¼ qZ=qU and are chosen such that FðUðWÞÞ, GðUðWÞÞ, HðUðWÞÞ and UðWÞ are homogeneous functions of W of degree b, i.e. there is a constant b such that for all f -20 -20 the conservative variable vector and the flux satisfy:

UðfWÞ¼fbUðWÞ, FðfWÞ¼fbFðWÞ. ð2:6Þ -40 -40 The homogeneity property implies that

F W W ¼ bFðUðWÞÞ,

U W W ¼ bU, ð2:7Þ 020 020 (a)101x101 Grid (b) 101x101 Grid where U W ¼ qU=qW is the transformation matrix from conservative variables to entropy variables and UW ¼ Fig. 4. Vortex pairing: comparison of normalized temperature qFðWÞ=qW is the flux Jacobian with respect to entropy contours for a ¼À3 with the un-split approach at time t ¼ 160 on a 101 Â 101 grid with k ¼ 0:7 for the nonlinear fields and variables. The splitting of the flux derivative F x is k ¼ 0 for the linear fields using ACM66. qF 1 1 F ¼ ¼ ðF WÞ þ F W x qx b þ 1 W x b þ 1 W x ison of Fig. 4 demonstrated improvements of the b 1 ¼ F x þ F W W x. ð2:8Þ numerical solutions for vortex pairing computed with b þ 1 b þ 1 entropy splitting and the unsplit scheme. For the value The entropy variables vector is a ¼À3 ðb ¼ 4Þ, used in [71] the improvements of the 2 3 entropy split scheme are evident. A comparison of 2 3 a À 1 W 1 e þ p computations obtained with entropy split and unsplit 6 7 6 g À 1 7 6 W 7 6 7 schemes for shock wave impingement on a spatially 2 n 6 Àru 7 6 7 p 6 7 evolving mixing layer is shown in Fig. 5. Deterioration is 6 W 7 6 7 W ¼ 6 3 7 ¼ 6 Àrv 7. (2.9) obtained for a ¼ 10 ðb ¼À28:5Þ corresponding to 6 7 r 6 7 4 W 4 5 4 Àrw 5 103.6% conservative portion, but for a ¼À10 ðb ¼ W 5 r 21:5Þ corresponding to less than 100% conservative portion the solution shows reduction of spurious noise. Expressions for FðUðWÞÞ and F W appearing in Increase of the conservative portion beyond 80% splitting of the flux derivative of Eq. (2.8) can be found often defeats the purpose of using splitting since any in [71]. gains in stability are diminished by the expense of added Introduction of entropy splitting increases signifi- CPU computation required by splitting. For additional cantly the computing cost of the inviscid part. It was comments and details on the choice of the splitting shown however in [71] that the increase in computing parameters b, see Yee et al. [71]. cost is justified because for turbulent flowsimulations, for example, that involve long time integrations and 2.3. Time integration of the Euler and NS equations contain weak shock waves, entropy splitting can minimize the use of numerical dissipation due to its The time scales of turbulence are very small and nonlinear stability property. For example the compar- the evolution of turbulent structures is very rapid. ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 201

p at t = 120 Un−split (min= 0.311290 max = 2.89300) 2.3.1. Explicit temporal schemes 20 Time integration can be obtained with the method of 10 lines by considering the governing equations as a system 0 of ordinary differential equations ODEs in time where -10 the right-hand side contains the space discretization. -20 (a) 0 50 100 150 200 Possible choices for explicit time stepping of these ODEs are the third- or higher-order explicit linear multistep p at t = 120 Un−split α=−10 (min= 0.317116 max = 2.87823) 20 methods (LMMs) Gear [78], Lambert [79] and Bucher 10 [80]. For non-stiff or moderately stiff multidimensional 0 problems, high-order accuracy (higher than second) -10 temporal accuracy may be obtained with the Runge– -20 Kutta (RK) methods. RK methods advance in time the 0 50 100 150 200 (b) following semi-discrete form of the governing equations p at t = 120 α =10 (min= 0.310554 max = 2.85710) 20 Ut ¼ RðUÞ,, where R is the spatial discretization 10 operator that is discussed in detail in the following 0 sections. -10 A general RK can be written in the form -20 0 50 100 150 200 XiÀ1 (c) ðiÞ ðkÞ ðkÞ U ¼ aikU þ DtbikRðU Þ; i ¼ 1; ...; m, Fig. 5. Shock–shear-layer interaction: comparison of the k¼0 pressure contours for a ¼Æ10 with the un-split approach using (2.10) ACM66 at t ¼ 120 on a 321 Â 81 grid with k ¼ 0:35 for the nonlinear ﬁelds and k ¼ 0:175 for the linear ﬁelds. where Uð0Þ ¼ U n; U ðmÞ ¼ Unþ1

As a result, in DNS and LES of transitional and for aikX0, bikX0 Eq. (2.10) is just a convex combination turbulent flows the time step is not determined by of Euler forwardP operator with Dt replaced by kÀ1 stability considerations but from time resolution re- ðbik=aikÞDt since k¼0 aik ¼ 1. The general RK method quirements. For this class of problems, explicit schemes, of Eq. (2.10) is TVD under the CFL condition DtpcDt if which are computationally efficient, easy to parallelize, c ¼ mini;kðaik=bikÞ, and aikX0; bikX0. In many CFD and provide high accuracy in time, are the methods of applications the third-order (TVD) Runge–Kutta meth- choice. In many cases, use of explicit methods is also od, RK3-TVD [81] that is compatible with TVD, ENO necessary for wave propagation applications, such as or WENO schemes was used. This RK method is TVD aeroacoustics, where dissipation and dispersion must be in the sense that the temporal operator itself does not kept at very lowlevel. For this class of problems, increase the total variation of the solution. The TVD optimized explicit schemes have been developed. property of the time integration scheme plays an In many CFD applications, steady-state solutions are important role for time marching of nonlinear hyper- needed. Convergence to a steady state is often obtained bolic problems. It was shown in Ref. [81] that the TVD using explicit and/or implicit methods with multigrid property is achieved with a four-stage, fourth-order acceleration or with implicit methods. For time- TVD RK-4 method which is, however, quite intensive depended aerodynamic problems, where time accuracy computationally because during the second and third is of interest, such as rotor aerodynamics, time integra- stage two additional Re’s, the adjoint operator of R, must tion of the Euler and Navier–Stokes equations is often be computed. In addition, a fifth-order accurate RK obtained with implicit methods that are second order method was presented by Shu and Osher in [32]. For accurate in time. Implicit methods [75] are more order of accuracy higher than four, however, the intensive computationally compared to explicit methods number of stages is larger than the order of accuracy and they do not offer high accuracy in time. However and RK methods with order of accuracy higher than they do not have limitations in time step due to four become very intensive computationally. For multi- numerical stability for highly stretched meshes required dimensional calculations storage is usually of impor- for viscous flowsimulations. Another class of methods tance. Therefore, in [82,83] lowstorage RK method, for time integration of time-dependent viscous flow which only require two storage units, have been problems is explicit methods with multigrid acceleration developed. and dual time stepping. Multigrid methods [76,77] for Numerical schemes for computations of wave propa- convergence acceleration is a wide topic and will not be gation have special requirements. In computational presented here. A brief overviewof explicit and implicit aeroacoustics, for example, numerical schemes that have method used in CFD is given in the following sections. minimal dissipation and dispersion errors in both time ARTICLE IN PRESS

202 J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 and space discretization are desired because the acoustic order upwinding using Steger–Warming flux vector waves are non-dispersive and non-dissipative in their splitting [89] for numerical solution with upwind propagation. High-order spatial discretization is applied discretizations of the right-hand side. The BW algorithm in aeroacoustics while time integration is often per- is formally second accurate in space and compatible with formed with RK methods. In the multistage RK centered discretizations of right-hand side. For higher- methods of Eq. (2.10), the coefficients are chosen such order discretizations of the right-hand side, high-order that the maximum possible order of accuracy is obtained accuracy of the implicit operators was obtained [90] for a given number of stages. It is possible, however, Hu where the derivatives in the LHS operator were evaluated et al. [84], to choose the coefficients of the RK method with a fourth-order accurate, three-point, compact so as to minimize the dissipation and dispersion errors stencil. Using a fourth-order accurate compact scheme for the propagation of waves, rather than to obtain the for the evaluation of the right-hand side the algorithm maximum possible formal order of accuracy. The has fourth-order formal accuracy in space. The resulting optimized schemes are referred to in [84] as low- high-order accurate in space implicit, compact algorithm dissipation, low-dispersion RK (LDDRK) schemes. was tested and analyzed in [90,91]. It was found that it is Optimization strategies of numerical schemes for wave unconditionally stable in two dimension as the original propagation has been conducted in several other studies Beam–Warming second-order accurate algorithm. [85,86]. The LDDRK schemes have certain advantages Furthermore, it was shown that it yields accurate solu- because they are applicable to different spatial discreti- tions of time dependent problems with fewer subitera- zation methods. The optimization is carried out only for tions and converges faster to the steady state [90]. the resolved wavenumbers in the spatial discretization, Starting from the control volume formulation Yoon they preserve the frequency in the time integration and and Jameson [92] obtained an unfactored implicit thus are dispersion relation preserving in the sense of scheme from the nonlinear implicit scheme by linearizing [86], and they are lowstorage. Other optimized schemes the flux vectors about the preceding time step and are the multistage Runge–Kutta-type time marching dropping terms of second order or higher. Recently, the methods for Mac Cormack-type schemes developed by LU-SGS scheme was further extended by Zhang and Hixon and Turkel [87]. Wang [93] for time accurate solutions by introducing dual time-stepping. 2.3.2. Implicit schemes An implicit time marching method that was used 2.3.3. Time accurate solutions with multigrid successfully in computations where high order of spatial Implicit methods in the finite-volume context algo- accuracy was implemented with finite differences is the rithm can be found in Refs. [95,94]. After applying approximately factored Beam-Warming algorithm [88] finite-volume discretization and using the method of augmented with Newton-like subiterations in order to lines [94], the following coupled system of ordinary achieve second-order time accuracy. The factored im- differential equations is obtained plicit Beam-Warming algorithm, as well as other implicit dU þ RðUÞ¼0 (2.12) time marching schemes, assumes second-order accurate dt in time linearization of the fluxes FðUÞ as follows: applying three-point backward difference approxima- qF n tion for the time derivative in Eq. (2.12) obtain F nþ1 ¼ F n þ DU n þ OðDtÞ2, qU 3 2 1 U nþ1 À U n þ UnÀ1 þ RðU nþ1Þ¼0. (2.13) ¼ F n þ AnDU n þ OðDtÞ2, ð2:11Þ 2Dt Dt 2Dt where DU n ¼ U nþ1 À Un. As a result, these schemes can For time accurate computations with multigrid [94], achieve second order of accuracy in time at most, this equation is treated as a steady-state equation by Ã provided that linearization errors are eliminated during introducing a pseudo-time variable t and using a the time integrations process. Therefore, for time accu- multigrid strategy to solve the following nonlinear Ã rate solutions with the Beam-Warming algorithm and system to steady state using local time steps Dt . other implicit algorithms, which use the linearization of qW þ RÃðWÞ¼0, (2.14) Eq. (2.11), require one or more Newton-type subitera- qtÃ tions in order to eliminate linearization and/or factor- where W is the approximation to Unþ1 and the residual ization errors and obtain second-order accuracy in time. RÃðWÞ defined by The approximately factorized Beam-Warming (BW) algorithm is compatible with centered finite difference 3 RÃðWÞ¼ W þ RðWÞÀSðU n; UnÀ1Þ, discretizations of right-hand side and is often used in 2Dt CFD application. Central differences on the right-hand 2 1 SðUn; U nÀ1Þ¼ U n À UnÀ1. ð2:15Þ side of the BW-factored algorithm were replaced by first- Dt 2Dt ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 203

Time integration of Eq. (2.14) is obtained with a fourth-order explicit scheme was used in CFD. Explicit multistage Runge–Kutta method that performs the role fourth-order, finite-difference formulas are often used to of the smoother in the multigrid process. discretize the second derivatives in the viscous terms by taking the first derivative twice. In order to reduce the stencil width the inner derivative is evaluated at half- 3. High-order finite-difference schemes points. The first systematic attempt to develop high-order In the past, efforts were made towards developing accurate, narrowstencil, finite-difference schemes ap- high-order finite-difference (FD) methods in the areas propriate for problems with a wide range of scales was direct and large eddy simulations. For nonlinear presented by Lele [100]. Compared to the traditional FD problems, straightforward application of high-order approximations the compact schemes presented by Lele accurate central difference schemes is not possible, [100] provided a better representation of the short-length because the spurious modes that develop from the scales. As a result, compact high-order schemes are unresolvable by the numerical discretization high- closer to spectral methods and at the same time maintain frequency modes lead to instabilities. Rai and Moin the freedom to retain accuracy in complex stretched [96] found that high-order upwind schemes are more meshes. Emphasis in the development of compact promising to simulate turbulent flows. However, early schemes was given on the resolution characteristics of attempts to apply high-order finite differences were the difference approximations rather than formal often frustrated because of lack of robustness of the accuracy (i.e. truncation error). The notion of resolution proposed high-order (FD) schemes compared to spectral was quantified by Lele [100] using a Fourier analysis of methods. In spite of the difficulties, some success was the differencing scheme [101,102]. This analysis com- achieved for the computation of incompressible [96] and pares the resolving power of different schemes based on compressible [56] flows with high order, upwind FD a more general notion of intervals per wavelength of schemes. The fifth-order accurate derivatives in [96] were Swartz and Wendroff and Kreiss and Oliger. Using computed with upwind-biased formulas based on the these ideas, Lele [100] analyzes the resolution character- sign of the velocity as istics of the schemes based on the accuracy with which the difference approximation represents the exact result 1 ui ¼À120 ½6uiþ2 þ 60uiþ1 þ 40ui À 120uiÀ1 over the full range of length scales that can be realized þ 30uiÀ2 À 4uiÀ3 for ui40, for a given mesh. 1 ui ¼ 120 ½4uiþ3 À 30uiþ2 þ 12uiþ1 À 40ui À 60uiÀ1 þ 6uiÀ2 for uio0. ð3:1Þ 3.1. Centered compact schemes Upwinding alleviated some of the problems encountered Of particular interest in recent applications has been with centered schemes and yielded some promising the class of centered schemes that require small stencil results for both incompressible [96] and compressible support. These ‘‘compact’’ schemes presented by Lele flow [56] direct numerical simulations. [100] can be derived from Taylor series expansions and Upwind biased, high-order accurate stencils (up to compute simultaneously the derivatives along an entire fifth-order) for the evaluation of the derivatives were line in a coupled fashion. The main advantage of also used successfully for the computation of complex compact schemes is simplicity in boundary condition incompressible flows [97,98] and wave propagation [99]. treatment and smaller truncation error compared to Upwind-biased schemes, however, based only on formal their noncompact counterparts of equivalent order. accuracy (truncation error) inherently introduce some Compact schemes are, however, more intensive compu- form of artificial smoothing that makes them inap- tationally compared to explicit schemes because they propriate for long time integration and direct simulation require matrix inversion. or large eddy simulation or turbulence. A seven-point wide stencil, finite-difference discretiza- Application of Taylor series expansion yields cen- tion of any first-order derivative f 0 of a scalar pointwise tered, explicit and compact finite difference formulas for discrete quantity, f, in the governing equations, such as space differentiation on equally spaced meshes. Central- metric terms or flowvariables, is obtained in the difference schemes gained popularity in the simulation computational domain on an equally spaced mesh by of wave propagation phenomena because in contrast to upwind methods, which in addition to dispersion 0 0 0 0 0 Bf jþ2 þ Af jÀ1 þ f j þ Af jþ1 þ Bf jþ2 naturally introduce artificial dissipation, the dominant f À f f À f f À f error in centered discretizations is depressive. The ¼ a jþ1 jÀ1 þ b jþ2 jÀ2 þ c jþ3 jÀ3 , ð3:2Þ stencils for the fourth-, sixth-, and eight-order accurate 2h 4h 6h symmetric, explicit, centered schemes are five-, seven- where A; B; a; b and c determine the spatial accuracy of and nine-point wide, respectively. As a result, only the the discretization. Different values of the coefficients in ARTICLE IN PRESS

204 J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 the formula of Eq. (3.2) yield schemes of different π 1.5 accuracy ranging from the fourth-order explicit method Exact (E4) to the compact tenth-order accurate scheme (C10). E2 E4 0.75 The values of the coefﬁcients in Eq. (3.2) for schemes of 3π/4 E6 C4 0.75 1.5 2.25 different order of accuracy are shown in Table 1. C6 In Table 1, C8/3 refers to the eight-order compact C8 scheme that requires tridiagonal matrix inversion and π/2 C8/5 refers to the eight-order compact scheme that requires pentadiagonal matrix inversion. Compact approximations for the second-order deri- 00 π/4 vative f are obtained from the following general form: Modified wave number

00 00 00 00 00 Bf jÀ2 þ Af jÀ1 þ f j þ Af jþ1 þ Bf jþ2 π a b π/4 π/2 3π/4 ¼ ðf þ À 2f þ f À Þþ ðf þ À 2f þ f À Þ h2 j 1 j j 1 4h2 j 2 j j 2 wave number c þ ðf jþ3 À 2f j þ f jþ3Þ. ð3:3Þ Fig. 6. Wave space resolution of explicit and compact centered 9h2 schemes for the first-order derivative. Explicit schemes can also be used for the evaluation of the viscous terms because application of the compact compact and non-compact schemes is shown in Fig. 6. schemes is more expensive computationally. Further- Using the scaled wavenumber o ¼ 2pkh=l where l is the more, there are no very significant improvements in wavelength and the number of intervals or grid points wavespace resolution with the use of compact schemes per wavelength is 2pk=o. Therefore, the lower the [100]. scheme’s resolving ability the higher is the number of The wavespace resolution of various explicit and points per wavelength required to resolve accurately compact schemes is obtained using Fourier analysis certain predetermined portion of the range ½0; 2p. [100–102] for Eqs. (3.2) or (3.3). Considering that the Stable, accurate formulas for the boundary points can ikh exact result is a sinusoidal function f j ¼ e , where h ¼ be found in [103]. Numerical solutions of nonlinear Dxj is a uniform grid spacing and k is the wavenumber, hyperbolic equations with central-difference methods 0 the exact value of the derivative is f j ¼ ikf j. The develop spurious modes arising from unresolvable scales derivative computed with finite difference formulas, on and inaccuracies in the application of boundary condi- b0 b b the other hand, is given by f j ¼ ikf j, where k the tions. Spectral-type [28] or characteristic-based [1] filters modified wavenumber, which depends on the form of that can be used to stabilize numerical solutions the FD formulas used for the evaluation of the first- performed with central-difference methods are presented order derivative. The difference between the true in the following sections. wavenumber k and the modified wavenumber kb is a measure of the scheme’s resolving ability. The modified 3.2. Boundary closures for high-order schemes wavenumber of various finite difference schemes can be 0 0 Æik obtained using standard shift operators f jÆn ¼ f je . The primary difficulty in using higher-order schemes For example, the modified number of the fourth-order is identification of stable boundary schemes that accurate explicit scheme of Eq. (3.2) is given by preserve their formal accuracy. Boundary closures for b k ¼ ið8 sin k À sin 2kÞ=6, with analogous expression various explicit and compact high-order centered for the other methods. A comparison of the modified schemes were presented by Carpenter et al. [103]. The wavenumbers of the first derivative for several central stability characteristics of compact fourth- and sixth- order spatial operators with boundary closures were Table 1 assessed [103] with the theory of Kreiss [104] and Schemes with five-point stencil of Eq. (3.2) Gustafsson, Kreiss and Sundstrom (GKS) [105] for the Scheme ABa b c semidiscrete initial value problem. Numerical solutions of hyperbolic systems preserve E4 0 0 4 1 0 3 À3 their formal spatial accuracy when an Nth-order inner C4 1 0 3 00scheme is closed with at least an ðN À 1Þth-order 4 2 C6 1 0 14 1 0 boundary scheme. Furthermore, determination of the 3 9 9 C8/5 4 1 40 25 0 numerical stability of a fully discrete approximation 9 36 27 54 C8/3 3 0 75 1 1 (including boundary schemes) for a linear hyperbolic 8 48 5 À80 C10 1 1 17 101 1 partial differential equation is a difficult task. Fourier 2 20 12 150 100 techniques are not straightforward to apply and do not ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 205 provide sufficient conditions for numerical stability. (4, 5-6-5, 4) schemes are used for full discretization with Gustafsson et al. [105] developed stability analysis the sixth-order compact scheme. techniques based on normal modal analysis. The theory The third order closure at j ¼ 1 is: of [105], generally referred to as GKS stability theory, 0 0 1 established conditions that the inner and boundary 2U1 þ 4U 2 ¼ ðÀ5U1 þ 4U 2 þ U3Þ. schemes must satisfy to ensure stability. The GKS Dx theory was used by Carpenter et al. [103] to assess the The fourth-order closure at j ¼ 1is stability of boundary schemes proposed for the fourth- 0 0 1 and sixth-order compact schemes. 6U1 þ 18U 2 ¼ ðÀ17U 1 þ 9U2 þ 9U 3 À U4Þ Consider the finite-difference representation of the Dx 0 continuous derivative Ux ¼ Uj on an equally spaced and the fifth-order closure at j ¼ 2 is accomplished by 0 mesh. The discrete form of the first derivative Uj 0 0 0 1 involves functional values U j; j ¼ 1; ...; N at discrete 3U1 þ 18U 2 þ 9U3 ¼ ðÀ10U1 À 9U 2 þ 18U3 þ U 4Þ. points. For an explicit uniformly fourth-order accurate Dx in space scheme, the spatial discretization with boundary The two key issues encountered with higher-order closures is obtained by Carpender et al. [103] as finite-difference schemes used in CFD are boundary 1 treatments and grid uniformity. The boundary treat- U 0 ¼ ðÀ25U þ 48U À 36U þ 16U À 3U Þ, 1 12Dx 1 2 3 4 5 ment was carefully addressed by Carpenter et al. [103].It 1 was found that the effect of boundary closures to the U 0 ¼ ðÀ3U À 10U þ 18U À 6U þ U Þ, 2 12Dx 1 2 3 4 4 overall resolution is indeed small [106] even for highly 1 accurate DNS. The grid non-uniformity issue is, U 0 ¼ ðU À 8U þ 8U À U Þ, j 12Dx jÀ2 jÀ1 jþ1 jþ2 however, more important and its effect on the overall j ¼ 3; ...; N À 2, accuracy of higher-order finite-difference schemes on non-uniform grids was recently assessed in [107]. 1 The use of non-uniform grids in turbulent flow U 0 ¼ ðÀU þ 6U À 18U NÀ1 12Dx NÀ4 NÀ3 NÀ2 simulations is inevitable. The typical ratio of the maximum to the minimum grid spacing is about 100. þ 10U NÀ1 þ 3U N Þ, The behavior of the second- and fourth-order explicit 1 centered schemes and the fourth- and sixth-order U 0 ¼ ð3U À 16U þ 36U N 12Dx NÀ4 NÀ3 NÀ2 compact schemes was assessed in [107] for smooth stretched grids. It was found that grid quality has À 48UNÀ1 þ 25U N Þ. ð3:4Þ stronger effects on the higher-order compact schemes The fourth-order, compact FD scheme for the than on the explicit schemes. Furthermore, an accuracy 0 approximation of Uj that has narrower stencil requires deterioration of higher-order compact schemes with low closures only at j ¼ 1 and j ¼ N. The fourth-order grid density was observed for non-uniform meshes. compact scheme with boundary closures is

0 0 1 3.3. Simultaneous evaluation of the first and second U 1 þ 3U2 ¼ ðÀ17U1 þ 9U 2 þ 9U3 À U 4Þ, 6Dx derivative with compact schemes 0 0 0 1 U jÀ1 þ 4Uj þ Ujþ1 ¼ ðÀ3U jÀ1 þ 3Ujþ1Þ, Dx A more general version of the standard compact schemes presented by Lele [100] was developed by 0 0 1 3U þ U ¼ ðUNÀ3 À 9U NÀ2 Mahesh [108]. These schemes are symmetric and differ NÀ1 N 6Dx from the standard compact schemes in that the first and À 9U þ 17U Þ. ð3:5Þ NÀ1 N second derivatives are evaluated simultaneously. In The sixth-order compact scheme has a five point-wide addition, for the same stencil width the schemes stencil and utilizes information from all five points proposed by Mahesh are two orders higher in accuracy, explicitly and three points implicitly (tridiagonal sys- they have significantly better spectral representation, tem). Boundary closures must be provided at two points and the computational cost for the evaluation of both at each end of the domain j ¼ 1; 2 and j ¼ N À 1; N: derivatives is shown to be essentially the same as To ensure formal sixth-order formulas, e.g. the opti- standard compact schemes. As a result, the proposed mal scheme, in shorthand nomenclature, would be schemes appear to be attractive alternative to standard (5, 5-6-5, 5), e.g. fifth order at the boundaries j ¼ 1; 2 compact schemes for the Navier–Stokes equations that and j ¼ N À 1; N and sixth order in the interior. include second-order derivative evaluation in the viscous Formally sixth-order accurate GKS stable schemes terms. The schemes that compute simultaneously the are difficult to find. Therefore, the (3, 5-6-5, 3) and the first f 0 and the second derivative f 00 of a function f given ARTICLE IN PRESS

206 J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 at uniform mesh, Dx ¼ h, are defined by [86,111,112]. The rationale for optimizing numerical schemes for short waves is that for long waves, even a f 0 þ a f 0 þ a f 0 þ h ðb f 00 þ b f 00 þ b f 00 Þ 1 jÀ1 0 j 2 jþ1 1 jÀ1 0 j 2 jþ1 lower-order schemes can do well. The short waves, 1 however, require high-resolution in order to obtain ¼ ðc1f þ c2f þ c0f þ c3f þ c4f Þ. ð3:6Þ h jÀ2 jÀ1 j jþ1 jþ2 accurate representation of the broadband acoustic Enforcing symmetry for the coefficients and considering waves. The optimized FD scheme of Tam and Webb a0 ¼ 1 and b0 ¼ 1 Mahesh [108] obtain from Eq. (3.6) [86], for example, referred to as the dispersion relation a sixth- and an eight-order compact scheme for the preserving (DPR) scheme, uses central differences to simultaneous evaluations of the first- and second-order approximate the first derivative. The approximation is derivative therefore, non-dissipative in nature. The maximum For example, the sixth-order scheme is formal order of accuracy of the centered scheme [86] 0 0 0 00 00 for certain stencil size is again sacrificed in order to 7f jÀ1 þ 16f j þ 7f jþ1 þ hðf jÀ1 À f jþ1Þ optimize resolution of the high wavenumbers. Although 15 non-dissipative schemes are ideal for aeroacoustics, ¼ ðf À f Þ h jþ1 jÀ1 numerical dissipation is often required to damp non- 0 00 00 À 9f jÀ1 À 9f jþ1 À hðf jÀ1 À 8f j þ f jþ1Þ physical waves generated by boundary and/or initial 24 conditions. In many practical applications, therefore, ¼ ðf À 2f þ f Þ. ð3:7Þ high-order dissipative terms were added to the centered h jÀ1 j jþ1 scheme of Ref. [86]. To remedy this problem optimized Comparing the newscheme of Eq. (3.7) withthe DPR schemes were developed by Zhuang and Chen standard compact sixth-order compact schemes for the [111] and Lockard et al. [110]. In the following sections, first- and second-order derivative of the previous section the upwind high-order schemes of Zhong [109] and the 7 1 DPR scheme of Tam and Webb [86] are presented. f 0 þ 3f 0 þ f 0 ¼ ðf À f Þþ ðf À f Þ, jÀ1 j jþ1 3h jþ1 jÀ1 12h jþ2 jÀ2

12 3.4.1. Upwind high-order schemes 2f 00 þ 11f 00 þ 2f 00 ¼ ðf À 2f þ f Þ A family of finite-difference high-order upwind jÀ1 j jþ1 h2 jÀ1 j jþ1 compact and explicit schemes for the discretization of 3 convective terms was derived in [109]. The general þ 2 ðf jÀ2 À 2fj þ f jþ2Þ. ð3:8Þ 4h compact and explicit finite-difference approximation of It is evident that the Mahesh scheme [108], which qu=qx ¼ u0 is computes the first and second derivative in a coupled fashion, uses a more narrowstencil compared to the XM0 1 XN0 b u0 ¼ a u , (3.9) standard compact scheme. In addition, to the sixth- iþk iþk h iþk iþk k¼ÀMþM þ1 k¼ÀNþN þ1 order compact scheme an eight-order compact scheme 0 0 was developed in [108]. Boundary closures were also 0 where h is the uniform grid spacing, uiþk is the numerical applied for both the sixth- and eight-order compact approximation of the first derivative at the ði þ kÞth grid schemes for simultaneous computation of the first and point, and N0, M0 are biases with respect to the base second derivative. point i. The family of upwind compact and explicit schemes with central grid stencils N ¼ 2N0 þ 1, 3.4. Modified high-order finite-difference schemes M ¼ 2M0 þ 1, was considered in [109]. The coefficients aiþk and biþk of these upwind schemes were determined The idea of modifying or optimizing a finite difference such that the order of the schemes is one order lower schemes by calculating values of the coefficients that than the maximum achievable order for the standard introduce upwinding or minimize a particular type of central stencil. As a result, the orders of the upwind error instead of the truncation error has been used schemes are always odd integers p ¼ 2ðN0 þ M0ÞÀ1, successfully in the design of newschemes withdesired and there is a free parameter y in the coefficients aiþk properties. Some form of upwinding is often needed for and biþk. The value of y is set to be the coefficient of the computation of flows with discontinuities and the leading truncation term, which is an even order nonlinearities. Modifications of standard centered, derivative. explicit and compact schemes were carried out by Zhong XM XN [109]. For the modified schemes, the formal order of the 0 1 0 b u0 ¼ a u scheme for certain stencil size was sacrificed and high- iþk iþk h iþk iþk k¼ÀM k¼ÀN order upwinding with low dissipation was introduced. 0 0 y qUpþ1 Other optimized schemes have also been developed À p þÁÁÁ ð Þ h pþ1 3:10 [110] in the field of computational aeroacoustics ðp þ 1Þ! qx i ARTICLE IN PRESS

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Schemes of Eq. (3.10) are ðp þ 1Þth order accurate for Clearly the quantity kb ¼Àike is effectively the wave- y ¼ 0 and pth order accurate for ya0. The choice of y number of the Fourier transform for the finite-difference affects the magnitude of numerical dissipation and the schemes of Eqs. (3.11) and (3.12). stability of the scheme. The third-, fifth- and seventh- The Fourier transform of the finite difference scheme order explicit and compact schemes based on this idea is a good approximation of that of the partial derivative were derived by Zhong [109]. over the range of wavenumbers jkDxjpp=2 when the coefficients aj are chosen to minimize the integral of the 3.4.2. Dispersion-relation-preserving (DPR) scheme error defined by Z Numerical solutions of the linearized Euler equations p=2 b with high-order finite-difference schemes are used to E ¼ jkDx À kDxj2 dðkDxÞ assure that the computed results have the same number p=2 Z of wave modes (acoustic, vorticity and entropy), the same p=2 XM ¼ iK À a eijK dK. ð3:14Þ propagation characteristics (isotropic, non-dissipative j Àp=2 j¼ÀN and non-dispersive), and the same wave speeds as those of the solution of these equations, which govern acoustic The condition of the minimum is disturbance propagation. These conditions are satisfied qE by the numerical solution if the discrete equations have ¼ 0; j ¼ÀN; ...; M. (3.15) qa the same dispersion relation with the continuous j equation. Finite difference schemes which yield the same The solution of the algebraic system given by Eqs. dispersion relations as the original partial differential (3.15) determines the values of the coefficients aj that give equations are referred to as dispersion-relation-preser- a good approximation of the derivative for jkDxjpp=2. ving (DPR) schemes. A way to construct time marching In Ref. [86], the condition of Eq. (3.12) was imposed DPR schemes was proposed in [86] by optimizing the for n ¼ M ¼ 3 (fourth-order accuracy) and a1 was left finite difference approximations of the derivatives in the as free parameter. Minimization of Eq. (3.14) yielded wavenumber and frequency space. The new optimized, the following values for the coefficients a0 ¼ 0, a1 ¼ high order, finite difference scheme [86] designed with ÀaÀ1 ¼ 0:79926643, a2 ¼ÀaÀ2 ¼À0:18941314, a3 ¼ these criteria meets the usual conditions of consistency, ÀaÀ3 ¼ 0:02651995. Therefore, the formal accuracy of stability, and hence convergence. In addition, it supports, the scheme with the coefficients from Eq. (3.14) was in the case of small amplitude waves, wave solutions sacrificed, since a fourth-order accurate scheme was which have the same characteristics as those of the obtained for a seven-point-wide stencil, but the resolu- linearized Euler equations as nearly as possible. tion in wavespace was improved. Consider the approximation of the first derivative qf =qx at the jth node on a uniform grid using N values 3.5. Spectral-type filters of f to the left. The finite difference approximation is qf 1 XM High-order accurate centered schemes are non-dis- ’ aj f . (3.11) sipative and they are particularly suitable for convection qx Dx iþj i j¼ÀN of small-scale disturbances governed by the linearized Euler equations. Non-dissipative, central-difference dis- The usual way to determine the coefficients aj in Eq. (3.11) is to expand in Taylor series and equate cretizations for nonlinear problems, however, produce coefficients of the same powers in Dx. In Ref. [86],itwas high-frequency spurious modes that originate from mesh proposed to determine the coefficients by requiring the non-uniformities, inaccuracies of the boundary condi- Fourier transform of the finite difference scheme on the tions, and nonlinear interactions. In order to prevent right of Eq. (3.11) to be a close approximation of the numerical instabilities due to growth of high-frequency partial derivative. modes while retaining the high-order accuracy of the The finite difference representation of Eq. (3.11) can compact or non-compact central discretizations, filtering be written as of the computed solution is required. Filtering of the solution with explicit-type filters was proposed by Lele qf 1 XM [100]. More recently, high-order compact filters were ’ a f ðx þ jDxÞ. (3.12) j introduced by Gaitonde and Visbal [28]. These compact qx Dx j¼ÀN filters are applied on the components of the computed Considering the Fourier transform of the left and right solution vector. Denoting by f, the computed value, the of Eq. (3.12) obtain filtered value fb is obtained by solving the system ! XM XN e 1 ikjDx e ee b b b an ikf ’ aje f ¼ kf . (3.13) a f þ f þ a f ¼ ðf þ f Þ. (3.16) Dx f jÀ1 j f jþ1 jþn jÀn j¼ÀN n¼0 2 ARTICLE IN PRESS

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The compact filter of Eq. (3.16), which was proposed in 3.6. Characteristic-based (ACM) filters Ref. [28], provides a 2Nth-order accurate formula on a 2N þ 1 point stencil. Application of the compact filter Characteristic filters of Yee et al. [1] can be applied makes possible high-resolution, low-diffusion numerical instead of spectral-type filters with implicit and explicit solutions of flows without discontinuities. The coeffi- methods for time discretization of Section 2. Character- cients an of the filter are functions of the filtering istic-based filters remove spurious oscillations and in parameter af . These coefficients for different order filters addition can be used for shock capturing. They can be are given in Table 2. applied at every stage of an RK method or after each The filtering ability of the filters given by Eq. (3.16) is Newton-type subiteration of an implicit-type time determined by the transfer function in wavespace. integration scheme for flows with strong shock interac- Therefore, the compact filters of Eq. (3.16) are referred tions. For computational efficiency, however, the filter is to from nowon as spectral-type filters. The transfer often applied at the end of the full RK step or at the final function for the spectral-type filter of Eq. (3.16) is update of an implicit time scheme. given by Let Lf be the filter operator defined as P N n n 1 en en an cos ðnoÞ L ðF ; G Þ ¼ ½F À F SðoÞ¼ n¼0 . (3.17) f i; j Dx iþ1=2; j iÀ1=2; j 1 þ 2af cos ðoÞ 1 en en þ ½Gi; jþ1=2 À Gi; jþ1=2, ð3:18Þ The parameter, af , which is in the range 0:5oaf o0:5 Dy determines the filtering properties. High values of the en en where F iþ1=2; j and Gi; jþ1=2 are the dissipative numerical parameter af yield less dissipative filters. fluxes of the filter operator to be discussed below. Then At the boundary points, the order of the filter must be the newtime level n þ 1 (or next stage p þ 1 for implicit dropped in order to reduce the stencil size. It was schemes with subiterations) is defined as demonstrated [28] that the filtering ability of the low- order filters could approximate the filtering performance nþ1 bnþ1 n n U ¼ U þ DtLf ðF ; G Þi; j, (3.19) of high-order filters by varying the value of the filtering en n parameter a . The variation of the spectral function of where the filter numerical fluxes F iþ1=2; j and Gi; jþ1=2 f bnþ1 Eq. (3.17) for constant value of the filtering parameter are evaluated at U . The simplest form for Lf is the af ¼ 0:45 for the second up to eighth-order filters is linear filter proposed by Gustafsson and Olsson [113] shown in Fig. 7. The spectral function of Eq. (3.17) for where a switch similar to that of Harten [35] was used. the second-, and eight-order filter is plotted in Fig. 8 The filter numerical flux may be written in the form for different values of the filtering parameter similar to TVD schemes, as in [72,114–116] as follows af ¼ 0:45 À 0:49. It is evident that as the value of the e 1 F iþ1=2; j ¼ 2 ½F iþ1; j þ F i; j þ Di; j, (3.20) filtering parameter af becomes larger low-pass filtering is 1 obtained even with the second-order filter. where 2 ½F iþ1; j þ F i; j is the central difference portion of Centered schemes with spectral-type filters are not the numerical flux, which is substituted by a high-order appropriate for computations of flows with shock waves centered approximation in the schemes proposed by Yee and other discontinuities. High-order accurate compu- et al. [1], and the term Di; j is the nonlinear dissipation. tation of transonic or supersonic flows with compact For characteristic-based methods, e.g. methods where centered schemes can be obtained with the application of the dissipative term is evaluated in characteristic characteristic-based filters [1] described in the following variable space Di; j ¼ Riþ1=2; jFiþ1=2; j where Riþ1=2; j is section. the right eigenvector of the flux Jacobian matrix qF=qU.

Table 2 Coefﬁcients for the spectral-type ﬁlter of Eq. (3.16)

F2 F4 F6 F8 F10

1 3a 5a 70a 193À126a a0 þ a 5 f 11 f 93 f f 2 f 8 þ 4 16 þ 8 128 þ 128 256 1 1 17a 18a 105À302a a1 þ a þ a 15 f 7 f f 2 f 2 f 32 þ 16 16 þ 16 256 1 a 3a 14a À15þ30a a2 0 À þ f À3 f À7 f f 8 4 16 þ 8 32 þ 32 64 1 a 1 a 45À90a a3 00 À f À f f 32 16 16 8 512 À1 a À5þ10a a4 00 0 þ f f 128 64 256 1À2a a5 00 0 0 f 512 Order of accuracy 2nd 4th 6th 8th 10th ARTICLE IN PRESS

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1 where Riþ1=2 is the right eigenvector matrix of the ﬂux Jacobian A ¼ qF=qU at Roe’s approximate average 0.9 n state and the elements of the matrix Fiþ1=2 denoted by nl 0.8 fiþ1=2 are 1 0.7 nl l l fiþ1=2 ¼ kyiþ1=2fiþ1=2. (3.22) 0.999 0.6 l

) The function kWiþ1=2 in Eq. (3.22) is the key ω 0.5 0.998 mechanism for achieving high accuracy of the fine scale S( flow structures as well as capturing of shock waves in a 0.4 π n 0 /2 stable manner. The elements of Fiþ1=2 can be identified 0.3 af = 0.45 as the nonlinear dissipation portion of a TVD, ENO, or 2nd order WENO scheme with the exception that they are 0.2 4th order n 6th order premultiplied by kWiþ1=2. Yee et al. [1] defined Fiþ1=2 0.1 8th order using a TVD scheme. Garnier et al. [36], on the other n hand, defined Fiþ1=2 using the dissipative part of ENO 0 π π/4 π/2 3π/2 or WENO schemes. ω The main disadvantage with characteristic-based filters is that the parameter k is problem depended. Fig. 7. Transfer function for filters of different order for Numerous simulations and tests were carried out by Yee constant value of the filtering parameter Af ¼ 0:45. et al. [1] for different flows. It was found, however, that different examples require a different value of k. The suggested range of k in [1] was 0:03pkp2 where larger 1 values of k are used for flows with discontinuities and 0.9 smaller values are required for smooth flows including 1 complex features, such as vortex convection or vortex 0.8 pairing. In order to remedy this problem, Sjogreen and 0.7 Yee [117] used a regularity estimate obtained from the 0.995 wavelet coefficients of the solution to obtain a better 0.6 value for the filter sensor.

) l

ω The function y in Eq. (3.22) is the Harten 0.5 iþ1=2

S( 0.99 switch. This switch for a general 2m þ 1 points scheme π π π 0.4 /4 /2 3 /2 is given by S2,af=0.480 S2,a =0.485 0.3 f l b b S2,af=0.490 yiþ1=2 ¼ maxðyiÀmþ1; ...; yiþmÞ, (3.23) S2,a =0.495 0.2 f S8,af=0.475 S8,af=0.485 l l p 0.1 S8,a =0.495 l jeiþ1=2jÀjeiÀ1=2j f by ¼ . (3.24) i jel jÀjel j 0 π iþ1=2 iÀ1=2 π/4 π/2 3π/2 ω In Eq. (3.24), p is a second parameter that determines the performance of the filter and can be varied to better Fig. 8. Transfer function of the second and eighth-order filter capture the particular physics instead of varying k. The for different values of the filtering parameter Af . higher the parameter p the less is the amount of numerical dissipation added to the numerical solution. For pX1 the order of accuracy of the dissipation term is In order to introduce some upwinding the elements essentially increased. For all numerical examples in [1,117], a constant value p ¼ 1 was used. Furthermore, of Riþ1=2 are computed at Roe’s approximate average state. in order to keep the stencil of the scheme compact the The artificial compression method (ACM) of Harten’s Harten’s switch was computed as idea [35] was generalized in [1] to achieve a low- yl ¼ maxðyl; yl Þ. (3.25) dissipative high-order shock-capturing scheme by nearly iþ1=2 i iþ1 maintaining the accuracy to high order. The ACM filter l À1 e In Eq. (3.24), eiþ1=2 are elements of Riþ1=2ðUiþ1 À U iÞ numerical flux F iþ1=2; j in Ref. [1] has the form À1 where Riþ1=2 is the left eigenvector of the flux Jacobian n qF=qU that transforms back to the conservative Fe ¼ 1 R Fn , (3.21) iþ1=2; j 2 iþ1=2 iþ1=2 variable space the filter operator. ARTICLE IN PRESS

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l The elements fiþ1=2 in Eq. (3.20) are evaluated based In addition, the switch in Eq. (3.24) is modified to avoid on TVD schemes[115,116,118]. Choosing Harten–Yee division by zero upwind TVD for example obtain l l jje þ jÀje À jj yl ¼ i 1=2 i 1=2 . (3.31) l 1 l l l l l i jel jþjel jþe fiþ1=2 ¼ 2 cðciþ1=2Þðgiþ1ÞÀcðciþ1=2 þ giþ1=2Þeiþ1=2, iþ1=2 iÀ1=2 (3.26) Recently, Sjogreen and Yee [121] proposed further improvements to the estimation of the filter numerical where flux introducing regularity estimates from the wavelet 8 < l l l l a coefficients of the solution. This recent development is as ðgiþ1 À giÞ=eiþ1=2 eiþ1=2 0; l 1 l follows. Considering the lth element of the filter ¼ ð Þ¼ l giþ1=2 c ciþ1=2 : l e 2 0 eiþ1=2 ¼ 0: numerical flux function f iþ1=2 (see Eq. (3.22)) as the l product of the sensor oiþ1=2 and a nonlinear dissipation (3.27) l function fiþ1=2 8 l fe ¼ ol fl (3.32) > j l jjl jX iþ1=2 iþ1=2 iþ1=2 <> ciþ1=2 ciþ1=2 d1; l l l 2 the sensor o that in Eq. (3.24) is modified while the cðc þ Þ¼ c þ d (3.28) iþ1=2 i 1=2 > iþ1=2 1 l numerical dissipation portion f of Eq. (3.26) of : jciþ1=2jod1: iþ1=2 2d1 Harten and Yee TVD scheme remains the same. In Eq. (3.26), el are the characteristic speeds of the iþ1=2 3.6.1. ENO and WENO ACM filters flux Jacobian matrix qF=qU evaluated at the Roe’s A recent improvement of ACM filters is application of average state [119], c is an entropy correction [120] ENO and WENO procedure in the evaluation of the where 0od51, and gl is a limiter function. i dissipative fluxes [36]. The dissipative numerical fluxes Examples of commonly used limiter functions are: for TVD-MUSCL schemes (see Eq. (3.21)) are

l l l nM ðaÞ gj ¼ minmodðejÀ1=2; ejþ1=2Þ, e 1 nM F iþ1=2 ¼ Riþ1=2Fiþ1=2, (3.33) l l l l l l l 2 ðbÞ g ¼ðe e þje e jÞ=ðe þ e Þ, M j jþ1=2 jÀ1=2 jþ1=2 jÀ1=2 jþ1=2 jÀ1=2 where the elements fl of Fn are given by l l l 2 l l 2 iþ1=2 iþ1=2 ðcÞ g ¼fe À ½ðe þ Þ þ d2þe þ ½ðe À Þ þ d2g= j j 1=2 j 1=2 j 1=2 j 1=2 l l l l l 2 l 2 fiþ1=2 ¼ kWiþ1=2jciþ1=2jeiþ1=2, (3.34) ½ðejþ1=2Þ þðejÀ1=2Þ þ 2d2, l l l 1 l l where ðdÞ gj ¼ minmodðejÀ1=2; 2ejþ1=2; 2 ðejþ1=2 þ ejÀ1=2ÞÞ, l À1 R L l l l eiþ1=2 ¼ Riþ1=2ðUiþ1=2 À U iþ1=2Þ ðeÞ gj ¼ S max½0; minð2jejþ1=2j; S Á ejÀ1=2Þ, l l and cl are the eigenvalues of the flux Jacobian minðjejþ = j; 2S Á ejÀ = Þ; iþ1=2 1 2 1 2 R L l Uiþ1=2, Uiþ1=2 are the upwind-biased interpolation of the S ¼ sgnðejþ1=2Þ. neighboring Ui values with the slope limiters imposed. The MUSCL approach can be extended to rth order Here d2 is a small dimensionless parameter to prevent accurate ENO schemes of Ref. [1] as follows. The l l division by zero and sgnðejþ1=2Þ¼sgnðejþ1=2Þ. In prac- dissipative numerical flux is written as tical calculations 10À7 d 10À5 is a commonly used p 2p nENO l l l e nENO range. For ejþ1=2 þ ejÀ1=2 ¼ 0; gj is set to zero in (b). F iþ1=2 ¼ Riþ1=2Fiþ1=2, (3.35) The minmod function of a list of arguments is equal to ENO where the element fl of Fn is obtained from the the smallest number in absolute value if the list of iþ1=2 iþ1=2 arguments is of the same sign, or is equal to zero if any dissipative part of the ENO scheme, which results (see arguments are of opposite sign. Section 4) by subtracting an mth-order accurate, To facilitate computer implementation the entropy centered scheme from an rth-order accurate ENO approximation as correction cðcÞ in Eq. (3.28) is evaluated as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XrÀ1 l l r À1 2 2 1 fiþ1=2 ¼ yiþ1=2 ck;pRiþ1=2F iÀrþ1þkþp cðcÞ¼ ðc þ d Þ; d ¼ 16 (3.29) p¼0 ! l XmÀ1 and giþ1=2 in Eq. (3.27) is computed by m À1 À cm R F iÀmþ1þmþp , ð3:36Þ 2 ;p 2 p¼0 l l l l cðciþ1=2ÞðWiþ1=2 À gi Þeiþ1=2 gl ¼ ; e ¼ 10À7. (3.30) where cr are the reconstruction coefficients of the ENO iþ1=2 2ðcl Þ2 þ e k;p iþ1=2 reconstruction, and k is the stencil index selected among ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 211 the r candidate stencils Sk that are defined as context for two- and three-dimensional flows, similar to the FD formulation, is however based on one-dimen- SðkÞ¼fI ; ...; I ; ...; I g iÀrþkþ1 i iþk sional physics. Namely, the multidimensional FV for- ¼ðxiÀrþkþ1; ...; xi; ...; xiþkÞ; k ¼ 0; ...; r À 1. mulation is based on the solution of the one-dimen- ð3:37Þ sional Riemann problem that describe the interaction between two fluid cells by finite-amplitude waves normal The mth-order accurate centered scheme is a sub- to their interface. The inadequacy of this approach class of ENO coordinate stencils with k ¼ m=2 however clearly shows up, irrespective of the order of and m even. All choices mX2l, l ¼ 1; 2; ... are valid accuracy, when the numerical solution contains shocks for the construction of error-dissipative terms. How- or shear waves not aligned with the grid. Roe [122] ever, in order to keep the accuracy of the base recognized this deficiency as early as 1985. The remedy scheme the same with order of the dissipative terms to one-dimensional upwinding is incorporation of m ¼ q since larger values of m do not improve genuinely multidimensional physics in upwind algo- the formal accuracy and increase the computing rithms [123,124]. Numerous efforts were made in the cost. past to apply Riemann solvers in several physi- An increased order of accuracy can be achieved by cally appealing directions. Closer to the genuinely exploiting the WENO idea in the construction of the l multidimensional approach is the work of Rumsey f þ dissipative terms as follows: i 1=2 " et al. [125] where multidimensional wave models XrÀ1 XrÀ1 were developed that minimize wave strengths and l l r À1 fiþ1=2 ¼ ok yiþ1=2 ck;pRiþ1=2F iÀrþ1þkþp require only two inputs states, just as a classic high- k¼0 p¼0 !# order solver. XmÀ1 In support to quasi-, and genuinely multidimensional À cm RÀ1 F . ð3:38Þ m=2;p iþ1=2 iÀmþ1þm=2þp approaches, aimed at putting better physics in to p¼0 interface fluxes some investigators have dedicated efforts The WENO approach (see Section 4) achieves in achieving more accurate process of flux calculation by ð2r À 1Þth-order of accuracy by performing linear, improving Godunov scheme [126] in the reconstruction convex combination with weights ok of the r possible step that is the most important task in the solution rth order ENO stencils. process. Piecewise linear approximations based on adjacent values of the solution that do not produce 3.7. Finite-volume compact difference-based schemes spurious oscillations near discontinuities and regions of high solution gradients are obtained with the MUSCL Finite-volume (FV) methods offer several advantages schemes developed by van Leer [127,128] and/or compared to finite-difference (FD) formulation espe- quadratic approximations by Colella and Woodward cially for the numerical simulation of nonlinear phe- [129]. Furthermore, for applications on structured-type nomena. The advantage of the FV formulation with grids high-order accuracy may be introduced into the respect to the FD formulation is that the former is based finite-volume formulation performing reconstruction by on the integral form of the conservation laws. As a means of primitive function as is described in the next result, flux conservation is enforced even on arbitrary section. meshes since the fluxes collapse telescopically by The solution procedure of the multidimensional construction. Furthermore, analysis of the finite-volume problem with the finite-volume method consists of the methods shows that they have superior performance following steps: in the high wavenumber range and that they exhibit lower truncation error. The main disadvantage 1. Reconstruction: Given the average values of the of the FV formulation, especially when high- solution, reconstruct a polynomial approximation order accuracy is required, is the significantly higher to the solution in each control volume. This computing cost of the FV methods compared to FD polynomial may vary discontinuously from control formulation. volume to control volume. For complex aerodynamic simulations, there is little 2. Flux quadrature: Using the piecewise polynomial place left for tuning parameters that regulate accuracy reconstruction of the solution approximate the flux and stability of the computed solutions. The upwind- integral or in the discrete form the summation of differencing schemes used in CFD, although computa- fluxes by numerical quadrature. tionally more intensive than finite differences were 3. Evolution and projection: simply referred to as proven unsurpassed in their computational accuracy evolution where a Riemann solver and an appro- and robustness. The properties of these schemes are well priate temporal discretization scheme are used to justified and explained only for one-dimensional flows. evolve the numerical approximation of the flux Application of upwind-differencing schemes in the FV integral. ARTICLE IN PRESS

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Barth [130] describes in detail several reconstruction The crucial step in this finite-volume formulation is the methods. Among them the k-exact reconstruction reconstruction, e.g. approximation of the pointwise approach of Barth and Frederickson [131] is most quantity ubiþ1=2 at the cell interfaces to the desired commonly used for high-order accurate numerical accuracy using known average values at the ui cell solutions with the finite-volume method. centers so that the accuracy requirement of Eq. (3.44) is High-order (fourth- and sixth-order) compact differ- satisfied, e.g. determine ubiþ1=2 such that ence-based schemes were developed by Gaitonde and ub ¼ u þ ðhkÞ. (3.45) Shang [132] in the finite volume context. The formula- iþ1=2 iþ1=2 O tion of these schemes utilizes the primitive function The first step for the solution of this reconstruction approach. Optimization of the schemes for better linear problem is to form a primitive function U of uðxÞ wave propagation characteristics is performed by mini- defined as Z mizing dispersion and isotropy errors. The one-dimen- x sional advection equation is used as the prototype for U ¼ uðxÞ dx. (3.46) scheme development. 0 Then (see Section 5 for more details) qu qf þ ¼ 0; f ¼ cu; c40. (3.39) qt qx uiþ1=2 ¼ uiÀ1=2 þ uih; i ¼ 1; ...; N (3.47)

For equally spaced mesh Dx ¼ h the finite-volume and the desired point values ubiþ1=2 are obtained by formulation is dU k ubiþ1=2 ¼ þ Oðh Þ. (3.48) qu h h dx h þ fuxþ ; t À fuxÀ ; t ¼ 0, qt 2 2 Using a three- or five-point compact scheme for the 0 (3.40) evaluation of U ¼ dU=dx (see Section 3.1) one obtains fourth or sixth order accuracy for the evaluation of where u is the cell average ubiþ1=2. Z 1 xþh=2 u ¼ u dx. (3.41) 3.8. Results with ACM filters h xÀh=2 A mesh is introduced (see Fig. 9) with cells Application of the ACM filters with discretizations 1; 2; ...; i; ...; N and cell interfaces 1=2; 3=2; ...; obtained from fourth- and sixth-order accurate in space i À 1=2; i þ 1=2; ...; N þ 1=2. The discrete approxima- compact schemes showed good performance [1] for long tion of Eq. (3.40) on this mesh is time convection of isentropic vortices. Numerous examples presented in [1] demonstrate the superior qui 1 b b þ ½f iþ1=2 À f iÀ1=2¼0, (3.42) performance of high-order discretizations with ACM qt h filters. The effect of parameters, which are involved in b here f is the numerical flux function approxi- the filter (see Eqs. (3.22) and (3.24)), on the performance mating f with reconstructed values ubiþ1=2 at the cell of the ACM filter was tested extensively. interfaces The first complex flowproblem considered in [1] is vortex pairing in time-developing mixing layer. The base b b b b f iþ1=2 ¼ f ðuiÀ1; ui; uiþ1Þ¼f ðuiþ1=2Þ. (3.43) parallel flow, u ¼ tanhð2yÞ=2, is disturbed with a normal 0 The order of accuracy, k, is introduced in Eq. (3.43) as velocity perturbation v given by X2 b b k 0 Ày2=b f ðuiþ1=2Þ¼f ðuiþ1=2ÞþOðh Þ. (3.44) v ¼ ak cosð2pkx=Lx þ fkÞe , k¼1

where Lx ¼ 30, b ¼ 10, a1 ¼ 0:01, f1 ¼ p=2, and a2 ¼ 0:05, f2 ¼Àp=2. Fig. 10 shows the temperature 1 2i N-1 N field obtained from fourth order in space and time computation with the ACM filter at t ¼ 40, 80, 120 and 160. This computation was performed on a fine 201 Â 201 stretched mesh with k ¼ 0:7 and the recommended i-1/2∆x = h i+1/2 1 value of the parameter d ¼ 16 was used. The effect of the ACM filter on the resolution of the i-1 i i+1 computed solution is shown in Fig. 11. It appears that higher resolution is obtained with the ACM filter and the advantage of using high-order methods for the Fig. 9. One-dimensional finite volume mesh. computation of complex flowfeatures is evident. This ARTICLE IN PRESS

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T at t=40 T at t=80 T at t=120 T at t=160

40 40 40 40

20 20 20 20

0 0 0 0

-20 -20 -20 -20

-40 -40 -40 -40

020 020 020 020 201x201 Grid 201x201 Grid 201x201 Grid 201x201 Grid

Fig. 10. Four stages in the vortex pairing, at times t ¼ 40, 80, 120, 160, showing the temperature contours for a 201 Â 201 grid with k ¼ 0:7 for the nonlinear ﬁelds and k ¼ 0:35 for the linear ﬁelds using ACM44.

T at t = 160, ACM 44 TVD22 TVD44 TVD66

40 40 40 40

20 20 20 20

0 0 0 0

-20 -20 -20 -20

-40 -40 -40 -40

020 020 020 020 (a) 101x101 Grid (b) 101x101 Grid (c)101x101 Grid (d) 101x101 Grid

Fig. 11. Effect of order of accuracy on TVD methods (TVD22, TVD44, and TVD66), compared with the ACM44 solution at t ¼ 160, illustrated by temperature contours at t ¼ 160 for a 101 Â 101 grid. latter effect is further demonstrated in Fig. 12 where example, demonstrate that the value of d cannot increase of the order of accuracy with increasingly be reduced very much without introducing spurious higher order base schemes results into crisper resolution oscillations. of fine flowfeatures in the vortex region whilethe Another example used in [1] was the shock wave resolution of smooth flowremains the same. The effect impingement on a spatially evolving mixing layer. The of the limiter is shown in Fig. 13 where four different mixing layer was again generated with a hyperbolic limiters were tested with the same ACM44 scheme. The tangent profile with u1 ¼ 3 and u2 ¼ 2 and perturbed in effect of the smoothing parameter d in Eq. (3.27) was the normal to the main flowdirection. A strong oblique also investigated. The computations of Fig. 14, for shock at b ¼ 12 impinged on the shear layer and the ARTICLE IN PRESS

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T at t = 160, ACM22 T at t = 160, ACM44 T at t = 160, ACM66

40 40 40

20 20 20

0 0 0

-20 -20 -20

-40 -40 -40

020 020 020 (a) 101x101 Grid (b) 101x101 Grid (c) 101x101 Grid

Fig. 12. Effect of order of accuracy on ACM methods (ACM22, ACM44, and ACM66) at t ¼ 160 for a 101 Â 101 grid compared with the reference solution (ACM44, 201 Â 201 grid) using k ¼ 0:7 for the nonlinear ﬁelds and k ¼ 0:35 for the linear ﬁelds.

T at t= 160, lim1 T at t= 160, lim2 T at t= 160, lim4 T at t= 160, lim5

40 40 40 40

20 20 20 20

0 0 0 0

-20 -20 -20 -20

-40 -40 -40 -40

020 020 020 020 (a) 101x101 (b) 101x101 (c) 101x101 (d) 101x101

Fig. 13. Effect of ﬂux limiters in the ACM44 scheme on the solution resolution, illustrated by temperature contours at t ¼ 160.

flowafter the interaction remained supersonic through- simple shear layer case, the effect of the parameter d in out up to the outflowboundary. The main features of Eq. (3.27) on the stability is shown in Fig. 17. the computed flowfield withthe ACM44 method A viscous shock tube problem was considered in [36]. (fourth-order accurate for the inviscid terms plus Computed flowfields withdifferent base schemes and fourth-order accurate in space viscous terms) is shown grid densities are shown in Fig. 18. It is demonstrated in Fig. 15. The effect of the order of accuracy is shown in that the centered schemes with ACM filter reach the Fig. 16 for computations obtained with second-, fourth- same resolution and solution quality as the well- and sixth-order accurate schemes. Similarly to the established WENO schemes. Comparisons of solutions ARTICLE IN PRESS

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smu=0.25 smu=0.15 smu=0.1 smu=0.0

40 40 40 40

20 20 20 20

0 0 0 0

-20 -20 -20 -20

-40 -40 -40 -40

020 020 020 020 101x101 Grid 101x101 Grid 101x101 Grid 101x101 Grid (a)(b)(c)(d)

Fig. 14. Effect of d, ((smu)2) in the ACM44 scheme on the solution resolution, illustrated by temperature contours at t ¼ 160.

at t = 120 (min = 0.314432 max = 2.83920), 641×161 20 10 0 -10 -20 (a) 050100 150 200 p at t = 120 (min = 0.218441 max = 0.718442), 641×161 20 10 0 -10 -20 (b) 0 50 100 150 200 T at t = 120 (min = 0.196550 max = 1.16033), 641×161 20 10 0 -10 -20 (c) 0 50 100 150 200

Fig. 15. The reference solution for the shock–shear-layer interaction problem at t ¼ 120.

computed with ﬁner and ﬁner meshes and evaluation of original ACM sensor of Eq. (3.27) are shown in Fig. 20. the ACM sensor using wavelets shown in Fig. 19 Comparisons of the computed density distribution on demonstrate that at all grid densities the performance the wall obtained from numerical solutions with is very good. For comparison, computations with the different methods is shown in Fig. 21. ARTICLE IN PRESS

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at t = 120 (min = 0.284828 max = 2.95068), ACM22, 321×81 20 10 0 -10 -20 (a) 0 50 100 150 200 at t = 120 (min = 0.319307 max = 2.89339), ACM44, 321×81 20 10 0 -10 -20 (b) 0 50 100 150 200

at t= 120 (min = 0.319319 max = 2.70419), ACM66, 321×81 20 10 0 -10 -20 (c) 0 50 100 150 200

Fig. 16. Comparison of density contours at t ¼ 120 for the shock–shear-layer test case.

at t = 120 (min= 0.321908 max = 2.70169), ACM44, smu = 0.15, 321×81 20 10 0 -10 -20 (a) 0 50 100 150 200

at t = 120 (min= 0.321352 max = 2.71471), ACM44, smu = 0.1, 321×81 20 10 0 -10 -20 (b) 0 50 100 150 200

at t = 120 (min= 0.319385 max = 2.73695), ACM44, smu = 0.0, 321×81 20 10 0 -10 -20 (c) 0 50 100 150 200

Fig. 17. Effect of d, ((smu)2) in the ACM44 scheme on the solution resolution, illustrated by density contours at t ¼ 120. ARTICLE IN PRESS

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Fig. 18. Grid reﬁnement comparison of ACM66-RK4 and WENO5-RK4 for Re ¼ 200. Density contours. (a) ACM66-RK4, 250 Â 125 grid; (b) WENO5-RK4, 250 Â 125 grid; (c) ACM66-RK4, 500 Â 250 grid; (d) WENO5-RK4, 500 Â 250 grid; (e) ACM66-RK4, 1000 Â 500 grid; (f) WENO5-RK4, 1000 Â 500 grid.

Fig. 19. Grid reﬁnement of WAV66-RK4 for Re ¼ 1000. Fig. 20. Grid reﬁnement of ACM66-RK4 for Re ¼ 1000. Density contours, 2000 Â 1000 grid. Density contours, 2000 Â 1500 grid.

High-order accurate solutions with ACM filters were obtained for the computation of the flowwithvery also obtained in [117] for more complex problems complex flowfeatures is shownin Fig. 23. For containing finite-rate chemistry. The flowfield resulting comparison, in Fig. 24 the computed flowfields with from the interaction of a planar Mach 2 shock in air the WENO fifth-order accurate scheme on the same with a circular zone of hydrogen was considered. meshes is shown. It appears that the high order in space- The computed flowfield at different times is shownin centered scheme with the ACM filter yields the same Fig. 22. The improvements with grid refinement solution quality as the WENO scheme. ARTICLE IN PRESS

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Fig. 21. Density along lower wall on different grids for Re ¼ 1000. WAV66-RK4. Fig. 23. Grid reﬁnement of the ACM66-RK4 scheme. Hydro- gen mass fraction contours at time 60 ms.

Fig. 22. Density contours of ACM66-RK4 on a 500 Â 250 grid. Fig. 24. Grid reﬁnement of the WENO5-RK4 scheme. Hydro- gen mass fraction contours at time 60 ms.

3.9. Results with compact schemes and filters was performed with the fourth-order Runge–Kutta scheme. The computational domain extended to two Application of compact centered schemes found acoustic wavelengths in all directions. A major numer- widespread application in the numerical simulation of ical consideration in the aeroacoustic computations of noise sources from compressible flowsuch as vortices [133] was the choice of boundary conditions. The and jets. Several examples of computations obtained appropriate boundary conditions for the analysis with compact schemes are presented in the following of flows in free space are non-reflecting boundary paragraphs. The far-field sound generated by compres- conditions which allow waves to freely leave the sible co-rotating vortices was computed by the numer- domain. The zeroth-order boundary conditions of ical solution of the two-dimensional, compressible [134] were used for the computations. It was found in Navier–Stokes using the sixth-order compact scheme [133] that the combination of spatial and temporal to evaluate the spatial derivatives [133]. Time marching schemes with these boundary conditions has negligible ARTICLE IN PRESS

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Fig. 25. The far-ﬁeld pressure (ambient removed) at 2 tc0=R ¼ 185. The contour levels are from P=ðr0c0Þ¼Æ4 Â À5 2 À5 10 with DP=ðr0c0Þ¼0:05 Â 10 . The ‘cross’ pattern is a plotting illusion caused by the high grid densities near the x- and y-axis.

numerical damping and preserves the physical pro- 1 Fig. 26. Far-ﬁeld pressure traces at (a) r=l ¼ 2 and (b) r=l ¼ 2 perty that, in the absence of viscosity waves pro- showing the results of the simulation (—), and the prediction of pagate unattenuated. For the vortices separated by Mohring’s equation (- - -). Both measurement points are located distance 2R, the initial vorticity distribution was on the positive y-axis. Gaussian as

U 0 2 o ¼ 3:57 eÀ1:25ðr=r0Þ r 0 Fig. 27 for different frequencies. A comparison of À1 with circulation G0 ¼ 2pð0:7Þ U0r0 and r0 the distance the computed results with an acoustic analogy is shown from the vortex core where the maximum Mach number in Fig. 28. is attained for each vortex. The numerical results of [133] A third example of noise source calculation from free were compared with exact results from the acoustic shear flowis the investigation of sound generation analogue by Mohring [135]. The computed far-field mechanisms in a Mach M ¼ 0:9, Re ¼ 3600 turbulent jet pressure is shown in Fig. 25. Comparisons of the [137]. The numerical solution of the governing equations computed solutions [133] with the theoretical predic- was obtained for a cylindrical coordinates system. The tions of Mohring [135] are shown in Fig. 26. The azimuthal derivatives were computed with Fourier agreement of the computations with the theory for the spectral methods and the radial and axial derivatives far-field sound from compact low-Mach-number flows is were computed with sixth-order compact finite differ- very good. ences. Again the fourth-order Runge–Kutta method was A second example of sound generation computation is used to advance the solution in time. Some examples the sound in a mixing layer [136]. The sound generated from the computation of [137] are presented. The by vortex pairing in a two-dimensional compressible instantaneous contours of vorticity are shown in Fig. mixing layer was computed with DNS of the Navier– 29. A visualization of the sound field is shown in Fig. 30. Stokes equations for the near-field and the acoustic field. Comparisons of the computations with measurements Fourth- and sixth-order compact schemes were used for are shown in Figs. 31–33. A comparison of the the computation of the derivatives. Time marching computed data with Lighthill’s equation is shown in was performed with the fourth-order Runge–Kutta Fig. 34. Further examples of jet noise prediction with method. The computed acoustic field [136] is shown in explicit and compact finite-difference schemes can be ARTICLE IN PRESS

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Fig. 27. Contours of the real part of the DFT of the dilatation away from the sheared region for several frequencies: (a) f =4, (b) f =2, (c) f, (d) 3f =2.

found in recent publications [138–142] and references Fig. 37. The fluctuating pressure of Fig. 37b clearly therein. shows the dipole radiation pattern of the sound Sound generation flowlowMach number flowover a generated by the cylinder unsteady wake. Comparisons cylinder was computed by DNS of the two-dimensional of the DNS with the theory are shown in Fig. 38. The Navier–Stokes equations [143]. The governing equations agreement with the theory of Curle is very good. Further were again solved with a sixth-order accurate compact applications of high-order finite-difference method to scheme and time integration was performed with the aeroacoustics can be found in recent publications fourth-order accurate Runge–Kutta method. Exam- [144–146] and references therein. ples from the computations of [143] are shown in Comparisons of the numerical solution obtained with Figs. 35–38. Fig. 35 shows the time-dependent vorticity fourth-order accurate centered schemes with other field. Fig. 36 shows the development of the fluctuating solutions obtained by second-order centered schemes pressure. Different views of pressure field are shown in and upwind methods were carried out in [147].Itwas ARTICLE IN PRESS

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Fig. 28. Comparison of the magnitude of the acoustic waves (dilatation) predicted by solving the acoustic analogy (solid lines) and from the DNS (open circles).

6 4 2 o

r 0 r -2 -4 -6 0 5 10 15 20 25 30

Fig. 29. Instantaneous contours of vorticity magnitude: levels are or0=Uj ¼ 0:35, 1, 2, 3, 4 with lighter contours representing larger values. Peak vorticity magnitude (not shown) was or0=Uj ¼ 11.

found that the discretization using higher-order approximation for all terms was substantially more accurate than the others. It produced less than two percent numerical error in lift and drag components on grids Fig. 30. Visualization of the far-ﬁeld sound: black is Y ¼ with less than 13,000 nodes for subsonic cases and less rÁuo À 0:0005a0=r0 and white is Y40:0005a0=r0. The gray than 18,000 nodes for transonic cases. It was also scale varies continuously between these extrema. The jet is concluded [147] that the higher-order discretization visualized with contours of vorticity magnitude. ARTICLE IN PRESS

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Fig. 31. Second moments of velocity: , x ¼ 20r0; B, x ¼ 25r0; &, x ¼ 30r0; —, low-Mach-number experimental data from Panchapakesan and Lumley [268]; and - - - -, low-Mach-number experimental data from Hussein et al. [269]. The Panchapakesan and Lumley [268] profiles were digitized and re-plotted from the least-squares fits in their publication; the Hussein et al. [269] profiles are the curve fits they provided. The simulation data were averaged in time for the entire simulation history and in space over a streamwise band of width Dxave ¼ 2r0.

produces solutions of a given accuracy much more Spiral vortex breakdown above a slender delta wing efficiently than the others. with a sweep angle of 75 at 32 angle of attack was also Shedding past a cylinder at ReD ¼ 1000 was com- computed in [148]. The free stream Mach number was puted in [148] with the second- and sixth-order compact M ¼ 0:2, and the Reynolds number based on centerline schemes with tenth order spectral type filters (CD6F10 chord is ReC ¼ 9200. Fig. 41a and b showthe schemes). It was found that the higher-order scheme is instantaneous y component of vorticity on a vertical well behaved for this general configuration, which plane cutting through the vortex core for the second- was discretized with a high-aspect-ratio, stretched mesh. order method. At this angle of attack, the breakdown Fig. 39 shows that even at this low Reynolds number, location is highly sensitive to details of the numerical results with the higher-order compact algorithm are simulation. Thus, on the coarse mesh the second-order superior in preserving the vortex street behind the scheme results in a premature break down of the vortex. cylinder. A quantitative comparison made at RD ¼ 100 By contrast, the CD6F10 solution shown in Fig. 41c with other highly resolved computations displayed good displays a breakdown location similar to the second- agreement the Strouhal number and maximum lift and order results on the fine mesh. The second-order method drag coefficients for the high-order schemes. Preliminary on the coarser mesh (Fig. 41a) shows relatively little direct numerical simulations of the three-dimensional detail with only two smeared concentrations of azi- wall-jet transition process are performed in [148] with muthal vorticity in the vortex wake. On the other hand, the implicit time-integration scheme on a mesh of size CD6F10 results (Fig. 41c) exhibit a much richer 230 Â 125 Â 101 with both the second and CD6F10 structure corresponding to a stronger, tightly wound schemes. In the computations the jet is forced at the spiral in far better agreement with the grid-converged inflow plane with a two-dimensional forcing function. results of Fig. 41b obtained with the second order Fig. 40 shows the computed instantaneous flow struc- method with a fine grid. ture in terms of contours of vorticity magnitude on a Several test of vortex convection were performed in plane parallel to the plate at a distance of 1:3h. [149]. It was shown that for long-time propagation it is ARTICLE IN PRESS

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Fig. 34. Comparison of dilatation on a ray from x ¼ 18r0 and inclined at 45 from the jet axis: —, direct numerical simulation; - - - -, Lighthill’s equation.

Fig. 32. Overall sound pressure level (and acoustic intensity in brackets) directivity on an arc at 60r0 from the nozzle and with a measured from the jet axis: ——, Dp~curleRe ¼ 3600 present study; , Re ¼ 3600, Stromberg et al. [270] experimental data; 5 , Re ¼ 2 Â 10 , Mollo-Christensen et al. [271] experimental 5 data; n, Re ¼ 6 Â 10 , Lush [272] experimental data (adjusted by 12 dB from 240 jet radii to 60).

Fig. 35. Time development of a vorticity ﬁeld. M ¼ 0:2, Re ¼ 150. The contour levels are from omin ¼À1:0toomax ¼ 1:0 with an increment of 0.02: ____, o40;–––,oo0. (a) t ¼ 1930, (b) t ¼ 1945.

approach for a non-trivial column vortex convection on a 3-D dynamic mesh was considered in [149]. The axis of the vortex placed along z ¼ 0. Inviscid calculations were performed with the E2 and C6 schemes using the RK4 method and a time step Dt ¼ 0:002. A cross-section of the vortex on a z ¼ constant plane at T ¼ 3:25 is shown in Fig. 42a. At this instant, the grid has already Fig. 33. Far-ﬁeld pressure spectrum at a ¼ 30: —, present experienced more than three cycles of the imposed simulation; ...... , measurements of Stromberg et al. oscillation with frequency o ¼ 1. As the contours of velocity magnitude indicate, despite the signiﬁcant important to compute the metric quantities with the unsteady mesh deformations, the high-order method is same high order of accuracy as the base scheme for the capable of preserving the axisymmetric character of the convective terms. The performance of the high-order vortex even on this relatively coarse discretization ARTICLE IN PRESS

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Fig. 36. Time development of a ﬂuctuation pressure ﬁeld, Dp~.

M ¼ 0:2, Re ¼ 150. The contour levels are from Dp~min ¼ 2:5 2:5 À0:1M to Dp~max ¼ 0:1M with an increment of 0:0025M2:5.—,Dp~40;----,Dp~o0. (a) t ¼ 1925, (b) t ¼ 1930, (c) t ¼ 1935, (d) t ¼ 1940, (e) t ¼ 1945, (f) t ¼ 1950, (g) t ¼ 1955, (h) t ¼ 1960.

(approximately 10 points across the vortex). A comparison of the computed and exact solutions is shown in Fig. 42b in terms of the v component of velocity. The Fig. 37. Three different views of a pressure ﬁeld at M ¼ 0:2, Re ¼ 150; —, positive pressure; – – – –, negative pressure. (a) C6F10 results are observed to be in excellent agreement Total pressure, Dp,att ¼ 2010, (b) ﬂuctuation pressure, Dp0, with the theoretical answer. This case demonstrates t ¼ 2010, (c) mean pressure, Dp . the advantage of the high-order methodology over mean standard low-order approaches even for 3-D applica- Further demonstration of the high-order, dynamic- tions in which the mesh is subjected to severe dynamic mesh technique was carried out in [149] with the deformation. simulation of the aeroelastic interaction arising from ARTICLE IN PRESS

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Fig. 38. Comparison of pressure distribution between DNS and Curle’s solutions. M ¼ 0:2, Re ¼ 150, t ¼ 2000. The contour levels are from À0:1M2:5 to 0:1M2:5 with an increment of 0:0025M2:5. —, Positive pressure, - - - -, negative pressure. (a) Dp (DNS), (b) Dp~ (DNS), M D D (c) Dp~ (DNS), (d) Dp~curle þ Dpmean, (c) Dp~curle, (f) Dp~curle.

viscous laminar flowover a flexible surface. A schematic flexible panel was assumed to be fixed at the free of the configuration considered is shown in Fig. 43a. The stream value p1. The flowfield wascomputed using free-stream Mach number and Reynolds number (based the sixth-order scheme (C6) and the second-order, 5 on panel length, L) were M ¼ 0:9 and Re ¼ 1:0 Â 10 , implicit Beam-Warming method (with D ¼ 0:01 and respectively. The computed boundary-layer thickness at four subiterations). At each subiteration of the implicit the leading-edge of the flexible panel was approximately time-marching method, the shape of the deforming d ¼ 0:04L. The pressure in the cavity underneath the panel is updated by the structural solver. Based on ARTICLE IN PRESS

226 J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 the newboundary coordinates, the fluid dynamic tuations result in a significant acoustic radiation mesh is evolved by propagating the panel deforma- pattern above the vibrating panel, shown in Fig. 43cin tions into the entire field. Following initial transients, terms of a snapshot of the instantaneous pressure. a limit-cycle-oscillation with an approximate non- Corresponding contours of vorticity are shown in dimensional frequency St ¼ fL=u1 ¼ 1:62 was achieved Fig. 43d, with an enlarged scale (by a factor of 8) in by the combined fluid/structural system. A representa- the y direction for the purpose of clarity. Vorticity waves tive plot of the instantaneous panel deflection is are clearly visible in the boundary layer and appear to shown in Fig. 43b. From this and many other roll up and interact with the wall resulting in the instantaneous realizations of the panel shape (not formation of secondary incipient separation regions. shown), it became apparent that the panel dyna- High-order accurate computations of compressible flows mics comprises a first-mode mean downward deflec- with centered schemes were carried out in numerous tion upon which a high-mode, high-frequency vertical recent publication. See for example, [150–154], for more fluctuation is superimposed. These high-frequency fluc- details.

0.2 Breakdown 2nd (Grid 1) 0.15 2 2nd order 0.1

Z/C 0.05 1 0 0 Y/D (a) 0.4 0.6 0.8 1 -1 0.2 2nd (Grid 2) -2 0.15 0.1 0246810 Z/C X/D 0.05 0 2 (b) CDF10 0.4 0.6 0.8 1 1 0.2 CD6F10 (Grid 1) 0 0.15 Y/D 0.1

-1 Z/C 0.05 -2 0 02468 0.4 0.6 0.8 1 X/D (c) X/C

Fig. 39. Computed Karman vortex street behind cylinder at Fig. 41. Azimuthal component of vorticity on vertical plane ReD ¼ 1000. passing through the core of the broken-down vortex.

20 20

15 15

5 5 Z/h 10 Z/h 10

0 0

-5 -5

-10 -10 01020 30 0 1020 30 (a) X/h(b) X/h

Fig. 40. Vorticity magnitude contours on y/dimensional wall-jet instability with (a) second order and (b) CD6F10 schemes. ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 227

0.01 E2 C6 Exact 0.005

-0.005

-0.01 (a) (b) -4 -2 0 2 4

Fig. 42. Computed vorticity and comparison with the exact solution for the convection of a vortex in a three-dimensional deforming grid.

y -0.001 U∞ boundary layer X -0.002

flexible panel Panel deflection

(a) L -0.003 0 0.5 1 (b) X/L

Fig. 43. Computed solution of the ﬂowover a ﬂexible panel.

4. ENO and WENO schemes The ENO idea, proposed by Harten and Osher [155] and Harten et al. [156], is the first successful attempt to ENO and WENO schemes are high-order accurate obtain no-mesh size-dependent (self-similar), uniformly finite-difference or finite-volume numerical methods high-order accurate, yet essentially non-oscillatory designed for the solution of hyperbolic problems with interpolation for piecewise smooth functions using an piecewise smooth solutions containing discontinuities. adaptive local stencil that satisfies certain measures of The key idea of these methods is the design of the locally local smoothness. ENO offered significant improve- smoothest stencil that avoids crossing discontinuities as ments over earlier approaches, which attempt to much as possible. eliminate or reduce spurious oscillations generated at ARTICLE IN PRESS

228 J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 discontinuities by the ﬁxed stencil, second- or higher- k−1 = ∑ ui+1/2 crjuir+j order accurate methods, which used artiﬁcial viscosity or j=0 limiters. ENO and WENO schemes were proven suitable for r+s=k−1 = CFD of compressible turbulence, aeroacoustics, and S(i) {Ii−r,...,Ii...,...,Ii+s} other applications where the solution contains both = S1{i} {Ii} smooth but complex features and discontinuities. xi−r xi−r−1/2 xi−r−1 xi−1/2 xi xi+1/2 xi+s−1 xi+s−1/2 xi+s Several of ENO and WENO applications shown at the ~ = end of this section demonstrate the enhanced resolution Sm+1{i} {xi−1/2,...,xi+m−1/2} of these schemes and their potential to replace tradi- s tional methods in large-scale computations. The pre- r sentation of ENO and WENO schemes starts with a reviewof interpolation and approximation theory used Fig. 44. ENO r þ s ¼ k À 1 wide stencil for the kth order with these schemes. accurate approximation of uiþ1=2 from cell average values u¯i.

4.1. High-order reconstruction dures are presented. The implementation of ENO and WENO schemes is given and recent improvements of A basic approximation problem encountered in the these schemes are discussed. numerical solution of hyperbolic conservation laws is to obtain high-order (second-order accurate or higher) 4.2. Approximation in one dimension reconstruction (finite-volume approximation) or conservative approximation of the derivatives (finite-difference The basic information about polynomial interpolation approximation) from cell averaged or point values of the and approximation is reviewed. This background is state variables, respectively. Numerical algorithms for fundamental for the understanding of ENO interpola- conservation laws that have been extensively investi- tion and other numerical methods. The presentation is gated in the past three decades laid the foundation for given in one space dimension. The formulation of the the development of modern schemes, [127,128,157] basic approximation problem is as follows: including TVD schemes [72] and the piecewise parabolic Given a mesh in the interval between a and b method [129] that reduce or eliminate spurious numerical oscillations at discontinuities. In the following a ¼ x1=2ox3=2o ...xiþ1=2 ...oxNÀ1=2oxNþ1=2 ¼ b sections, the basic ideas of ENO approximation and with subintervals or cells I i ½xiÀ1=2; xiþ1=2, cell centers reconstruction, and the conservative approximation of 1 xi 2 ðxiÀ1=2 þ xiþ1=2Þ and cell size Dxi xiþ1=2À the derivatives with an adaptive, smooth stencil, which xiÀ1=2; i ¼ 1; 2; ...; N, (see Fig. 44), consider the follow- uses the smoothest possible data for the reconstruction, ing reconstruction problem. Given the cell averages of a are explained. Based on this idea, ENO and WENO function uðxÞ at the cell centers xi schemes have a distinct advantage over TVD schemes Z 1 xiþ1=2 that near every extrema, even smooth ones, degrade to u uðxÞ dx; i ¼ 1; 2; ...; N (4.1) i Dx first-order accuracy to suppress any spurious numerical i xiÀ1=2 oscillations. ENO schemes, on the other hand, which are find a polynomial p ðxÞ, of degree k À 1 at most, such based on high-order accurate conservative approxima- i that p ðxÞ is the kth order accurate approximation to the tions of derivatives (FD) or polynomial reconstructions i function uðxÞ inside each cell I , e.g. a polynomial from the average state (FV), are high-order accurate and i satisfying essentially non-oscillatory up to the discontinuity; i.e. k the numerical oscillations, if any, decay with the order of piðxÞ¼uðxÞþOðDx Þ; x 2 Ii; i ¼ 1; 2; ...; N. (4.2) the truncation error. The approximations of the function uðxÞ with the The one-dimensional high-order approximation pro- polynomial p ðxÞ at the I cell boundaries, i À 1=2 and blem of a function from given cell average values is i i i þ 1=2, are also kth order accurate, e.g. described in Section 4.2. ENO obtains approximation k with polynomials. This polynomial approximation uiÀ1=2 ¼ piðxiÀ1=2Þ¼uðxiÀ1=2ÞþOðDx Þ, procedure is the basis of finite volume ENO schemes for arbitrary grid spacing in one dimension. The k uiþ1=2 ¼ piðxiþ1=2Þ¼uðxiþ1=2ÞþOðDx Þ conservative approximation of the derivative, which is i ¼ 1; 2; ...; N. ð4:3Þ also based on the one-dimensional polynomial approximation, is the basis of the finite-difference ENO The procedure to solve this problem is: Given the schemes and presented in Section 4.3. In the following location I i and the order of accuracy k, choose a stencil sections, the ENO and WENO reconstruction proce- SðiÞ¼fI iÀr; ...; I iþsg including I i itself, r cells to the ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 229

− left of Ii, and s cells to the right, with r þ s þ 1 ¼ k, (see k 1 ^ = ∑ + = − ui+1/2 crjuir+j r s k 1 Fig. 44), and use the average values ui in SðiÞ to obtain j=0 = the polynomial piðxÞ that yields the kth order accurate S(i) {Ii−r,...,Ii...,...,Ii+s} = approximation of uðxÞ given in Eq. (4.2). S1{i} {Ii} For a smooth function uðxÞ in the region covered by x − − xi−1/2 x + the stencil SðiÞ, there is a unique polynomial of degree i r 1/2 i 1/2 xi−r x − − x k À 1 ¼ r þ s at most whose cell average in each cell of i r 1 xi i+1 xi+s−1 xi+s ~ = SðiÞ agrees with the average of uðxÞ, e.g. Sm+1{i} {xi,...,xi+m} Z 1 xjþ1=2 S pðxÞ dx ¼ u ; j ¼ i À r; ...; i þ s. (4.4) r Dx i i xjÀ1=2 The polynomial pðxÞ is the kth order approximation we Fig. 45. ENO r þ s ¼ k À 1 wide stencil for the kth order are looking for. Furthermore, since the mappings from accurate approximation of the numerical flux uîþ1=2 from point values. the given cell average values uj in the stencil SðiÞ to the values uiÀ1=2 and uiþ1=2 at the cell boundary are linear, there exist constants crj, which depend on the left shift r, on the order of accuracy k and on the cell size Dxi such that averages (see Eq. (4.7)) one can use Lagrange interpolation polynomials to obtain the constants crj in Eq. (4.5). XkÀ1 k The interpolation polynomials and the values of the uiþ1=2 ¼ crjuiÀrþj ¼ uðxiþ1=2ÞþOðDx Þ. (4.5) constants crj for k ¼ 1, up to k ¼ 4 can be found in j¼0 Appendix A. The constants crj are obtained using the primitive function UðxÞ of uðxÞ defined as Z 4.3. One-dimensional conservative approximation of the x derivative UðxÞ uðxÞ dx. (4.6) À1 For finite-difference schemes, the basic problem of The primitive function can be expressed by cell average high-order accurate discretization is the conservative values u of uðxÞ using the definition of Eq. (4.1) to i approximation of the derivative of a function uðxÞ obtain Z ui uðxiÞ; i ¼ 1; 2; ...; N (4.9) Xi xiþ1=2 Xi Uðx Þ¼ uðxÞ dx ¼ u Dx . (4.7) iþ1=2 j j from given point values at the nodes xi of the mesh. x j¼À1 jÀ1=2 j¼À1 In the ENO and WENO framework, this is accom- equation shows that the cell averages ui yield the exact plished through the use of a numerical flux function values of the primitive function UðxÞ at the cell ubiþ1=2 at the cell centers or half-nodes xiþ1=2 boundaries. ub ubðu ; ...; u Þ; i ¼ 0; 1 ...; N. (4.10) Considering the unique polynomial PðxÞ of degree k iþ1=2 iÀr iþs which interpolates the primitive function at the k þ 1 This numerical flux depends on r point values on the left cell boundaries xiÀrÀ1=2; ...; xiþsþ1=2 and denoting its and s point values on the right (see Fig. 45) such that the derivative P0ðxÞ by pðxÞ it is easy to verify that pðxÞ is the flux difference approximates the derivative u0ðxÞ¼ polynomial we are looking for and satisfies Eq. (4.4) as duðxÞ=dx to kth order accuracy follows: Z 1 0 k x ðub À ub Þ¼u ðx Þþ ðDx Þ, 1 jþ1=2 iþ1=2 iÀ1=2 i O pðxÞ dx Dxi Dxj x i ¼ 0; 1; ...; N. ð4:11Þ jÀ1=Z2 xjþ1=2 1 0 ¼ P ðxÞ dx For a uniform grid Dxi ¼ Dx, this problem can be solved Dxj x Z jÀ1=2 with the same technique used to obtain kth order 1 xjþ1=2 accurate reconstruction, which was presented in Section ¼ uðxÞ dx ¼ uj; j ¼ i À r; ...; i þ s ð4:8Þ 4.2. The assumption of uniform grid spacing for the Dxj x jÀ1=2 conservative evaluation of the derivative in the in addition, have finite-difference formulation is essential. However, the reconstruction for the finite-volume formulation of P0ðxÞ¼U 0ðxÞþ ðDxkÞ8x 2 I . O i Section 4.2 is valid for non-uniform grid spacing. Since the exact value of the primitive function UðxÞ of Assuming that a numerical flux function hðxÞ exists, uðxÞ at the cell boundaries are obtained from the cell and that this function depends on the uniform grid ARTICLE IN PRESS

230 J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 spacing Dx, such that For k odd, the best approximation is obtained by one Z point upwind-biased stencils r ¼ s or r ¼ s À 2. For 1 xþDx=2 ui ui ¼ hðxÞ dx (4.12) example, third-order upwind fluxes for k ¼ 3 are Dx xÀDx=2 1 3 ubiþ1=2 ¼ ½ÀuiÀ1 þ 5ui þ 2uiþ1þOðDx Þ, then 6 1 3 1 Dx Dx ub ¼ ½Àu À þ 5u À þ 2u þOðDx Þ u0ðxÞ¼ hxþ À hxÀ (4.13) iÀ1=2 6 i 2 i 1 i Dx 2 2 which yield a third order accurate conservative approx- therefore, the numerical flux function we are looking for imation of the derivative must satisfy 1 b b 1 k ½uiþ1=2 À uiÀ1=2¼ ½uiÀ2 À 6uiÀ1 þ 3ui þ 2uiþ1 ubiþ1=2 ¼ hðxiþ1=2ÞþOðDx Þ. (4.14) Dx 6 ¼ u0ðx ÞþOðDx3Þ. It is not straightforward to find hðxÞ because i Eq. (4.12) defines the unknown function hðxÞ implicitly. The selection of the most suitable stencils among However, observing that the known function uðxÞ is the different possible stencils of Sections 4.2 and 4.3 for cell average of hðxÞ, (see Eq. (4.12)), one can use the fixed k is accomplished through the ENO or WENO same reconstruction procedure of Section 4.2 to reconstruction procedures that are described next. approximate hðxÞ. Considering the primitive HðxÞ of hðxÞ, where HðxÞ 4.4. ENO reconstruction satisfies Z x Approximation of discontinuous solutions that occur HðxÞ¼ hðxÞ dx (4.15) in hyperbolic conservation laws is accomplished with À1 piecewise smooth functions. These functions uðxÞ have then Eq. (4.12) implies derivatives at all points except at discontinuities where Z Xi xjþ1=2 Xi the function and its derivatives are assumed to have Hðxiþ1=2Þ¼ hðxÞ dx ¼ Dx ui. (4.16) finite left and right limits. For such piecewise smooth j¼À1 xjÀ1=2 j¼À1 functions, the order of accuracy is determined by the Summarizing: The conservative approximation of the local truncation error in smooth regions of the definition derivative in the finite-difference context becomes of the function. A fixed stencil high-order approxima- equivalent to the reconstruction problem in the finite- tion of a piecewise smooth function is not adequate near discontinuities, because stencils that contain discontin- volume formulation when the given point values ui are identified as cell averages of an unknown function hðxÞ uous cells cause oscillations (Gibbs phenomenon) in the that satisfies Eq. (4.12). The primitive function of hðxÞ, numerical solution. HðxÞ, is then exactly known at the cell interfaces (see The basic idea of the ENO approximation is to avoid including discontinuous cells in the stencil as much as Eq. (4.16)), or half-points xiþ1=2 from the point values ui at the nodes. Using the same approximation procedure possible. This is accomplished by the ‘‘adaptive stencil’’ described in Section 4.2 obtain the kth order approx- where the left shift changes with the location xi. In the ENO approximation, this is achieved by using the imation to hðxiþ1=2Þ, which according to Eq. (4.14) is the Newton divided differences of the interpolation poly- numerical flux ubiþ1=2 we are looking for. For a stencil SðiÞ around the point i (see Fig. 45), nomial as a smoothness indicator of the stencil. r points to the left and s points to the right, The jth degree divided differences F½xiÀ1=2; ...; xiþjÀ1=2 of the primitive function FðxÞ,off ðxÞ (see Eq. (4.15)), ðxiÀr; ...; xi; ...; xiþsÞ, where r þ s ¼ k þ 1 the numerical flux is expressed as that is defined at the cell faces or half- points xiÀ1=2; ...; xiþjÀ1=2 are given by the recursive XkÀ1 formula ubiþ1=2 ¼ crjuiÀrþj, (4.17) j¼0 F½xiÀ1=2¼FðxiÀ1=2Þ where the values of the constants crj are given in Table A.1 of Appendix A. F½xiÀ1=2; ...; xiþjÀ1=2 For a globally smooth function uðxÞ, the best F½xiþ1=2; ...; xiþjÀ1=2ÀF½xiÀ1=2; ...; xiþjÀ3=2 . approximation is obtained for even k by centered xiþjÀ1=2 À xiÀ1=2 approximation r ¼ s À 1. For example, the fourth-order ð4:18Þ accurate centered flux approximation from Eq. (4.17) and k ¼ 4 is obtained by A similar formula defines the divided differences of the cell averages f f ½xi of the function f ðxÞ defined at the 1 4 i ubiþ1=2 ¼ ½ÀuiÀ1 þ 7ui þ 7uiþ1 À uiþ2þOðDx Þ. 12 cell centers xiþ1=2; ...; xiþ1=2þj (FV), or nodes xi; ...; xiþj ARTICLE IN PRESS

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2 (FD) approach. The divided differences for f i are (3) Obtain higher-order interpolation polynomials PR 2 and PS by expanding the stencil to the left including f ½xiþ1; ...; xiþjÀf ½xi; ...; xiþjÀ1 f ½x ; ...; x ¼ . xiÀ3=2 or to the right including xiþ3=2, respectively. i iþj x À x iþj i 2 1 R PR ¼ P ðxÞþU½xiÀ3=2; xiÀ1=2; xiþ1=2 The function f ðxÞ and its primitive FðxÞ¼ x f ðxÞ dx À1 Âðx À x Þðx À x Þ, are related by iÀ1=2 iþ1=2 Z Xi xjþ1=2 Xi 2 1 PS ¼ P ðxÞþU½xiÀ1=2; xiþ1=2; xiþ3=2 Fðxiþ1=2Þ¼ f ðxÞ dx ¼ f jDxj (4.19) j¼À1 xjÀ1=2 j¼À1 Âðx À xiÀ1=2Þðx À xiþ1=2Þ. ð4:24Þ therefore 2 The deviations from the linear approximation of PR 2 Fðx ÞÀFðx Þ and PS depend on the divided differences F½x ; x ¼ iþ1=2 iÀ1=2 ¼ f . (4.20) iÀ1=2 iþ1=2 i U ½xiÀ3=2; xiÀ1=2; xiþ1=2 and U ½xiÀ1=2; xiþ1=2; xiþ3=2, xiþ1=2 À xiÀ1=2 respectively. As a result Eq. (4.18) can be expressed in terms of f , However, a divided difference is a measure of since the ﬁrst degree divided differences of FðxÞ are the smoothness of the function in the stencil because zeroth degree divided differences of f , and the computa- for a smooth function UðxÞ have U½xiÀ1=2; ...; tion of the primitive function F can be completely xiþjÀ1=2¼VðjÞðxÞ=j for some xiÀ1=2oxoxiþjÀ1=2, avoided. while if UðxÞ is discontinuous inside the stencil j Using the above deﬁnitions, the kth degree interpola- U ½xiÀ1=2; ...; xiþjÀ1=2¼Oð1=Dx Þ. tion polynomial PðxÞ that interpolates the primitive Therefore, if function FðxÞ at k þ 1 points is expressed with divided jU½x ; x ; x j jU½x ; x ; x j differences by iÀ3=2 iÀ1=2 iþ1=2 o iÀ1=2 iþ1=2 iþ3=2 (4.25) Xk PðxÞ¼ F½xiÀrÀ1=2; ...; xiÀrþjÀ1=2 select the stencil ¼ j 0 e YjÀ1 S3ðiÞ¼fxiÀ3=2; xiÀ1=2; xiþ1=2g

Â ðx À xiÀrþmÀ1=2Þð4:21Þ or S2ðiÞ¼fIiÀ1; I igð4:26Þ m¼0 and the polynomial p ðxÞ¼P0ðxÞ is expressed as otherwise, select the stencil e Xk S3ðiÞ¼fxiÀ1=2; xiþ1=2; xiþ3=2g pðxÞ¼ F½x ; ...; x iÀrÀ1=2 iÀrþjÀ1=2 or S2ðiÞ¼fIi; I iþ1g. ð4:27Þ j¼0 XjÀ1 YjÀ1 e Â ðx À xiþrþlÀ1=2Þ, ð4:22Þ (4) Continue this process until the stencil SkðiÞ with the m¼0 l¼0 lam desired number of points is reached. where again pðxÞ can be expressed by divided differences Note again that all divided differences of the primitive of f . are computed in terms of averages. For uniform mesh, The ENO selection process of the smoothest stencil of the divided differences are replaced by undivided k þ 1 consecutive points that include x and x is iÀ1=2 iþ1=2 differences. The finite-volume and finite-difference based on Eq. (4.21) and is performed with the following ENO algorithms based on the approximation and steps: reconstruction procedures of Sections 4.2–4.4 are e summarized next. (1) Start with the two-point stencil S2ðiÞ¼ fxiÀ1=2; xiþ1=2g (see Figs. 44 and 45) of the primitive 4.5. 1D finite-volume ENO scheme function U of u, which has a corresponding single- cell stencil SðiÞ¼fI g in terms of u (see Eq. (4.20) i From the cell average values fuig of the function uðxÞ, and Figs. 4.1 and 4.2). (see Fig. 44) obtain a piecewise polynomial reconstruc- (2) Obtain the linear (first degree) interpolation poly- tion of uðxÞ of degree k À 1 at most as follows: 1 e nomial P on the stencil S2ðiÞ using Newton forms as (1) Using fuig compute the divided (or undivided P1ðxÞ¼U½x þU½x ; x differences for uniform mesh) of the primitive iÀ1=2 iÀ1=2 iþ1=2 function UðxÞ for degrees 1 to k with Eqs. (4.18) Âðx À xiÀ1=2Þ. ð4:23Þ and (4.20). ARTICLE IN PRESS

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(2) Start with the two-point stencil for the primi- Stencil S e 1 tive S2ðiÞ¼fxiÀ1=2; xiþ1=2g which is equivalent u ≡ u u + to the one-point stencil S1ðiÞ¼fIig for the cell ui−2 ui−1 i i ui+1 i 2 average. ui

(3) Add one of the two neighboring points to the stencil ˆˆ uˆ + ˆ e ui−3/2 ui− 1/2 i 1/2 ui+3/2 Sl ðiÞ, l ¼ 2; ...; k following the ENO procedure of Eqs. (4.25)–(4.27). Stencil S (4) Use the Lagrange form, Eq. (A.1) of Section 4.2, or 0

Newton form Eq. (4.22) to obtain the polynomial Stencil S2 piðxÞ of degree k À 1 the most that satisfies the accuracy requirement. Fig. 46. Candidate stencils for ð2k À 1Þth order accurate WENO; WENO5 for k ¼ 3. The nodal values ui, i À roioi À In practice, however, once the stencil is known, it is r þ k À 1 are identified as cell averages for the computation of more convenient to find the approximation at the cell the WENO reconstruction of the numerical fluxes uiÀ1=2 at half- point nodes. boundaries using Eq. (4.5)

XkÀ1 uiþ1=2 ¼ crj uiÀrþj, j¼0 of divided differences. However, ENO reconstruction in where the values of the constant crj are given by practice may face the following problems: Eqs. (A.2) and (A.3), in Section 4.2–4.4 for non-uniform and uniform mesh, respectively. (1) For finite-volume reconstructions, round-off error 4.6. 1D finite-difference ENO scheme perturbations may change the stencil because they can result in sign change of divided differences even The finite-difference ENO reconstruction is valid in the smooth regions of the solution [159,160]. This only for fixed mesh size and includes the following stencil-free adaption due to round-off error results steps: into a non-smooth numerical flux in finite-difference ENO. (1) Compute the numerical flux ubiþ1=2 using the given (2) The reconstruction in FV or the numerical flux point values fujg (see Fig. 45) and all k points fixed evaluation in FD ENO of kth order accuracy uses stencils, where r þ s ¼ k À 1, by only one of the k candidate stencils that cover 2k À 1

XkÀ1 cells. However, if all candidate stencils were used, then ð2k À 1Þth order accuracy could be achieved. ubiþ1=2 ¼ crjuiÀrþj. j¼0

Note that in the finite-difference ENO approxima- It was proposed in [160,161] to remedy the free tion the given point values fuig are identified as cell adaption problem using a biasing strategy for the stencil averages of another function hðxÞ, which has a selection process. However, the most recent improve- primitive HðxÞ exactly known at the cell interfaces ment of ENO is the WENO (weighted ENO) approx- xiþ1=2, and the kth order approximation hðxiþ1=2Þ is imation [33,162]. The basic idea of WENO is to use a b the numerical flux uiþ1=2. convex combination of all candidate stencils to form the (2) Perform steps 1–4 of the finite-volume ENO reconstruction. The WENO approximation is explained reconstruction treating fuig as cell averages to select more precisely in this section. the smoothest stencil. Suppose that the k candidate stencils SrðiÞ¼ (3) Obtain a conservative kth order accurate approx- fxiÀr; ...; xiÀrþkÀ1g (see Fig. 46 for k ¼ 3) that produce imation of the derivative as k different reconstructions for the value uiþ1=2 at the cell 1 0 k ðub À ub Þ¼u ðx ÞþOðDx Þ. interface or the numerical flux ubiþ are available. In x iþ1=2 iþ1=2 i 1=2 D both cases the reconstruction is obtained by

XkÀ1 4.7. WENO approximation ðrÞ uiþ1=2 ¼ crjuiÀrþj; r ¼ 0; ...; k À 1, (4.28) j¼0 In the previous sections, it was shown that the ENO reconstruction is uniformly high-order accurate right up where the superscript r denotes the shift to the left. to the discontinuity, and achieves this by adaptively Reconstruction with WENO considers a convex ðrÞ choosing the smoothest stencil using the absolute values combination of all uiþ1=2 as a newapproximation at ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 233 the cell boundary as follows: replaced by the numerical flux function ubiþ1=2 at half- point nodes. XkÀ1 ðrÞ Implementation of WENO schemes is more conve- uiþ1=2 ¼ or uiþ1=2, (4.29) r¼0 nient in practice with the formulation of Jiang and Wu [163]. For example, for k ¼ 3, fifth-order scheme, the XkÀ1 numerical flux ubiþ1=2 is taken as the weighted average of or ¼ 1; orX0 (4.30) the numerical fluxes in the three substencils S0, S1, and r¼0 S2 of Fig. 46. The third-order accurate approximations when the function uðxÞ includes a discontinuity in one or bs uiþ1=2, s ¼ 0; 1; 2 are more of the stencils SrðiÞ then the corresponding weight b0 1 7 11 or must be essentially zero in order to followas closely uiþ1=2 ¼ 3 uiÀ2 À 6 uiÀ1 þ 6 ui, as possible the successful ENO idea. Furthermore, the 1 1 5 1 ub ¼À uiÀ1 þ ui þ uiþ1, weights should be smooth functions of the cell averages iþ1=2 6 6 3 b2 1 5 1 (FV) or the point values (FD) approximation. All these uiþ1=2 ¼ 3 ui þ 6 uiþ1 À 6 uiþ2, ð4:36Þ considerations [33] lead to the following forms of u u i the weights: where i i denotes the point value at the nodes for the FD WENO formulation. a ¼ P r ¼ À The fifth-order accurate WENO approximation of the or kÀ1 ; r 0; ...; k 1, (4.31) S¼0 aS numerical flux is b b0 b1 b2 dr uiþ1=2 ¼ o0uiþ1=2 þ o1uiþ1=2 þ o2uiþ1=2, (4.37) ar ¼ 2 (4.32) ðe þ brÞ where the ð2k À 1Þ¼fifth-order approximation to ubiþ1=2 À6 where e40 taken as e ¼ 10 in [33] to avoid division by is based on the five-point stencil i À 2pkpi þ 2. zero and b are the smoothness indicators of the stencil 1 6 3 r For o0 ¼ 10, o1 ¼ 10, o2 ¼ 10, which are referred to as SrðiÞ. The smoothness indicators in [33] are obtained by optimal weights in Balsara and Shu [34], the approxima- minimizing the total variation of the ðk À 1Þth degree tion of Eq. (4.37) becomes reconstruction polynomial pðxÞ constructed on each b 1 13 47 9 1 SrðiÞ, which if evaluated at xiþ1=2 yields the kth order uiþ1=2 ¼ 30 uiÀ2 À 60 uiÀ1 þ 60 ui þ 20 uiþ1 À 20 uiþ2 (4.38) approximation of uðx Þ. iþ1=2 or Then the smoothness indicators br are defined by Z b 1 7 7 1 XkÀ1 m 2 uiþ1=2 ¼À uiÀ1 þ ui þ uiþ1 À uiþ1 xiþ1=2 q p ðxÞ 12 12 12 12 2mÀ1 r 1 br ¼ Dx m dx. (4.33) À ðÀu þ 4u À 6u þ 4u À u Þ. q x 30 iÀ2 iÀ1 i iþ1 iþ2 m¼1 xiÀ1=2 The smoothness indicators for k ¼ 2 and 3 obtained ð4:39Þ from Eq. (4.33) are The last form of Eq. (4.39) shows that the WENO Smoothness indicators for k ¼ 2, third-order WENO approximation of the numerical flux ubiþ1=2 is a sum of a reconstruction centered flux 2 b0 ¼ðuiþ1 À uiÞ , bc 1 uiþ1=2 ¼ 12ðÀuiÀ1 þ 7ui þ 7uiþ1 À uiþ2Þ 2 b1 ¼ðui À uiÀ1Þ . ð4:34Þ plus a dissipative portion of the WENO scheme. Smoothness indicators for k ¼ 3, fifth order WENO Replacing o1 by o1 ¼ 1 À o0 À o2 and using reconstruction Eqs. (4.36) in Eq. (4.37) obtain 13 2 1 2 b0 ¼ 12ðui À 2uiþ1 þ uiþ2Þ þ 4ð3ui À 4uiþ1 þ uiþ2Þ , b 1 uiþ1=2 ¼ 12ðÀuiÀ1 þ 7ui þ 7uiþ1 À uiþ2Þ 1 13 2 1 2 þ 3ðuiÀ2 À 3uiÀ1 þ 3ui À uiþ1Þo0 b1 ¼ ðuiÀ1 À 2ui þ uiþ1Þ þ ðuiÀ1 À 4uiþ1Þ , 12 4 1 1 þ 6ðuiÀ1 À 3ui þ 3uiþ1 À uiþ2Þðo2 À 2Þ 13 2 1 2 1 b3 ¼ 12ðuiÀ2 À 2ui þ uiÞ þ 4ðuiÀ2 À 4uiÀ1 þ 3uiÞ , ¼ 12 ðÀuiÀ1 þ 7ui þ 7uiþ1 À uiþ2Þ (4.35) À fw ðDuiÀ3=2; DuiÀ1=2; Duiþ1=2; Duiþ3=2Þ 1 where for k ¼ 2 obtain ð2k À 1Þ¼third-order accuracy ¼ 12ðÀuiÀ1 þ 7ui þ 7uiþ1 À uiþ2Þ and for k ¼ 3 obtain ð2k À 1Þ¼ fifth-order accuracy. 1 À 3 o0ðD0 À 2D1 þ D2Þ The WENO idea presented for FV reconstruction carries 1 1 þ ðo2 À ÞðD1 À 2D2 þ D3Þ, ð4:40Þ over to the finite-difference context once the cell 6 2 averages ui are replaced by nodal values ui and the where DuiÀ3=2 ¼ D0 ¼ uiÀ2 À uiÀ1; D1 ¼ uiÀ1 À ui; approximation at the cell boundaries of Eq. (4.29) are D2 ¼ ui À uiþ1; D3 ¼ uiþ1 À uiþ2. ARTICLE IN PRESS

234 J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300

The weights are defined as 4.8.1. Finite-volume reconstruction for Cartesian mesh Multidimensional reconstruction and approximation a0 a2 o0 ¼ ; o2 ¼ (4.41) without loss of generality is considered in two dimen- a0 þ a1 þ a2 a0 þ a1 þ a2 sions. The ideas described carry over to three dimension k ar ¼ dr=ðSI r þ eÞ; r ¼ 0; 1; 2. as well. The two-dimensional ENO reconstruction Summarizing: problem is as follows. Given the cell Iij and the order For k ¼ 3 the centered stencil of the ð2k À 1Þ, fifth- of accuracy k, choose a stencil Sði; jÞ based on kðk þ order scheme is 1Þ=2 neighboring cells, and find a polynomial pðx; yÞ of c 1 4 degree k À 1 at most whose cell average in each of the ub ¼ ðÀuiÀ1 þ 7ui þ 7uiþ1 À uiþ2ÞþOðDx Þ (4.42) iþ1=2 12 cells in the stencil Sði; jÞ agrees with uðx; yÞ the optimal weights are d ¼ 1 ; d ¼ 6 ; d ¼ 3 . Z Z 0 10 1 10 2 10 1 yjþ1=2 xiþ1=2 For the smooth regions the numerical flux is evaluated uij ¼ pðx; ZÞ dx dZ, DxiDy using optimal weights by j yjÀ1=2 xiÀ1=2 I 2 Sði; jÞ ð : Þ b 1 13 47 ij . 4 46 uiþ1=2 ¼ 30 uiÀ2 À 60 uiÀ1 þ 60 ui 9 1 5 The multidimensional ENO reconstruction is more þ u þ À u þ þ OðDx Þ. ð4:43Þ 20 i 1 20 i 2 complex than the one-dimensional case because there The dissipative WENO part is are many more candidate stencils Sði; jÞ and has the following difficulties: Some candidate stencils f3 ¼ 1 o ðD À 2D þ D Þ o 3 0 0 1 2 cannot be used to obtain the polynomial pðx; yÞ that 1 1 þ 6ðo2 À 2ÞðD1 À 2D2 þ D3Þ. ð4:44Þ satisfies Eq. (4.46). Some of the polynomials do not 3 3 satisfy the accuracy condition pijðx; yÞ¼uðx; yÞþ The smoothness indicators br ¼ SI r , r ¼ 0; 1; 2 are k given by O ðD Þ; ðx; yÞ2I ij; i ¼ 1; ...; Nx; j ¼ 1; ...; Ny. These difficulties are more profound for unstructured 3 2 SI 0 ¼ D0ð4D0 À 11D1Þþ10D1, meshes [164]. For rectangular meshes [165] the tensor 3 2 products of one-dimensional polynomials are used and SI ¼ D1ð4D1 À 5D2Þþ4D , 1 2 ð Þ 3 2 p x; y is written as SI 2 ¼ D2ð10D2 À 11D3Þþ4D3. ð4:45Þ XkÀ1 XkÀ1 n m Using this notationP the high-order WENO recon- pðx; yÞ¼ amnx y . (4.47) b iþkÀ1 k structions uiþ1=2 ¼ r¼iÀkþ2 cr ur of Balsara and Shu [34] m¼0 n¼0 for k ¼ 4 and 5, which yield the seventh- and ninth- Furthermore, by restricting the search in the following order accurate WENO schemes can be obtained. tensor product stencils (see Fig. 47) The component-wise application of the finite-volume ENO and WENO schemes is straightforward. Further- Srsði; jÞ¼fImn : i À rpmpi þ k À 1 À r, À1 more, the characteristic variables wi ¼ R ui for the j À spnpj þ k À 1 À sg divided differences (or undivided differences for equal grid spacing) for all cell averages can be used for ENO and WENO approximation. The local characteristic field is then used to perform scalar ENO or WENO reconstruction for each component of the characteristic Æ variables w to obtain the reconstruction wiþ1=2. Trans- Æ Æ forming back into physical space uiþ1=2 ¼ Rwiþ1=2 apply on exact or approximate Riemann solver to compute the b flux f iþ1=2. The finite-difference ENO or WENO for uniform meshes follows the same steps considering the point values ui as cell averages. Using then ui compute the undivided differences for the fluxes f ðuiÞ.

4.8. Multidimensional ENO and WENO reconstruction

The multidimensional ENO and WENO schemes are based on the preliminaries of the previous sections. For fully unstructured meshes, however, the identiﬁcation of suitable stencils is not straightforward. Presentation of multidimensional algorithms starts from the ENO in Cartesian meshes. Fig. 47. Possible reconstruction stencils in a Cartesian mesh. ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 235

second-order linear reconstruction, k ¼ 1 the stencil formed by T i plus two immediate neighbors is admissible for most triangulations. For third-order recon- ia struction k ¼ 3, (quadratic polynomial k À 1 ¼ 2is ja needed), m ¼ 6, some of the stencils consisting of T i and five of its neighbors may not be admissible. For i P1 fourth-order reconstruction k ¼ 4, (cubic polynomial), j m ¼ 10, the stencil consists of T i plus nine of the ib P2 immediate neighbors shown in Fig. 48. The most robust way for third- and fourth-order reconstruction is the jb least-squares reconstruction procedure suggested by Barth and Frederickson [131] where the polynomial p is determined by requiring that p has the same cell k kb ka average as u on T 0 and also has the same cell average as u on the collection of the neighboring triangles but only in a least-squares sense. The reconstruction problem Fig. 48. Stencil definition for reconstruction in a triangular becomes extremely time consuming for high-order mesh. reconstructions in three dimensions. For completeness, the k-exact reconstruction of Barth and Frederickson [131] that is used in ENO and WENO the reconstruction proceeds as in one dimension. The finite-volume sachems is briefly summarized. The basic computing cost for two-dimensional reconstruction is idea for arbitrary order (k-exact) reconstruction in two high because for each point the reconstruction cost is dimensions is to determine the following reconstruction double. Casper and Atkins [165] give details about polynomials for each subdomain Oi: possible reconstruction stencils (see Fig. 47). X k u ðx; yÞi ¼ aðm;nÞPðm;nÞðx À xc; y À ycÞ, (4.50) mþnpk 4.8.2. Two-dimensional reconstruction for triangles m n The reconstruction problem for triangular meshes is: where Pðm;nÞðx À xc; y À ycÞ¼ðx À xcÞ ðy À ycÞ and ðx ; y Þ is the centroid of the volume. The k-exact Given the cell averages ui of uðx; yÞ on a triangulation c c reconstruction strategy optimizes efficiency by precom- T i Z puting the weights W i in each cell Oi using a neighbor 1 u ¼ uðx; ZÞ dx dZ, (4.48) set Ni as follows i X jT ij Ti aðm;nÞ ¼ W ðm;nÞui, (4.51) j j where T i is the area of the triangle T i find a polynomial i2Ni piðx; yÞ of degree k À 1 at most for each triangle T i such that the kth order accurate approximation of uðx; yÞ where aðm;nÞ are the polynomial coefficients in Eq. (4.50). For k-exact reconstruction in two dimensions, the set N inside the triangles T i is i of the control volume neighbors must contain at least k piðx; yÞ¼uðx; yÞþOðD Þ; ðx; yÞ2T i i ¼ 1; ...; N, ðk þ 1Þðk þ 2Þ=2 members. The reconstruction is com- (4.49) plemented by a monotonicity enforcing procedure as described in [130]. where D denotes a typical length of the triangle for Friedrich [166] proposed to use a weighted sum of all example the longest edge. For finite-volume schemes, the reconstruction polynomials p ; ...; p as follows: approximation of uðx; yÞ at the triangle boundaries (see 1 m Xm points P1, P2 in Fig. 48) is needed to apply quadratures as in Eq. (4.49) yields the approximation of uðx; yÞ at p ¼ oipi, (4.52) these points. i¼1 Similar to the finite volume for Cartesian mesh, given where the weights of pi are chosen such that oi is lowif the triangle T i and the order of accuracy k, again choose the oscillation of pi is high. As computational cells in a stencil based on m ¼ kðk þ 1Þ=2 neighboring triangles, Friedrich formulation [166] were considered the dual which form the stencil SðiÞ. For SðiÞ, find a polynomial cells resulting from the lines joining the barycenters pðx; yÞ of degree k À 1 the most whose cell average in with the midpoints of the edges. The cell averages are each of the triangles in the stencil SðiÞ agrees with the defined as average of uðx; yÞ given by Eq. (4.48). This condition Z 1 yields on m Â m linear system, and if this system has a u ¼ uðxÞ dx, (4.53) k jO j unique solution then the stencil SðiÞ is admissible. For k Ok ARTICLE IN PRESS

236 J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300

bW where Ok are polygonally bounded by a finite number of (3) Project the numerical ðf LFÞ fluxes to the physical m line segments. Further details on the full reconstruction space by multiplying with the right eigenvectors ri algorithm the selection of admissible stencils and the to obtain the numerical fluxes fb in the conservative choice of smoothness indicators and the WENO recon- variable space struction weights oi in Eq. (4.49) can be found in [166]. Xm b m n bW n ðf Þiþ1=2 ¼ riþ1=2ðf LFÞiþ1=2. 4.8.3. Multidimensional finite-difference ENO n¼1 For finite-difference methods for hyperbolic conserva- (4) Compute the kth order accurate conservative tion laws, the problem is to obtain high order, approximation of the derivative f as conservative approximation of the derivative from point x b b values [32,167]. Similar to the one-dimensional case, in f iþ1=2 À f iÀ1=2 f ¼ þ OðDxkÞ. multidimensions the uniform mesh assumption is x Dx essential. For finite-difference methods in two dimensions the statement of the problem is: Repeat steps 1–4 for the other directions to obtain the flux derivatives g and h . Given the point uij values of uðx; yÞ on a uniform mesh y z i ¼ 1; 2; ...; N ; x 4.9. Optimization of WENO schemes uij uðxi; yjÞ (4.54) j ¼ 1; 2; ...; Ny;

find the numerical flux functions WENO schemes have been successfully applied to problems with shocks and complex smooth flow features ubiþ1=2; j ¼ ubðuiÀr;j;...; uiþkÀ1Àr;jÞ i ¼ 0; 1; ...; Nx, [168]. Direct application of WENO schemes to wave propagation problems, such as computational aeroa- ui; jþ1=2 ¼ ubðui; jÀs;...; ui; jþkÀ1ÀsÞ j ¼ 0; 1; ...; Ny ð4:55Þ coustics (CAA) and computational electromagnetics that obtain a kth order conservative approximation of (CEM), where resolution of short waves is important the derivative by Eq. (4.14). Therefore, the conservative is not optimal because WENO schemes are designed for approximation of the derivative from point values is high resolution of discontinuities and to achieve the accomplished in multidimensions as in the one- formal order of accuracy of the reconstruction. Efficient dimensional case, e.g. considering wðxÞ¼uðx; yjÞ and accurate resolution of short waves is achieved in 0 obtain uxðxi; yjÞ¼w ðxiÞ and vðyÞ¼uðxi; yÞ obtain CAA with the optimized schemes where the coefficients 0 vyðxi; yjÞ¼v ðyjÞ. of the scheme are altered to minimize a particular type An example of multidimensional ENO or WENO of error instead of the truncation error. These optimized scheme is the characteristic-wise ENO or WENO schemes [100,111,112,132,169,170] have been used very implementation for the Euler equations in the finite- successfully for better resolution of short waves in difference context on uniform meshes. The Lax– broadband acoustic wave propagation. Friedrich’s or Roe’s approximate Riemann solver can Recently, Wang and Chen [171] using ideas of CAA- be used to split the fluxes. For example, using the optimized schemes proposed modification of the WENO Lax–Friedrich’s flux as building block obtain smoothness measures and developed an optimized 1 WENO (OWENO) scheme. Following the practice of f LFðx; yÞ¼2½f ðxÞþf ðyÞÀlðy À xÞ, 0 the dispersion relation Preserving (DRP) scheme [86] l ¼ max jf ðuÞj. presented in Section 3.4.2, Wang and Chen [171] Perform the following steps in each of the ði; j; kÞ or achieved high resolution for short waves with OWENO. ðx; y; zÞ directions The OWENO schemes of Wang and Chen [171] optimizes all candidate stencils and finds the best (1) Project the positive and negative part of the flux in weights to combine them. The approach of Wang and the i direction, to the characteristic field in the i Chen [171] was a significant improvement over previous direction by multiplying with the left eigenvectors attempts to optimize WENO schemes [172], where only m the weights of the WENO schemes were optimized. li ðm ¼ 1 À 4 for 2D or m ¼ 1 À 5 for 3DÞ to obtain f W as follows: The development of the OWENO scheme is based on LF the DRP idea [86]. The OWENO scheme instead of Xm achieving the maximum order of accuracy k, compro- ðf W Þm ¼ ðf Þn ln . LF iþ1=2 LF iþ1=2 iþ1=2 mises the accuracy requirement by setting p k in the n¼1 1o conservative approximation of the derivative ðubr À ubr Þ (2) Perform kth order ENO or WENO reconstruction to qui iþ1=2 iÀ1=2 p W ¼ þ OðDx 1 Þ; r þ s þ 1 ¼ k b qx Dx obtain the numerical fluxes f LF in the characteristic field. (4.56) ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 237 and minimizes the difference between the numerical The newsmoothness indicators of the k ¼ 4, wavenumber and the actual wave number. The numer- OWENO scheme that satisfy "# ical wavenumber for the scalar wave equation u þ aux Z 2 XkÀ1 m ¼ 0 where the spatial derivative is evaluated as in Eq. xiþ1=2 q p ðxÞ b ¼ DxmÀ1 r , (4.56) becomes r qmxdx m¼2 k42 xiÀ1=2 r ¼ 0; ...; k À 1 ð4:62Þ Xs r Ài a ¼ cr;jþr expðimaDxÞ½1 À expðÀiaDxÞ instead of Eq. (4.33) are given by Dx m¼Àr 2 p ¼ð þ þ À Þ ¼ a þ O ðaDxÞ 1 . ð4:57Þ b0 2ui 5uiþ1 4uiþ2 uiþ3 2 Àðui þ 3uiþ1 À 3uiþ2 þ uiþ3Þ , Optimized coefficient crjPin the approximation of the kÀ1 2 b b ¼ðu À À 2u þ u þ Þ numerical flux uiþ1=2 ¼ j¼0 crjuiÀrþj are obtained by 1 i 1 i i 1 2 minimizing the L2 norm of the difference between the þðÀuiÀ1 þ 3ui À 3uiþ1 þ uiþ2Þ , numerical wavenumber of Eq. (4.57) and the actual wavenumber for a particular range ½ÀaDx; a0Dx. 2 b2 ¼ðuiÀ1 À 2ui þ uiþ1Þ These coefficients c are obtained by minimizing the rj þðÀ þ À þ Þ2 integral uiÀ2 3uiÀ1 3ui uiþ1 ,

2 Z b3 ¼ðÀuiÀ3 þ 4uiÀ2 À 5uiÀ1 þ 2uiÞ a0Dx r 2 2 Er ¼ l½Reða DxÞÀaDx þðÀuiÀ3 þ 3uiÀ2 À 3uiÀ1 þ uiÞ . ð4:63Þ Àa0Dx þð1 À lÞ½InðarDxÞ2dðaDxÞ. ð4:58Þ Taylor expansion of b0; b1; b2; b3 yields 00 2 000 3 2 6 br ¼ðui Dx Þþðui Dx Þ þ qðDx Þ, The OWENO scheme is then obtained in two steps: r ¼ 0; ...; 3 ð4:64Þ 00 000 therefore in the case ui ¼ ui ¼ 0 the OWENO scheme for k ¼ 4 does not have the formal accuracy of standard (1) For p1th order of accuracy with Eq. (4.56) the WENO scheme ð2k À 1Þ¼7 but achieves better resolu- following p1 linear equations must be satisfied tion of high wavenumbers. XkÀ1 bmjcrj ¼ zm; m ¼ 1; ...; p1, (4.59) 4.10. Compact WENO approximation j¼0 It was pointed out in Section 3 that two advantages of where bmj and zm are constants the rest k À p1 are compact schemes compared to non-compact (explicit) obtained from the minimization of Er in Eq. (4.58) counterparts are: (i) they yield better accuracy and wave- qEr space resolution for the same number of points in the ¼ 0; j ¼ p1; ...; k À 1. (4.60) qcrj stencil, and (ii) they offer an advantage in the implementation of boundary conditions because fewer boundary (2) Perform a convex combination of the k candidate points must be handled. These advantages are achieved at a moderate increase in computing cost resulting from the p1th order accurate stencils to obtain OWENO as matrix inversion. Recently, Pirozzoli [173], exploiting the XkÀ1 bOWENO br advantages of compact schemes and developed a narrow- uiþ1=2 ¼ hruiþ1=2 r¼o stencil, fifth-order accurate WENO compact scheme P kÀ1 referred to from nowon as CWENO5 scheme. so that hr ¼ 1, hrX0, p pk þ 1 r¼0 2 StartingwiththesameformulationasinSection4.3(see 1 qu Eq. (4.9)–(4.14)) considered the following compact repre- ðubOWENO À ubOWENOÞ¼ þ ðDxp1þp2 Þ. iþ1=2 iþ1=2 O sentation for the reconstruction of the numerical flux ub Dx qx i iþ1=2

(4.61) XL2 XM2 Al ubiþ1=2þl ¼ amuiþm. (4.65) l¼ÀL1 m¼M1 The k À 1 À p2 free weight parameters of Eq. (4.61) are p2pk À 1 then determined by minimizing again the The Taylor series expansion of u up to order K around r integralP of Eq. (4.58) but with a replaced by xiþ1=2 is kÀ1 r a ¼ r¼0 hra . KXÀ1 n n q u ðx À xiþ1=2Þ K The coefﬁcients of several high-order accurate uðxÞ¼ þ Oðh Þ (4.66) qxn n! OWENO schemes can be found in [171]. n¼0 iþ1=2 ARTICLE IN PRESS

238 J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 and recalling that the points values at the nodes, ui, which are considered as cell averages for the ﬁnite- difference ENO reconstruction, e.g. ui ui, satisfy

U þ þ À U þ À u ¼ i m 1=2 i m 1=2 , (4.67) iþm Dx where UðxÞ is the primitive function of uðxÞ satisfying

KXÀ1 n nþ1 q u ðx À xiþ1=2Þ Kþ1 UðxÞ¼ þ OðDx Þ (4.68) xn ðn þ Þ n¼0 q iþ1=2 1 ! therefore

KXÀ1 n q u 1 nþ1 nþ1 n uiþm ¼ ½m Àðm À 1Þ Dx xn ðn þ Þ n¼0 q iþ1=2 1 ! þ OðDxKþ1Þ. ð4:69Þ Fig. 49. Dispersion of compact WENO schemes; seventh-order Inserting the Taylor series expansions for ubiþ1=2 and uiþm WENO —, fifth-order compact WENO –Á–Á–Á, fifth-order from Eqs. (4.66) and (4.69), respectively, in Eq. (4.65) and WENO 2222 sixth-order centered —Á—ÁÁÁ, fourth-order matching the coefficients of like powers of h requiring kth centered – – –, exact ÁÁÁÁÁÁ. order accurate approximation (with kpK)obtain XL2 XM2 0 n nþ1 nþ1 ðn þ 1Þ All À am½m Àðm À 1Þ ¼0 l¼ÀL1 m¼ÀM1 -0.5 n ¼ 0; ...; k À 1. ð4:70Þ -1 Solving the system of Eq. (4.70) for L1 þ L2 þ 1, Al )

unknowns and M þ M þ 1, a unknowns obtain the Z 1 2 m -1.5

coefficients of the compact upwind scheme. The only Im( fifth-order accurate compact scheme ðk ¼ 5Þ for -2 L1 ¼ L2 ¼ M1 ¼ M2 ¼ 1, which involves tridiagonal matrix inversion is -2.5 b b b 1 19 10 3uiÀ1=2 þ 6uiþ1=2 þ uiþ3=2 ¼ 3 uiÀ1 þ 3 ui þ 3 uiþ1. -3 (4.71) 0 0.5123 1.5 2.5 ^ The numerical fluxes uiþ1=2 obtained from the point k values ui ui through the solution of the system of Eq. (4.71) yield a fifth-order accurate, conservative Fig. 50. Dissipation of compact WENO schemes: fifth-order compact WENO ____, sixth-order centered 2222, fourth- evaluation of the derivative as order centered ÁÁÁÁÁÁ. 1 0 k ðubiþ1=2 À ubiÀ1=2Þ¼u ðxiÞþOðDx Þ. (4.72) Dx a40 was considered for the evaluation of the resolving Furthermore, the following explicit WENO approxi- ability. The dispersion properties of the newschemes are mations were obtained fifth-order upwind-biased compared with the classical, symmetric compact approximation: schemes in Fig. 49. The upwind-biased approximations b 1 of Eqs. (4.68), (4.73), and (4.74) introduce in addition uiþ1=2 ¼ 60 ð2uiÀ2 À 13uiÀ1 þ 47ui þ 27uiþ1 À 3uiÀ2Þ. dissipation errors. The dissipation error is shown in (4.73) Fig. 50. It can be seen that the compact WENO scheme Seventh-order upwind-biased approximation: of Eq. (4.71) has very small dissipation error, smaller ub ¼ 1 ðÀ3u þ 25u À 101u than the fifth- and seventh-order explicit schemes of iþ1=2 420 iÀ3 iÀ2 iÀ1 b Eqs. (4.73) and (4.74), for the range 0okop=2. þ 319u þ 214u À 38u þ 4u Þ. ð4:74Þ i iþ1 iþ2 iþ3 For non-periodic domains, the following fourth-order The approximations of Eqs. (4.73) and (4.74) can be accurate boundary closures were proposed for the used as fifth- or seventh-order WENO schemes inn implementation of Eq. (4.71). smooth flowregions. ub ¼ 1 ð3u þ 13u À 5u þ u Þ, The resolution properties of the WENO approxima- 1=2 12 0 1 2 3 tions of Eqs. (4.71), (4.73), and (4.74) were demonstrated 1 ub ¼ ð25uN À 23uNÀ þ 13uNÀ À 3uNÀ Þ. (4.75) in [173]. The linear advection equation ux þ aux ¼ 0, Nþ1=2 12 1 2 3 ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 239

A Roe-type, characteristic-wise, finite-difference, imple- For siþ1=2 ¼ 1, Eq. (4.80) reduces to the compact mentation of the compact CWENO5 scheme developed scheme of Eq. (4.79) and for siþ1=2 ¼ 0 becomes a by Pirozzoli [173] was recently presented by Ren et al. WENO scheme. It is necessary, therefore, that the [174]. The formulation of Ren et al. [174] closely follow weight be directly related to the smoothness of the hybrid compact-ENO ideas introduced by Adams the numerical solution. This is achieved through the and Shariff [175]. The implementation proposed in [174] definition of a smoothness indicator riþ1=2 defined by for the scalar hyperbolic conservation law ut þ f xðuÞ¼0 Ren et al. [174] by is as follows. b riþ1=2 ¼ minðrj; rjþ1Þ, The numerical flux function f iþ1=2 for the evaluation 2jDf Df jþe of the conservative approximation of the derivative f x is iþ1=2 iÀ1=2 rj ¼ 2 2 , obtained by ðDf iþ1=2Þ þðDf iÀ1=2Þ þ e b b b Df ¼ f À f , 3f iÀ1=2 þ 6f iþ1=2 þ f iþ3=2 iþ1=2 iþ1 i 1 e X 0:9rc 2 ¼ 3 ðf iÀ1 þ 19f i þ 10f iþ1Þ if aiþ1=2 0, ð4:76Þ e ¼ x ; x40; rc ’ 1. ð4:81Þ 1 À 0:9rc b b b f iÀ1=2 þ 6f iþ1=2 þ 3f iþ3=2 Note that Pirozzoli simply defined the smoothness 1 e indicator as riþ1=2 ¼jf jþ1 À f jj and determined siþ1=2 by ¼ 3 ð10f i þ 19f iþ1 þ f iþ2Þ if aiþ1=2o0, ð4:77Þ 8 > 1ifr er and r er where ae is the numerical wave speed defined by < jÀ1=2p c jþ1=2p c iþ1=2 e 8 sjþ1=2 ¼ and rjþ3=2prc; (4.82) > b b :> > f iþ1 À f i 0 otherwise; <> if uiþ1 À uia0; u þ À u ae ¼ i 1 i (4.78) iþ1=2 > qf where rc is a problem-dependent threshold. Using the :> otherwise: definition of Ren et al. [174] for r the weight s is qu iþ1=2 iþ1=2 i computed by Denoting Siþ1=2 ¼ signðaeiþ1=2Þ Eqs. (4.76) and (92) can riþ1=2 be combined to siþ1=2 ¼ min 1; . (4.83) rc b b b Aiþ1=2 f iÀ1=2 þ f iþ1=2 þ Biþ1=2 f iþ3=2 ¼ biþ1=2, (4.79) 4.11. Hybrid compact-WENO scheme for the Euler where equations

1 Siþ1=2 1 Siþ1=2 A characteristic-wise approach is used in [174] with Aiþ1=2 ¼ þ ; Biþ1=2 ¼ þ , 3 6 3 6 the hybrid, compact WENO scheme for the numerical solution of the Euler equations. The numerical flux 1 þ Siþ1=2 1 19 5 biþ1=2 ¼ f iÀ1 þ f i þ f iþ1 evaluation is performed in the following steps: 2 1818 9 1 À Siþ1=2 5 19 1 þ f þ f þ f . (1) Compute an average state U iþ1=2 by the simple 2 9 i 18 iþ1 18 iþ2 mean U iþ1=2 ¼ðU i þ Uiþ1Þ=2 or the Roe average. The compact WENO scheme of Eq. (4.79) gives very (2) Compute the eigenvalues liþ1=2i þ 1=2 ¼ 1; 2; 3; 4 ðmÞ satisfactory results for smooth flowregions. However,at and the left eigenvectors ljþ1=2 at the average state. the discontinuities of the solution the Gibbs phenomen- (3) Perform local characteristic decomposition on will occur that contaminates the solution and m ¼ 1; 2; 3; 4; eventually leads to nonlinear instability. wðmÞ ¼ l F n iþ1=2 n n ¼ i À 1; ...; i þ 2: For these regions, a hybrid method is proposed in [174] where the compact scheme of Eq. (4.79) is coupled with the WENO procedure. The hybrid scheme is the (4) Define weighted average of the compact scheme in Eq. (4.79) ðmÞ ðmÞ and WENO of Section 4.7. siþ1=2 ¼ signðliþ1=2Þ, This hybrid scheme [174] has the following form: ðmÞ ðiÞ ðiÞ riþ1=2 ¼ minðrj ; riþ1Þ, b b ðmÞ ðmÞ siþ1=2Aiþ1=2f iÀ1=2 þ f iþ1=2 j2Dw Dw jþe rðmÞ ¼ iþ1=2 iÀ1=2 , b b i ðmÞ 2 ðmÞ 2 þ siÀ1=2Biþ1=2f iþ3=2 ¼ ciþ1=2, ð4:80Þ ðDw Þ þðDw Þ þ e iþ1 =2 !iÀ1=2 where s is the weight and rðmÞ ðmÞ iþ1=2 WENO siþ1=2 ¼ min 1; . ciþ1=2 ¼ siþ1=2biþ1=2 þð1 À siþ1=2Þf iþ1=2 . rc ARTICLE IN PRESS

240 J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300

(5) Apply the hybrid, compact-WENO scheme for the algorithms often degrade into first order for non-smooth local characteristic variables as grids. Analogous behavior should be expected for higher-order methods. The dependence on grid smooth- sðmÞ AðmÞ wðmÞ þ wðmÞ iþ1=2 iþ1=2 iÀ1=2 iþ1=2 ness of the numerical solutions obtained with ENO was ðmÞ ðmÞ ðmÞ ðmÞ investigated by Casper and Atkins [176]. þ siþ1=2Biþ1=2wiþ3=2 ¼ ciþ1=2, The two basic formulations, finite volume (FV) and þ sðmÞ À sðmÞ finite difference (FD), of ENO were considered by ðmÞ 2 iþ1=2 ðmÞ 2 iþ1=2 Aiþ1=2 ¼ ; Biþ1=2 ¼ , Casper and Atkins [176]. The FD and FV fourth-order 6 6 accurate algorithms were compared for accuracy, ðmÞ ðmÞ ðmÞ ðmÞWENO sensitivity to grid irregularities, wave resolution, and c ¼ s b þð1 À s Þw , iþ1=2 iþ1=2 iþ1=2 iþ1=2 iþ1=2 computational efficiency. It was found [176] (see Fig. 51) ! ðmÞ that for fourth-order design accuracy the two-dimen- 1 1 þ s bðmÞ ¼ iþ1=2 ðwðmÞ þ 19wðmÞ þ 10wðmÞ Þ sional FD ENO numerical solution is approximately iþ1=2 18 2 iÀ1 i iþ1 two times more efficient than the FV ENO numerical ! ðmÞ solution. The CPU time for three-dimensional imple- 1 1 À s ¼þ iþ1=2 mentation (see Fig. 52) shows more dramatic increase 18 2 for the FV formulation. For example, the CPU time ðmÞ ðmÞ ðmÞ required for FV ENO with fourth-order formal accuracy Âð10w þ 19w þ w Þ, ð4:84Þ i i iþ2 is approximately three times higher than the FD where wðmÞWENO is computed with the WENO ENO with equivalent formal accuracy. It was shown, scheme of Section 4.5. (6) Project Eq. (4.84) back to conservative variables and solve the following block-tridiagonal system of equations too obtain the numerical flux in the conservative variable space ½A F þ L F i1=2 iÀ1=2 iþ1=2 iþ1=2

þ½Biþ1=2Fiþ3=2 ¼ ciþ1=2, 2 3 sð1Þ Að1Þ lð1Þ 6 iþ1=2 iþ1=2 iþ1=2 7 ½A ¼ 4 5, iþ1=2 ð4Þ ð4Þ ð4Þ siþ1=2 Aiþ1=2 liþ1=2 2 3 ð1Þ sð1Þ Bð1Þ l 6 iþ1=2 iþ1=2 iþ1=2 7 ½Biþ1=2 ¼ 4 5, ð4Þ ð4Þ ð4Þ siþ1=2 Biþ1=2 liþ1=2 2 3 2 3 Fig. 51. Comparison of the required CPU time for FV and FD lð1Þ cð1Þ ENO in two dimensions. 6 iþ1=2 7 6 iþ1=2 7 6 7 6 7 6 ð2Þ 7 6 ð2Þ 7 6 l 7 6 c 7 6 iþ1=2 7 6 iþ1=2 7 Liþ1=2 ¼ 6 7; ciþ1=2 ¼ 6 7. 6 ð3Þ 7 6 ð3Þ 7 6 liþ1=2 7 6 ciþ1=2 7 4 5 4 5 ð4Þ ð4Þ liþ1=2 ciþ1=2 ð4:85Þ

4.12. Applications of ENO and WENO

It is well known in CFD that the solution error for a given problem strongly depends on the smoothness of the computational grid. Uniform or smoothly varying grids always yield smaller solution errors than non- uniform or non-smooth grids. For example, many Fig. 52. Comparison of the required CPU time for FV and FD second-order structured or unstructured grid CFD ENO in three dimensions. ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 241 however, in [176] that the formal accuracy of the FD for simple linear problems by Ekaterinaris [177]. The ENO can only be achieved with smooth grids. The finite- performance of centered and WENO schemes for volume implementation was found [176] less sensitive to aeroacoustics using the full Euler equations was also derivative discontinuities, whether in the computational considered. In addition, centered schemes with char- mesh or the solution. Therefore, for applications where acteristic based filters were compared with WENO the computational domain is known to be sufficiently schemes for problems with strong shocks in curvilinear smooth and can be suitably structured the FD ENO coordinates. The computed solutions were compared algorithm must be the method of choice. Taking, with available exact solutions. however, into account that the formal accuracy of the Long-time integration is important in many practical FD ENO can significantly degrade for non-smooth applications, such as aeroacoustics, LES and helicopter meshes [176] (see Fig. 53), for problems with complex rotor calculations where the tip vortex and the rotor geometries it may pay to use the more expensive FV wake need to propagate for long distances. Therefore algorithm. the ability of symmetric, centered compact and non- The resolving ability and the performance for long- compact schemes as well as several WENO high-order time integration of the WENO and other numerical accurate stencils was also evaluated in [177], for wave schemes presented in the previous sections was evaluated convection. Sufficiently accurate convection of simple

ENO-FD ENO-FV C3 Geometry C1 Geometry

10-3 Interior10-3 Interior

2.01

10-4 10-4

4.00 E ε 10-5 10-5

395 10-6 10-6

4.10

10-7 10-7 10-2 10-1 10-2 10-1 (a) Ax (b) Ax

-3 10-3 Wall10 Wall

10-4 10-4 2.01 3.23

-5 -5 E 10 E 10

10-6 10-6 2.00 3.54

10-7 10-2 10-1 10-2 10-1 (c)Ax (d) Ax

Fig. 53. L1 entropy errors, steady two-dimensional channel ﬂow. ARTICLE IN PRESS

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Table 3 L2 error at T ¼ 200 for the convection of sinðpx=6Þ with explicit schemes

Explicit Fourth-order Sixth-order Fifth-order Seventh-order schemes centered centered WENO WENO

L2 error 0.2E00 0:11E À 20:12E À 10:14E À 3

2 Àax 1 Gaussian pulses, uðx; t ¼ 0Þ¼e , with 12 points per 1 waveform (not shown here) was achieved even with 0.5 explicit fourth-order accurate in space methods. 0.8 Further accuracy test were shown for linear wave 0.6 0 convection by the one-dimensional wave equation 0.4 -0.5 ut þ cux ¼ 0. Numerical solutions with unit wavespeed, 198 199 200 201 202 c ¼ 1, were obtained for long propagation times. Time 0.2 marching was performed with the third-order accurate 0 Runge–Kutta method. A time step of Dt ¼ 0:1, which is belowthe stability limit of the method, wasused for all Amplitude,u(t) -0.2 Exact 4th ord symmetric tests in order to keep time integration errors at lowlevel. -0.4 6th ord symmetric 5th ord WENO The first test was propagation of a high-frequency 7th ord WENO -0.6 9th ord WENO sinusoidal wave u0ðxÞ¼sinðpx=6Þ. Convection of the sinusoidal wave was obtained with Dx ¼ 0:1 (12 points -0.8 160 180 200 220 240 per wavelength) and periodic boundary conditions. The X, Time mean square error obtained from explicit space discretizations with the symmetric fourth-, and sixth-order Fig. 54. Comparison of the exact solution u0ðxÞ¼ accurate schemes and with WENO schemes of fifth- and cosðajxjÞeÀbjxj at T ¼ 200 with results computed with explicit seventh-order accuracy, Eqs. (4.43), is shown in Table 3. schemes. It appears that only the schemes with formal accuracy more than five were capable to obtain sufficiently accurate solution for long-time integration. stencil. The computing cost of the eighth- and tenth- Further evaluation of high-order accurate symmetric order compact schemes that require pentadiagonal explicit and compact schemes and WENO stencils to matrix inversion was the highest and almost double perform linear wave convection was carried out using compared to the time required by the explicit schemes. Àbjxj the following modulated wave u0ðxÞ¼cosðajxjÞe However, use of very high-order centered methods may 3 1 with a ¼ 4 and b ¼ 10 as initial condition. For explicit be required for wave convection over long-time periods. schemes, time integration was performed until T ¼ 200. Symmetric schemes with spectral-type filtering and For compact schemes, which have increased resolving characteristic-based filters of Section 3, as well as ability, time integration was performed until the final WENO schemes off different order of accuracy were time T ¼ 500. A comparison of the solutions computed used to compute spread and reflection of a pressure using explicit high-order symmetric schemes and WENO disturbance. The full nonlinear Euler equations were stencils with the exact result is shown in Fig. 54. It can used for this test. At the far-field boundaries of the be seen that at least sixth order of accuracy is needed for domain a radiation boundary condition was used. On long-time propagation. The mean square error of the the solid surface the normal to the wall velocity solutions computed with different schemes is shown in component was set to zero while the density and Tables 4 and 5. It appears that the ninth-order accurate pressure were extrapolated from the interior assuming WENO stencil provides uniformly high order of that qr=qn ¼ qp=qn ¼ 0orqr=qy ¼ qp=qy ¼ 0. It was accuracy for smooth initial data. found that it was required to use high-order accurate The mean square error for long-time integration, approximations of the derivatives at the wall in order to T ¼ 500, of high-order compact schemes (see Table 5) retain the accuracy of the numerical solution. For also remains at lowlevels. It can be seen that compact example, the pressure was extrapolated using the schemes with formal order of accuracy more that four following one-seeded, fourth-order accurate approxima- perform adequately. The high-order accurate compact tion of the first derivative ðdp=dyÞ1 ¼ðÀ25p1 þ 48p2À schemes appear to be particularly suitable for linear 36p3 þ 16p4 À 3p5Þ=12. The computed results were aeroacoustic problems. The computing time of the compared with the exact solution which gives the explicit schemes was proportional to the width of the time variation of an initial pressure disturbance ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 243

Table 4 Àbjxj L2 error at T ¼ 200 for the convection of cosðajxjÞe with explicit schemes

Scheme Sixth-order Fifth-order Seventh-order Ninth-order centered WENO WENO WENO

L2 error 0:4E À 30:1E À 20:7E À 40:7E À 5

Table 5 Àbjxj L2 error at T ¼ 500 for the convection of cosðajxjÞe with compact schemes

Compact Fifth-order Sixth-order Eighth-order Tenth-order schemes WENO centered centered centered

L2 error 0:19E À 30:57E À 40:92E À 50:53E À 5

A grid-independent solution was obtained for the solution computed with the ninth-order accurate ðr ¼ 5Þ 1.004 WENO scheme since the error does not change for 1.003 computations performed with the baseline 100x50 point grid and reﬁned 200 Â 100 and 400 Â 200 point grids. 1.002 Therefore, the errors of the baseline grid are mainly due to the temporal integration scheme. The error of the 1.001 solutions obtained with different methods, along the

Pressure normal to the wall symmetry line, is shown in Figs. 56 1 and 57. Similarly to the WENO scheme that required Exact 0.999 9th order use of smoothness, characteristic-based filters were used 5th order for the computation of aeroacoustic pulse propagation refined 0.998 with the full nonlinear Euler equations. For the comparisons of Fig. 56, the same value of the ACM 0 1020304050 filter parameter ðk ¼ 0:1Þ was used. It can be seen that y the WENO schemes provide a comparable level of Fig. 55. Comparison of WENO scheme computations with the accuracy with the centered schemes. exact solution. The effect of the order of the ACM filter parameter on the accuracy of the computed solution is shown in 2 2 pðx; yÞ¼expfÀ ln 2½x Àðy À y0Þ g. The initial distur- Fig. 57. It can be seen that increase of k deteriorates the bance is located at y ¼ y0, and as it spreads, reflects accuracy of the solution. The comparisons of Fig. 57 from a solid wall at y ¼ 0. demonstrate that reduction of the ACM filter parameter A comparison of the solutions computed on an belowa certain level does not improve the solution. The artificially distorted mesh with fifth and ninth WENO solution obtained with the spectral-type filter has the schemes (r ¼ 3, and 5, respectively) is shown in Fig. 55. smallest error. Among the results compared in Fig. 57, It can be seen that the solution computed with the the solution computed with the spectral-type filter was ðr ¼ 5Þ ninth-order accurate WENO scheme on a the the most efficient. The results computed with the ACM baseline, 100 Â 50 point grid, which provides 12 points filter required approximately 20% more time compared per wave, is almost indistinguishable from the solution to the solution obtained with the spectral filter. The computed with the ðr ¼ 3Þ fifth-order accurate WENO solution obtained with the WENO scheme required scheme on a 200 Â 100 point grid, refined in both approximately 30% more time compared to the solution directions. It is important to note that the full WENO obtained with the spectral filter. Despite the longer scheme with the appropriate smoothness measures must computing time, both WENO schemes and the ACM be used for the propagation of the pressure disturbance filter provide shock capturing capability which is of with the nonlinear Euler equations. Numerical solutions interest in aeronautical applications. of the linearized Euler equations that describe the It can be concluded that both centered and WENO propagation of the acoustic-type pressure disturbance schemes of seventh order or higher are appropriate for may be possible only with the optimal WENO stencils of aeroacoustic computations of subsonic flows. For flows high-order WENO schemes. with shocks, however, WENO schemes appear to be ARTICLE IN PRESS

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5th o rder WENO baseline strated that the accuracy of the solution does not 5th o rder WENO refined deteriorate when the definition of the metrics is 7th o rder WENO baseline 4th o rder ACM, k=0.1 consistent and the metrics were computed with a high- 6th o rder ACM, k=0.1 order method. In previous comparisons, the third-order accurate 0.00015 Runge–Kutta of Ref. [81] was used, even though higher- order or optimized Runge–Kutta methods can further 0.0001 reduce temporal errors. Furthermore, it was found that sufficiently accurate computations of aeroacoustic phenomena can be obtained when implicit time marching is 5E-05 performed with the BW implicit algorithm with p ¼ 2 À 3 subiterations within each physical time step. The Error 0 error of the solutions computed with the explicit RK-3 method and the modified implicit Beam-Warming -5E-05 algorithm with n ¼ 2 is almost the same. The error of these computations is shown in Fig. 58. The performance of WENO and centered schemes -0.0001 with characteristic-based filters is evaluated for flows 0 1020304050with shocks. The oblique shock reflection problem at y Location M1 ¼ 2:9 is chosen as test case. The pressure at y ¼ 0:5 computed with WENO schemes of fifth, seventh, and Fig. 56. Comparison of the error obtained with WENO ninth-order accuracy is compared with the exact schemes and centered schemes with characteristic-based filters solution in Fig. 59. The computations were performed k ¼ 0:1. on a uniformly spaced 200 Â 50 point grid in a domain À2:0pxp2:0; 0pyp1:0. At the left inflowboundary, free stream was specified. At the right outflow boundary, 7th order WENO baseline 4th orde rACM, k=0.1 all quantities were extrapolated. On the solid wall 4th order ACM, k=0.2 y ¼ 4th order ACM, k=0.4 at 0, slip boundary condition was specified, and 6th order ACM, k=0.05 at the top the flowquantities werespecified as: 6th order ACM, k=0.2 6th order, compact filter r ¼ 1:69997; u ¼ 2:61934; v ¼À0:506; p ¼ 1:528. Suffi- cient number of ghost points, depending on the order 0.00015 of the scheme, were used at the edges of the domain in order to retain the formal order of the scheme. For 0.0001 example, computations with the seven-point wide WENO5 scheme required three ghost points while

5E-05 Error 0 0.0005 Implicit ,BW-24 Explicit, RK-3

-5E-05 0.00025

-0.0001

0 1020304050 Error 0 y Location

Fig. 57. Effect of the ACM filter parameter on the accuracy of the computed solution. -0.00025 more appropriate for aeroacoustics, because computa- 10 20 tion of these flows require high values of the ACM filter Normal Distance parameter ðk40:5Þ in order to prevent numerical oscillations. Numerical solutions obtained for the same Fig. 58. Error of the solutions computed with implicit and problem on artificially distorted meshes have demon- explicit time integration. ARTICLE IN PRESS

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1.2 4

3.5 1.1 inf 3

1 2.5 -1.2 -1.15 -1.1 2 Pressure, P / Exact 5th order 1.5 7th order 9th order 1 -2 -1 0 1 2 x

Fig. 59. Comparison of the computed pressure at y ¼ 0:5 with the exact solution. solutions with the sixth-order accurate ACM method require two ghost points. At steady state, an oblique incident shock and a Fig. 60. Pressure field computed with WENO fifth-order and reflected shock were generated. The comparison of the with the fourth-order centered scheme with characteristic-based computed pressure with the exact solution of Fig. 59 filter (ACM parameter k ¼ 0:7) on an artificially distorted mesh. shows that as the order of the WENO scheme increases the computed solution approaches the exact solution. For all computations, the shocks were captured within 250 500 750 1000 two cells. The pressure field obtained from the numerical BW-24, N=2 solutions with the fourth-order accurate compact -5 BW-22, N=4 -5 10 BW-22, N=2 10 centered scheme with k ¼ 0:7 and the fifth-order RK3 accurate WENO scheme on an artificially distorted 10-7 10-7 mesh is shown in Fig. 60. Both solutions were computed with the explicit time marching scheme. It can be seen that both methods can capture the oblique strong shock 10-9 10-9 without oscillations. Furthermore, the artificially distorted mesh does not cause oscillations. 10-11 10-11 The numerical solution for the same problem was also computed with the implicit BW time marching scheme. L2 Norm of Density -13 -13 The convergence rates of the numerical solutions 10 10 obtained for different number of subiterations is shown in Fig. 61. For reference, the convergence rate of the 10-15 10-15 solution obtained with the explicit third order Runge– 250 500 750 1000 Kutta method of Ref. [81] is shown in the same figure. Iteration At convergence all solutions were the same and Fig. 61. Convergence rate of solutions computed with implicit computed pressure and density obtained from implicit time marching. or explicit time marching were almost identical. A comparison of the computed pressure at y ¼ 0:5 from the solution obtained with the fifth-order accurate Computations of supersonic flows over a cylinder at WENO scheme and the solutions obtained with different various Mach numbers for Ref. [177] are shown next. values of the ACM parameter is shown in Fig. 62. These solutions were obtained using WENO schemes. It appears that the computed solution is sensitive An algebraically generated 181Â51 point grid was used to the selection of the ACM parameter. Furthermore, for this computation of supersonic flows over the the choice of the upwind TVD limiter affects the cylinder. Similarly to the shock reflection case sufficient solution. number of ghost points depending on the order of the ARTICLE IN PRESS

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4.5

4 WENO 5th order 4th order ACM =0.7

3.5 4th order ACM =0.9 4th order ACM =1.5 3

2.5

2 3.5 1.5 Pressure, P/ Pinf 3 1 2.5 0.5

Pressure,P/Pinf 2 1 1.05 1.1 1.15 0 -2 -1 0 1 2 x

Fig. 62. Effect of ACM parameter on the computed pressure at y ¼ 0:5.

Fig. 63. Computed entropy and pressure fields with the ninth- order accurate WENO scheme at M1 ¼ 5:0. scheme was used at the edges of the domain. At the inflowfree stream supersonic flowwasspecified. At the outflowall quantities wereextrapolated from the interior. On the cylinder solid surface the normal to the 10-2 surface velocity component was set to zero and all the WENO5, baseline grid, M=3 other quantities were extrapolated from the interior -3 WENO5, baseline grid, M=5 10 WENO5, refined grid, M=5 using high-order extrapolation and assuming that the WENO7, refined grid, M=5 normal derivative is zero. 10-4 WENO7, baseline grid, M=10 The computed pressure and entropy fields at M1 ¼ -5 5:0 are shown in Fig. 63. The resolution of the strong 10 shock generated by the high-speed flowis captured Residual 2 -6 without oscillations. Similar to the oblique shock 10 computations of the previous section the shock for the 10-7 supersonic cylinder flow(see Fig. 63) is captured within Density L two cells. The convergence rate obtained at different 10-8 Mach numbers, order of accuracy, and for baseline (91 Â 51) and refined (181 Â 101) grids is shown in 10-9 Fig. 64. All computations were obtained at the same time step and the third-order TVD Runge–Kutta 0 10000 20000 30000 method of Ref. [81]. For all cases, the convergence was Iteration satisfactory and the solution practically remained un- changed when the residuals drop four orders of Fig. 64. Convergence history at M1 ¼ 3:0, M1 ¼ 5:0, and M ¼ 10:0 with the baseline and refined grids. magnitude. A comparison of the computed pressure 1 distributions for the grid line on the symmetry axis that passes through the stagnation point is shown in Fig. 65. included three ghost points at the edges of the domain in It can be seen that the shock is captured within two order to use the WENO5 scheme for the entire domain computational cells and the solution is free from without dropping the stencil accuracy at the airfoil oscillations. surface and the wake. The supersonic and transonic flow Inviscid flows solutions over a NACA-0015 airfoil computations were obtained on the same grid. For both were computed using the fifth-order accurate WENO flowspeeds, a smooth solution is obtained on the highly scheme. The computed pressure fields at transonic and stretched, high aspect ratio C-type grid. supersonic speed are shown in Figs. 66 and 67. The The computed pressure field at M1 ¼ 2:0 is shown in solutions were computed with the explicit time marching Fig. 66. The computed pressure field shows adequate scheme on a 261 Â 51 point, C-type grid. The airfoil grid resolution of the leading edge bowshock and the two ARTICLE IN PRESS

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120 M = 10 M = 5 100 M = 4 M = 3

40 Pressure, P/ Pinf

0 Fig. 67. Computed pressure ﬁeld over a NACA-0015 airfoil at -2 -1.75 -1.5 -1.25 -1 M ¼ 0:8, a ¼ 0:1. Distance from stagnation point, y/r

Fig. 65. Pressure at the symmetry line for M ¼ 3, 4, 5, and 1 M ¼ 10.

0.5

-0.5 Computed Mesured SurfacePressureCoefficient,-Cp

-1 0 0.25 0.5 0.75 1 x / c

Fig. 68. Comparison of the computed pressure surface pressure coefficient for NACA-0012 airfoil at M ¼ 0:8, a ¼ 0:1 with the experiment. Fig. 66. Computed pressure field over a NACA-0015 airfoil at M ¼ 2:0, a ¼ 0:0. solution with the experiment is satisfactory and the shock is resolved within two cells. shocks at the trailing edge despite of the coarseness of Numerical solutions of two-dimensional viscous the grid. The same type of inflow/outflow and solid wall transonic flowover airfoils wereobtained using a boundary conditions as the supersonic cylinder flowwas second-order accurate FV ENO or WENO scheme by used. At the wake of the C-type grid averaging was used. Yang et al. [178]. The LU-SGS scheme of Section 4.3 For the transonic flowcomputation at M1 ¼ 0:8, the was used for time integration to avoid stability limita- shocks of the upper and lower surface shown by the tions from the highly clustered viscous meshes. The pressure contours of Fig. 67 are well resolved. For this computations of Yang et al. [178] showed sharp computation the inflowand outflowboundary condi- capturing of discontinuities (see Fig. 69) without tions were specified using one-dimensional Riemann oscillations. It was found (see Fig. 70) that after the invariants. A comparison of the computed surface residuals of the ENO2 scheme have decayed for three pressure coefficient for transonic flowover the NACA- orders of magnitude the convergence leveled off. In 0012 airfoil with the experimental data is shown in contrast, as expected (see remarks in Section 4.4) a Fig. 68. The overall agreement of the computed inviscid monotone convergence was achieved with the WENO2 ARTICLE IN PRESS

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0.95

0.9 1.1 0.8

1.2 0.7 1.2 1.1 0.93 0.8 1.6 0.2 0.4 0.6 0.5 0.1 0.3 0.7 0.6 0.7 2 0.0 0.6 0.9 0.5 0.4

0.8

0.9 0.8

Fig. 70. Convergence history for NACA 0012 airfoil at 6 Fig. 69. Mach number contours for NACA 0012 airfoil at M1 ¼ 0:799, a ¼ 2:26 and Rec ¼ 6:5 Â 10 . 6 M1 ¼ 0:799, a ¼ 2:26 and Rec ¼ 9 Â 10 ; WENO2-Roe scheme.

6 Fig. 71. RAE 2822 airfoil surface pressure distribution at M1 ¼ 0:731, a ¼ 2:51 and Rec ¼ 6:5 Â 10 ; comparison of WENO2 and ENO2 schemes. ARTICLE IN PRESS

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6 Fig. 72. Steady pressure distributions for ONERA M6 wing at M1 ¼ 0:8395, a ¼ 3:06 and Rec ¼ 2:6 Â 10 .

10-1

10-2

10-3

10-4

L 2 NORM OF RESIDUAL 10-5

10-6 0 1000 2000 3000 4000 5000 6000 ITERATIONS

Fig. 73. Upper surface contours for ONERA M6 wing at Fig. 74. Convergence history for ONERA M6 wing at 6 6 M1 ¼ 0:8395, a ¼ 3:06 and Rec ¼ 2:6 Â 10 . M1 ¼ 0:8395, a ¼ 3:06 and Rec ¼ 2:6 Â 10 . ARTICLE IN PRESS

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WENO5, h=1/240 WENO9, h=1/240 0.5 0.5 0.4 0.4 0.4 0.4 Y 0.2 0.2 0.1 0.1 0 0 2 2.25 2.5 2.75 2 2.25 2.5 2.75 X X

WENO5, h=1/480 WENO9, h=1/480 0.5 0.5 0.4 0.4 0.4

Y 0.4 0.2 0.2 0.1 0.1 0 0 2 2.25 2.5 2.75 2 2.25 2.5 2.75 X X

WENO5, h=1/960 WENO9, h=1/960 0.5 0.5 0.4 0.4 0.4 0.4 Y 0.2 0.2 0.1 0.1 0 0 2 2.25 2.5 2.75 2 2.25 2.5 2.75 X X

Fig. 75. Double Mach reflection problem. Blown-up region around the double Mach stems. Density r; 30 equally spaced contour lines 1 1 1 from r ¼ 1:5tor ¼ 22:9705. Left from top to bottom: fifth-order WENO results with h ¼ 240; 480; 960; right from top to bottom: ninth- 1 1 1 order WENO results with h ¼ 240; 480; 960. scheme. The computed surface pressure distributions has been used extensively in the literature as a test from the ENO and WENO algorithms (see Fig. 71) were for high-resolution schemes. A Mach 10, right moving almost indistinguishable. Very good agreement with shock at an angle of 60 is reflected from a wall (for the measured pressure coefficient distribution was more details see [179]). The flowis computed as inviscid found in [178] for three-dimensional flowover the and the results are displayed in Fig. 75 at t ¼ 0:2 ONERA M6 computed with the WENO2 scheme. in the domain ½0; 3Â½0; 1. Three different uniform 1 1 1 Comparisons of the surface pressure distribution are meshes h ¼ 240, h ¼ 480, and h ¼ 960 were used in [168]. shown in Fig. 72. The computed surface pressure The computed solutions are compared in Fig. 75. 1 distribution and the convergence rate are shown in It is clear that WENO9 with h ¼ 240 produces qua- 1 Figs. 73 and 74, respectively. litatively the same resolution as WENO5 with h ¼ 480. 1 The resolution and efficiency of high-order accurate The same is true for WENO5 with h ¼ 960 and 1 WENO schemes for computing flows containing both WENO9 with h ¼ 480. It is evident that the resolu- discontinuities and complex flowfeatures wasrecently tion increases consistently with the order of accu- demonstrated by Shi et al. [168]. The first representa- racy and mesh refinement. Furthermore, the ability to tive numerical example was the double Mach reflec- obtain the same resolution with half the mesh by tion, a problem that includes strong shock waves increasing the order of the method (compare WENO5 1 1 and very complex flowfeatures. This problem was with h ¼ 960 and WENO9 with h ¼ 480) is clearly initially proposed by Woodward and Colella [179] and demonstrated. ARTICLE IN PRESS

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1 1 1 1 WEN05, h = 1/240 WEN05, h = 1/480 WEN05, h = 1/960 WEN05, h = 1/1920 0.9 0.9 0.9 0.9

0.8 0.8 0.8 0.8

0.7 0.7 0.7 0.7

0.6 0.6 0.6 0.6 y y y 0.5 0.5 0.5 y 0.5

0.4 0.4 0.4 0.4

0.3 0.3 0.3 0.3

0.2 0.2 0.2 0.2

0.1 0.1 0.1 0.1

0 0 0 0 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 x x x x 1 1 1 1 WEN09, h = 1/240 WEN09, h = 1/480 WEN09, h = 1/960 WEN09, h = 1/1920 0.9 0.9 0.9 0.9

0.8 0.8 0.8 0.8

0.7 0.7 0.7 0.7

0.6 0.6 0.6 0.6

y 0.5 y 0.5 y 0.5 y 0.5

0.4 0.4 0.4 0.4

0.3 0.3 0.3 0.3

0.2 0.2 0.2 0.2

0.1 0.1 0.1 0.1

0 0 0 0 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 x x x x

Fig. 76. Rayleigh–Taylor instability. Density r; 15 equally spaced contour lines from r ¼ 0:952269 to r ¼ 2:14589. Top from left to 1 1 1 1 1 1 1 1 right: ﬁfth-order WENO results with h ¼ 240; 480; 960; 1920; bottom left to right: ninth order WENO results with h ¼ 240; 480; 960; 1920.

The second problem considered in [168] is the the comparable resolution obtained with the WENO9 Rayleigh–Taylor (RT) instability. This instability hap- scheme using half the number of grid points in each pens on an interface between fluids with different direction than WENO5 implies significant saving in densities due to the motion of the heavy fluid. The computing time since the WENO9 scheme needs only numerical solution of the RT problem demonstrates the approximately 30% more CPU time than the WENO5 ability of WENO schemes to consistently capture scheme. smooth, complex flowfeatures withincreased resolution The dynamics of shock–vortex interactions and the either by grid refinement or the increase of the order of coupling of counter-rotating compressible vortices the method. Inviscid flowin a computational domain interacting with a planar shock wave were investigated 1 ½0; 4Â½0; 1 was computed in [168] with the heavy fluid by Grasso and Pirozzoli in [180–182]. The numerical 1 with density r ¼ 2 below, the interface at y ¼ 2, and the solutions of the inviscid, compressible flowequations light fluid with r ¼ 1 above the interface. The solutions was obtained using a finite volume WENO scheme for of Fig. 76 were computed on uniform meshes with Cartesian-type meshes. The simulations for the single 1 1 1 1 h ¼ 240, 480, 960, and 1920. Again the WENO9 scheme vortex–shock wave interaction [180] initiated with a yields the same resolution with the WENO5 scheme at homoentropic Taylor vortex. Results from the simula- double grid resolution. In both cases of Figs. 75 and 76, tions of [180] are shown in Figs. 77 and 78. Fig. 77 shows ARTICLE IN PRESS

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the evolution of shock–vortex interaction with the computed Schlieren plot for a shock at Ms ¼ 1:2 and vortex Mach number Mv ¼ 0:8. Fig. 78 shows the computed pressure contour plots at the same times with Fig. 77. The dynamics of the interaction of shock wave at Ms ¼ 5:0 colliding with a counter-rotating vortex pair at Mv ¼ 0:26 from [181] are shown in Fig. 79. The numerical Schlieren for shock wave–colliding vortex pair interaction at Mv ¼ 0:9, Ms ¼ 1:2 [182] is shown in Fig. 80. The acoustic pressure field computed in [182] at Mv ¼ 0:1, Ms ¼ 1:2 is shown in Fig. 81. It can be seen that in all cases the dynamics of the interaction and the flowfield structure is very complex. The computed solutions in [180,181] were found in very good agreement with measurement and analytic solutions from acoustic analogies. These results showthat high-order WENO schemes are appropriate for the computation of noise. Further investigation of three-dimensional shock ring–vortex interaction was carried out by Pirozzoli [183] using the compact WENO scheme he developed in [173]. The computed complex interaction of the impact Fig. 77. Computed Schlieren for shock–vortex interaction.

Fig. 78. Computed pressure ﬁeld for shock–vortex interaction. ARTICLE IN PRESS

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Fig. 79. Computed pressure ﬁeld for shock counter–rotating vortex pair interaction. ARTICLE IN PRESS

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À Fig. 80. Shock wave–colliding vortex pair interaction: numerical Schlieren at Mv ¼ 0:9, Ms ¼ 1:2, d ¼ 4rv-Type-V interaction. SW, shock wave; SN, diffracted shock.

of a vortex ring Mv ¼ 0:25 on a planar shock Ms ¼ 1:5 fourth-order ENO scheme. The viscous terms in [175] is shown in Fig. 82. are discretized with the sixth-order accurate compact Adams [184] used the hybrid compact-ENO finite- finite-difference scheme of Lele [100]. Time marching of difference scheme of [175] to perform direct simulation the DNS in [184] is performed with a RK3 method. of the flowover a compression ramp. The scheme used Computed Schlieren from the DNS in Fig. 83 are in for this DNS is fifth-order accurate in smooth flow good agreement with similar measurement shown for regions and around the discontinuities becomes the comparison in Fig. 84. The computed density field is ARTICLE IN PRESS

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À Fig. 81. Shock wave–colliding vortex pair interaction: acoustic pressure ﬁeld at Mv ¼ 0:1, Ms ¼ 1:2, d ¼ 4rv-Type-I interaction. À0:1pp0p0:5, 72 contour levels. Dashed lines stand for p0p0, while solid lines stand for p0X0.

shown in Fig. 85. The sequence of shock evolution the numerical diffusion at a lowlevel. These are key obtained from the DNS of [175] is shown in Fig. 86. The ingredients required for accurate capturing of turbulent good agreement of the DNS of [184] with the experi- ﬂuctuations in compressible turbulence DNS and LES. ments demonstrates that high-order accurate ENO Additional computations with ENO and WENO discretization yields the necessary resolution and keeps schemes can be found in [185–190]. ARTICLE IN PRESS

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Fig. 84. Flow-ﬁeld experimental Schlieren visualization of a 25 compression ramp at M1 ¼ 2:9, Rey ¼ 9600, provided by A. Zheltovodov, ITAM, Novosibirsk.

Fig. 82. Impact of a vortex on a planar shock wave for

Ms ¼ 1:5, Mv ¼ 0:25, j0 ¼ 3p=4att ¼ 8. The gray shading represents the 0.01 isosurface of l2; the lighter shading represents the À0.1 isosurface of rÁu, which has been cut at the intersection with the symmetry plane.

Fig. 85. Density at the wall, in the plane x2 ¼ 2:9 and 4 cross-

ﬂowplanes; shock-surface with qxi ui ¼À0:4.

5. The discontinuous Galerkin (DG) method

Generation of three-dimensional meshes even in domains with moderate complexity, such as a wing body junction for example, is not trivial. On the other hand, the performance of most high-resolution accurate methods depends on the smoothness of the grid. The difficulty in generating smooth-structured grids for complex geometries has promoted the development of finite-volume algorithms for unstructured grids [191–195]. However, most of these unstructured grid methods are second-order accurate. High-order finite- volume schemes were pioneered by Barth and Freder- ickson [131] with the k-exact finite-volume scheme that can be used for arbitrary high-order reconstruction in triangular or tetrahedral meshes. The implementation of ENO for unstructured grids was developed by Abgrall [164], while WENO schemes for triangular meshes were developed by Friedrich [166] and Hu and Shu [196]. Theoretically, the k-exact approach [131,196], or other Fig. 83. Flow-field Schlieren imitation, krrk contours, (a) approaches [164] can be used to obtain arbitrarily high- instantaneous x2-average, (b) x2-average and time average order accurate finite-volume schemes with high-order using 100 samples. polynomial data reconstructions. However, in practice ARTICLE IN PRESS

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Fig. 86. Sequence of shock evolution (iso-contours) of krrk and velocity vectors at x2 ¼ 0:72 for a sequence of equal time steps between tl ¼ 309:00 and tr ¼ 325:34; sequence from bottom to top and left to right (numbers 1–12); the marker indicates the sensor position for Fig. 14; flowdirection is from bottom up. higher than linear reconstructions are not used in three [45,201]. For each time t 2½0; T the approximate dimensions because of the difficulty to construct non- solution, uh, of hyperbolic equations in conservation singular stencils and the large memory required to lawform, qtu þ div FðuÞ¼0, is sought in the following store the reconstruction coefficients. It was shown by finite-element space of discontinuous functions V h Delanaye and Liu [197] that for the third-order 1 (quadratic reconstruction) FV scheme in three dimen- Vh ¼ffh 2 L ðOÞ : fhjK 2 VðKÞ8K 2 T hg, (5.1) sions, the average size of the reconstruction stencils is where T h is a discretization of the domain O using about 50–70. The size of reconstruction stencils increases triangular or quadrilateral elements and VðKÞ is the nonlinearly with the order of accuracy and it was local space that contains the collection of polynomials estimated that for the fourth-order FV scheme the up to degree k. stencil size would be approximately 120. The development of the DG method starts from weak The high-order accurate conservative scheme called formulation of hyperbolic-type governing equations the discontinuous Galerkin (DG) method developed by Z Cockburn et al. in a series of papers [198–201] shows d uðx; tÞfðxÞ dx promise for high-resolution simulations in fully un- dt KZ structured meshes. The DG method assumes a high- order expansion for the distribution of the state ¼ Fðuðx; tÞÞ Á rðfðxÞÞ dx K variables for each element and solves for the coefficients X I of the expansion polynomials. The resulting state À Fðuðx; tÞÞ Á ne;K fðxÞ dG, ð5:2Þ variables are usually not continuous across the element e2qK e boundaries. Therefore, the fluxes through the element where fðxÞ is any sufficiently smooth function and ne;K boundaries are computed using an approximate Rie- denotes the outward, unit normal to the face or edge e. mann solver and the residual is minimized with a The stiffness matrix integral at the left-hand side of Galerkin approach. It is the use of Riemann fluxes Eq. (5.2) is evaluated numerically using Gauss–Radau across element boundaries that makes the DG method integration rules. The integrals on the right-hand side of fully conservative. Eq. (5.2) are evaluated using quadrature rules as follows: Z 5.1. DG space discretization Fðuðx; tÞÞ Á ne;K fðxÞ dG e The DG method is briefly described in this section. XL Further information and more details can be found in clFðuðxel ; tÞÞ Á ne;K fðxel Þjej, ð5:3Þ the original references [200,205,206] and the reviews of l¼1 ARTICLE IN PRESS

258 J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 Z linearized Euler equations the eigenvalues are constant Fðuðx; tÞÞ Á rðfðxÞÞ dx K and the derivatives are continuous for the nonlinear XJ case, however,p inffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi order to obtain continuous higher 2 ojFðuðxKj; tÞÞ Á rðfðxKjÞÞjKj. ð5:4Þ derivatives a ¼ 2 þ l with ¼ 0:05. The DG method j¼1 can be applied for discretizations on all triangular, The line integrals of Eq. (5.3) are computed using quadrilateral or mixed-type of elements. Expansion appropriate high-order Gaussian quadrature. For ex- bases of different order for triangular and quadrilateral ample, for a third-order polynomial basis a quadrature elements are shown next. rule that integrates exactly at least sixth-order poly- Large portion of computing resources during the nomial is used. In general, a kth order DG method implementation of the DG method is devoted for the (k À 1 order polynomial reconstruction) requires a 2kth evaluation of integrals, such as the mass and stiffness order quadrature formula for the surface (line) integrals matrices using quadrature rules. For linear systems of of Eq. (5.3), and a ð2k À 1Þth order quadrature formula hyperbolic equations, such as the linearized Euler for the volume integrals of Eq. (5.4). equations, the computing cost of the DG method can The data are assumed discontinuous across the be significantly reduced using the quadrature-free DG interfaces of the continuous domain and at each method. Then time integration of the semi-discrete DG interface two values are available. Therefore, the flux formulation with RK3 or RK4 yields the quadrature- Fðuðx; tÞÞ is replaced by a suitable numerical flux free Runge–Kutta DG approach of Atkins and e Shu [208]. F e;K ðx; tÞ for the approximate solution uh and the test e function fh 2 VðKÞ. Using F e;K ðx; tÞ in Eqs. (5.3) and (5.4) the approximate solution uh is given by 5.2. Element bases Z d For the general nonlinear case, the order of accuracy uhðx; tÞfhðxÞ dx dt K of the DG method (see Ref. [45] and references therein) XJ is at least k þ 1=2 if polynomials of degree at most k are ¼ ojFðuhðxKj; tÞÞ Á rðfhðxKjÞÞjKj used as basis functions. Furthermore, it was shown (see j¼1 Ref. [45]) that for linear problems that for canonical X XL semi-uniform triangular grids the order of accuracy is À c Fe ðuðx ; tÞÞ Á n fðx Þjej, l e;K el e;K el ðk þ 1Þ. For simplicity in the rest of this section, the e2qK l¼1 method is called ðk þ 1Þth order accurate if the basis 8f 2 VðKÞ8K 2 T , ð5:5Þ h h functions are polynomials of degree at most k. where time advancement of Eq. (5.5) is performed with The approximate solution within each element is the third-order accurate TVD Runge–Kutta method of expanded in a series of local bases functions (poly- [81]. nomials) as follows The major difference of the GD formulation with a Xd standard modal or node-based Galerkin finite-element k uhðx; y; tÞ¼ cjðtÞPj ðx; yÞ, (5.7) method is that the expansion in each element is local j¼1 without any continuity across the element boundaries. where c ðtÞ, j ¼ 1; 2; ...; d are expansion coefficients or The value of the numerical flux Fe ðx; tÞ at the edge of j e;K degrees of freedom for each element, to be evolved in the boundary of the element K depends on two values of time, and Pkðx; yÞ are polynomial bases of degree k the the approximate solution, one from the interior (right) j most. In two dimensions, the range of d, or number of of the element K, uR ¼ u ðxintðKÞ; tÞ, and the other from h polynomials in Eq. (5.7) (also the number of nodes on a the exterior (left) of the element K, uL ¼ u ðxextðKÞ; tÞ. h triangle for nodal expansions) is related to the degree of Any consistent, conservative exact or approximate the polynomial by d ¼ðk þ 1Þðk þ 2Þ=2. For example, Riemann solver can be used to obtain the numeri- cubic reconstruction k ¼ 3 requires d ¼ 10 e.g. 10 nodes cal flux Fe ðuðxintðKÞ; tÞ; uðxextðKÞ; tÞÞ or Fe ðuL; uRÞ as e;K e;K (see Fig. 87). follows The value of d for bases in two and three dimensions Fe ðuL; uRÞ¼1½FðuLÞÁn þ FðuRÞÁn (triangles or tetrahedra) is given by e;K 2 e;K e;K À F nðuL; uRÞ, ð5:6Þ n þ k ðn þ kÞ! d ¼ ¼ , (5.8) k n!k! where F nðuL; uRÞ is the dissipative part of the numerical k flux. The computationally efficient local Lax–Friedrichs where k is the order of the polynomial Pj and flux is used in many applications. The flux F is split as j ¼ 1; ...; d. It can be seen that in three dimensions the F ¼ F þ þ F À where F Æ ¼ F þ au where a ¼ l and l is third-order basis includes 20 polynomials and the fifth- the maximum eigenvalue of the flux Jacobian. For the order basis 56 polynomials. It appears therefore, that for ARTICLE IN PRESS

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(a) 1st order polynomial basis function

1/2 3

1/2 (b) 3rd order polynomial basis function

1/3 1/2

1/2 (c) 5th order polynomial 1 2 basis function Fig. 88. Reference element for ﬁrst-order polynomial basis.

1/5

1 3 5 Fig. 87. Nodal points for polynomial bases. P , P , and P y bases on triangular meshes. (0, 1) realistic, three-dimensional problems, it could be very intensive computationally to achieve accuracy higher that fourth order. In two dimensions, a first-order basis ðd ¼ 3Þ, 1 1 1 P1 ¼ 1 À 2y, P2 ¼ 2x þ 2y À 1, P3 ¼ 1 À 2x, can be used to achieve second-order accuracy. Each of these polynomials (see Fig. 88) takes unit value at one node, (2, 3, 3) y = 3 located in the middle of an edge, and zero value at the other nodes located at the middle of the other edges. The 1 1 1 first-orderR polynomials P1, P2, P3 are orthogonal ð PiPj ¼ 0; iajÞ and the mass matrix resulting from el t =3 the integration at the left-hand side of Eq. (5.2) is diagonal. For the first as well as the higher-order bases x all calculations of Eq. (5.5), except the numerical flux (0, 0) (1, 0) evaluation, are carried out on the reference element. The t numerical flux computation along the edges of the element, which depends on the neighboring element, can Fig. 89. Reference element and nodal points for fifth-order be carried out at the physical space. polynomial basis. Third- and higher-order polynomial nodal bases in two dimensions can be systematically generated with Lagrange interpolation in triangular coordinates is defined as x; y; t ¼ 1 À x À y. More details on polynomial bases can be found in [40,42,43]. The general form of yðy À 1=5Þ tðt À 1=5Þ P ðx; y; tÞ¼5x . (5.10) Lagrange interpolation in triangular coordinates is ð2;3;3Þ 2 2 ð25Þ ð25Þ given by The fifth-order polynomial basis is also non-orthogo- PLðx; y; tÞ¼LiðxÞLjðyÞLkðtÞ, (5.9) nal and the mass matrix is computed using Gauss Radau 1 3 integration. Note that the Pj and Pj bases can be where LmðxÞ are one-dimensional ðm À 1Þ order constructed using the procedure followed for the 5 Lagrange polynomials [43] and L ¼ Lði; j; kÞ (see construction of the Pj basis. The bases based on Fig. 89) denotes the nodal index. For example, in Lagrange interpolation for triangular elements have Fig. 89 at the nodal index Lð2; 3; 3Þ the base polynomial good condition number [43]. In addition, they offer ARTICLE IN PRESS

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Reference element Physical space isoparametric element yˆ (-1,1) (1,1)

4 11 7 3 (x3, y3)

8 16 15 10 xˆ

1 1 12 13 14 6 (x , y )

1 5 9 2 (x2, y2) (-1,-1) (1,-1)

Fig. 90. Reference element with nodal points for third-order polynomial basis and isoparametric mapping for the arbitrary shape quadrilateral element in physical space. implementation advantages compared to hierarchical Note that for the nonlinear case, the ADER-DG bases, which are more appropriate for multigrid [207], scheme requires Gaussian quadratures of suitable order because for these bases the nodal values are known. of accuracy. The ADER approach of Titarev and Toro Space discretization can be in addition be obtained [209] is based on the solution of the generalized using quadrilateral isoparametric elements of arbitrary Riemann problems (GRPs) at the cell boundaries and shape (see Fig. 90). The reference element is a square application of the Lax-Wendroff procedure for highly (see Fig. 90) with vertices at the points ðÀ1; À1Þ; accurate time integration of the numerical flux. ð1; À1Þ; ð1; 1Þ; ðÀ1; 1Þ.InFig. 90, the physical space It was shown numerically by Dumbser and Munz that arbitrary shape quadrilateral and the nodal points of the ADER-DG scheme is 3N þ 3 order accurate for Nth the third-order accurate polynomial basis on the order basis functions. k reference element are also shown. The coordinates xj , The main ingredients of the ADER-DG-Discretiza- k k k k j ¼ 1; 2; k ¼ 1; 2; 3; 4 ðx1 ¼ x ; x2 ¼ y Þ of the arbitrary tion are: shape quadrilateral elements in the physical space are related to the square reference element coordinates Taylor expansion in time of the solution. b b b b b xj ðx1 ¼ x; x2 ¼ yÞ through the map Use of the Lax-Wendroff procedure to replace time derivatives by space derivatives. X4 b b b k Solution of generalized Riemann problems (GRP) to xj ¼ f kðx1; x2Þxj , (5.11) k¼1 approximate space derivatives. where fb ðxb ; xb Þ are first-order Lagrange polynomials k 1 2 5.4. Analysis of the DG method for wave propagation b b b k b k b f kðx1; x2Þ¼ð1 þ x1 x1Þð1 þ x2 x2Þ=4. The basis polynomials Pkðx; yÞ2Qk for finite-element discretization with i Hu et al. [210] carried out a study of wave quadrilateral elements are tensor products of appro- propagation properties of the semi-discrete DG method priate order one-dimensional Lagrange polynomials. for conservation laws with linear flux given by For example, the basis with third-order accurate P eþ eÀ e d 3 Fnumðu1; u2; nÞ¼A u1 þ A u2, where A ¼ Aknk, polynomials is Pi ðx; yÞ¼LjðxÞLkðyÞ; i ¼ 1; ...; 16; av av k¼1 þ e e À e e j; k ¼ 1; 2; 3; 4. e AþajAj e AÀajAj the averages are defined as Aav ¼ 2 , Aav ¼ 2 , and a ¼ 0 yields a centered flux while a ¼ 1 yields the Roe flux. Considering the approximate solution uh is written 5.3. Arbitrary high-order DG schemes asP an expansion of the local basis set uhðx; tÞ¼ NÀ1 l¼0 clðtÞvl ðxÞ it was shown in [210] that for the one- The ADER (Arbitrary high-order scheme which dimensional scalar advection equation ut þ cux ¼ 0, utilizes the hyperbolic Riemann problem for the advec- with exact dispersion relation o ¼ ak, the numerical tion, of the higher-order DERivatives) can be applied dispersion relation for an upwind numerical flux ða ¼ 1Þ for DG numerical solutions of linear hyperbolic systems with the DG discretization is determined by to obtain a quadrature-free, explicit single-step method ÀiK of arbitrary order of accuracy in both space and time. detðÀiOQ þ 2e N1 þ 2N0Þ¼0. (5.12) ARTICLE IN PRESS

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The solution of the numerical dispersion relation yields Ihlenburg [215], Thomson and Pinsky [216]. More complex values for the numerical propagation fre- recently, high-order DG finite-element methods quency, O ¼ Or þ iOi. In the numerical propagation [45,217–220] were applied and analyzed for wave frequency, the imaginary part is negative and represents propagation. the numerical damping inherent in the DG discretization It was found that for the small wavenumber limit process. hk ! 0 the DG method gives a higher order of accuracy The solution of the numerical dispersion relation, Eq. than the standard Galerkin methods. Hu and Atkins (5.12) for a third-order DG method involves three [220], for example, concluded that the dispersion modes, one physical mode and two parasitic modes. It relation of the scalar advection equation for an Nth was found that the numerical dispersion relation order DG method is accurate to order 2N þ 3inhk for deviates from the exact one beyond K N, where N is the dispersion error and 2N þ 2 for the dissipation error. the order of the scheme. A more systematic study of the dissipative and Important conclusions of the analysis of Hu et al. dispersive behavior of the DG method that gives sharp [210] are: error estimates was carried out by Ainsworth [211]. Ainsworth’s estimates are very sharp and agree with Increase of the order of the scheme significantly a posteriori error estimates of Hu and Atkins [220]. reduces the dissipation error. The main conclusions of the analysis performed by The sixth-order scheme is optimal for scalar advec- Ainsworth [211] are tion in the sense that it is the minimal order for which the dispersion and dissipation errors are less than (1) As the order N is increased, the behavior of the error 0.5% for wavenumber K up to approximately N. passes through three different phases depending on The dissipation error imposes a relatively more the size N relative to hk. stringent condition on the accuracy of the scheme (2) Pre-asymptotic regime 2N þ 1ohk À OðhkÞ1=3. The than does the dispersion error. resolving ability of the method is inadequate and the relative error tends to oscillate without decay. For two-dimensional advection (two-dimensional (3) The transition zone where hk À OðhkÞ1=3o2N þ 1 2 2 þ ð Þ1=3 wave equation) ftt À c r f ¼ 0 the numerical disper- ohk O hk . The relative error is of order unity sion relation was found as and decreases at an algebraic rate NÀ1=3. (4) Asymptotic regime where N is large compared to hk ÀiK cos W iK cos W detðÀiOQ þ 2½N0 þ e NÀ1 þ e Nþ1 2N þ 14hk þ OðhkÞ1=3. The relative error reduces at ÀiK sin gW iK sin gW þ 2g½M0 þ e MÀ1 þ e Mþ1Þ ¼ 0, ð5:13Þ a super-exponential rate. where the value of O is function of the wavenumber K and the angle W. An important conclusion of this An exponential convergence on the envelope 2N þ analysis is that the wave propagation is anisotropic 1 hk was also found in Ref. [211]. Namely reduction 5 and the dependence on wave propagation angle is of the mesh-size to the limit hk 1 is not possible for stronger for the higher wavenumbers. high frequencies. It was shown, however, that increasing the order N on a fixed mesh is more effective than reducing the mesh size. Furthermore, it was concluded 5.5. Dissipative and dispersive behavior of high-order DG [211] that it is inefficient to increase the order N much method beyond the threshold 2N þ 14hk þ OðhkÞ1=2. A more practical approach to resolve problems where hkb1isto In a previous section, the dissipative and dispersive work on the envelope of the region where the super- properties of the DG discretizations were analyzed and exponential convergence sets in. For these cases, the demonstrated for linear aeroacoustic problems. A order of the method must be chosen so that 2N þ 1 systematic analysis of the DG method for linear wave khk where k41 is a constant. Ainsworth [211] showed propagation with very short-wave lengths was recently that the relative error r decays at an exponential rate presented by Ainsworth [211]. This analysis targeted N as N !1. electromagnetic wave propagation. Efficient and accurate resolution of electromagnetic waves without excessive numerical dissipation or dispersion is important 5.6. Limiting of DG expansions in the context of other high-frequency applications such as magneto-gas dynamics (MGD). The most promising Limiting operators LPh on piecewise linear DG approach for wave propagation is obtained with higher- expansions uh are constructed in such a way that they order schemes such as spectral element methods satisfy the following properties: [212,213]. Higher-order standard Galerkin finite-element methods were also used in the past by Astley et al. [214], (1) Accuracy:ifuh is linear, then LPhuh ¼ uh. ARTICLE IN PRESS

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(2) Conservation of mass: for every element K have The left and right eigenvector matrices RÀ1 and R, Z Z which diagonalize the Jacobian A ¼ qf ðuÞ=qu, are LPkuh dV ¼ uh dV. (5.14) evaluated at the average state ui; j in the element ij in k k the x and y directions as RÀ1AR ¼ L where L is the diagonal matrix that contains the eigenvalues of the (3) Slope limiting: The gradient of LPhuh is not bigger ﬂux Jacobian. The columns of R are the right À1 than that of uh for each element K. eigenvectors of A and the rows of R are the left eigenvectors. Theoretical analysis of the slope limiting operators can All quantities needed for limiting are transformed to be found in Cockburn and Shu [198] and Cockburn the characteristic ﬁeld by left multiplying by RÀ1. For et al. [200]. the system version of the limiter given by Eq. (5.17)

for example, the vectors uxi; j ; ðuiþ1; j À ui; jÞ; ðui; j À u Þ are transformed to the characteristic ﬁeld. 5.6.1. Rectangular elements iÀ1; j The limiter of Eq. (5.17) is applied on each The P1 expansion of the approximate solution component of the characteristic variables vector. uhðx; y; tÞ inside rectangular elements ½xiÀ1; xiþ1Â 2 2 The limited (replaced) values are transformed back ½yiÀ1; yiþ1 is 2 2 to the original conservative variables by multiplying uhðx; y; tÞ¼uðtÞþuxðtÞfiðxÞþuyðtÞcjðyÞ, (5.15) by R. where

x À xi y À yi 5.6.2. Triangular elements fiðxÞ¼ ; cjðyÞ¼ (5.16) 1 ðDxi=2Þ ðDyj=2Þ For the P case, the following expansion inside the triangle K is used for the approximate solutions and the degrees of freedom that are evolved in time are uhðx; y; tÞ uðtÞ; uxðtÞ; uyðtÞ. For the scalar equation, limiting for these quadrilat- X3 u ðx; y; tÞ¼ u ðtÞf ðx; yÞ, (5.20) eral element expansion is performed on ux and uy using h i i the difference of the means. i¼1 For example, ux is replaced by where the degrees of freedom or expansion coefficients u ðtÞ are the values of the numerical solution at the mðu ; u À u ; u À u Þ, (5.17) i x iþ1;j i; j i; j iÀ1; j midpoints of the edges. The basis function is a linear where m is the TVB corrected minmod function defined function that takes unit value at the midpoints mi of as the ith edge and zero value at the midpoints of the other two edges. mða ; a ; ...; a Þ 1 (2 m The slope limiting operator for triangular elements 2 a1 if ja1jpMðDxÞ ; is constructed as follows. Consider the triangle K0 of ¼ ð5:18Þ Fig. 91 where limiting is performed and the neighboring mða1; a2; ...; anÞ otherwise triangles K1; K2 and K3. Suitable choices of the triads of and m is the total variation diminishing (TVD) minmod the vectors c0m1; c0m2; c0m3; c0c1; c0c2; c0c3 as shown in function defined as Fig. 91 yields the following geometrical decomposition: ! ! ! mða ; a ; ...a ...; a Þ ðc0m1Þ¼a1ðc0c1Þþa2ðc0c2Þ. (5.21) 1 (2 n N s min1pnpN janj if s ¼ signða1Þ¼ÁÁÁ¼signðaNÞ; In this decomposition, the vector joining the barycenter ¼ 0 otherwise: c0 with the middle of an edge is between the vectors joining the barycenter c0 with the barycenters of the ð5:19Þ neighboring triangles. The parameters a1 and a2 are The TVB correction is introduced in order to avoid positive coefficients that depend on the geometry. unnecessary limiting near smooth extrema where the Furthermore, for any linear function have the expansion expansion coefficients (or degrees of freedom) ux; uy are on the order of OðDx2Þ; OðDy2Þ, respectively. The uhðm1ÞÀuhðc0Þ¼a1½uhðc1ÞÀuhðc0Þ numerical results are not usually sensitive to the choice þ a ½u ðc ÞÀu ðc Þ ð5:22Þ of the constant M. The suggested value for this constant 2 h 2 h 0 [206] is M ¼ 50. Similarly, uy is replaced (limited) by and the averages over the triangles are mðu ; u À u ; u ; Àu Þ. Z y i; jþ1 i; j i iÀ1; j 1 For systems, such as the Euler equations, limiting is uKi ¼ u dV ¼ u ðc Þ; i ¼ 0; 1; 2; 3. (5.23) jK j h h i performed in the local characteristic variables as follows: i Ki ARTICLE IN PRESS

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and obtain the limited value by

K X3 3 K0 b LPhuhðx; yÞ¼u þ Difiðx; yÞ, (5.27) i¼1 C3 where K 1 b ¼ þ ð ÞÀ À ð À Þ m3 m1 Di y max 0; Di y max 0; Di . (5.28) C C 0 1 For systems of equations on triangular meshes,

K0 limiting is performed on the local characteristic variables. For triangular meshes however the following flux Jacobian m 2 ! q mic0 f ðuK0 Þ (5.29) C2 qu jmic0j K e.g. the flux Jacobian A Á k~along the direction of the unit 2 ! vector, k ¼ mic0, must be diagonalized to evaluate the jmic0j À1 Fig. 91. Geometric definitions for arbitrary triangular mesh. left and right eigenvector matrices R and R.

5.6.3. Component-wise limiters The ﬁrst way to apply a limiter to each characteristic Therefore variable was presented in the previous sections for both quadrilateral and triangular elements. The other way is K0 K0 uh ðm1Þuhðm1ÞÀu to apply a limiter to each of the conservative variables. K1 K0 K2 K0 a1ðu À u Þþa2ðu À u Þ The characteristic-wise application of limiter for one- dimensional linear hyperbolic systems has the nice DuK0 ðm Þ. ð5:24Þ 1 property of naturally degenerating to the scalar case.

For a piecewise linear function uh in K0 and the three In multiple dimensions, the characteristic variables must midpoints m1; m2; m3 obtain be defined in a particular direction. For unstructured meshes, there is no coordinate direction to define a X3 characteristic variable and these variables are defined in uK0 ðx; yÞ¼ u ðm Þf ðx; yÞ h h i i the face normal direction. The design of characteristic- i¼1 X3 based limiters in multiple directions is difficult and time K0 K0 consuming. ¼ u þ uh ðmiÞfiðx; yÞ. ð5:25Þ i¼1 In this section, the component-wise approach for the limiter of the DG method is shown. This approach is The limited value LP uK0 ðx; yÞ is obtained by first h h expected to be more efficient than the characteristic computing the quantities approach of the previous sections. The component-wise

K0 K0 approach is based on the following numerical mono- Di ¼ mðu ðmiÞ; oDu ðmiÞÞ, (5.26) h tonicity criterion for each element: m where is the total variation bounded (TVB) modiﬁed min max minmod function of Eq. (5.18) and o41 is a weight ui ouiðrrÞoui , (5.30) taken o ¼ 1:5. min max where ui and ui are the minimum and maximum Then cell-averaged solutions among all its neighboring ele- ð Þ X3 X3 ments sharing a face with the triangle T i and ui rs is the K0 K0 solution at any of the quadrature points. Violation of if Di ¼ 0; LPhuh ¼ u þ Difiðx; yÞ, i¼1 i¼1 Eq. (5.30) for any quadrature point indicates that the X3 element is close to a discontinuity and the solution in the if Dia0 compute, element is forced locally linear, i.e. i¼1 X3 X3 uiðrÞ¼ui þruiðr À riÞ8r 2 T i, (5.31) pos ¼ maxð0; D Þ; neg ¼ maxð0; ÀD Þ, i i where ri is the position vector of the centroid of T i. The i¼1 i¼1 magnitude of the solution gradient is maximized subject neg pos yþ ¼ min 1 À ; yÀ ¼ min 1; to the monotonicity condition given in Eq. (5.30). The pos neg original DG polynomial basis is used to compute an ARTICLE IN PRESS

264 J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 initial guess for the gradient as defined as Z Kn n Kn n j nK qcl j nK qui qui ðu ; u j Þ¼ D ðu ; u j Þ r ¼ Dlm h h kp h h ui ; . (5.32) Kn qxk qx qy j r2 qc Â m dK, ð5:35Þ Using Eq. (5.31) and the gradient computed by qxp Kn Eqs. (5.32), (5.30) may not still be satisfied. Therefore, j n where uh is the expansion in the element Kj , uhðx; tÞ¼ uiðrÞ is limited by multiplying by a scalar limiter f 2 P Kn K b n n j k¼1ukðKj ÞckðxÞ and uh is the solution in the element ½0; 1 so that the solution vector obtained from n which connect to the element Kj . The stabilization operator must be applied in areas of uiðrÞ¼ui þ fruiðr À riÞ (5.33) discontinuities and regions where the residual is large satisfies Eq. (5.30). The scalar limiter f in Eq. (5.33) is due to insufficient grid resolution. This information is, obtained by examining the numerical solutions at all however, available in the DG discretizations, and are quadrature points as follows: coupled to the jump in the solution across element faces Denoting by and the element residual. In the regions of smooth solution both the jumps and the residual are on the Dur ¼ piðrÞÀui (5.34) order of the truncation error. Therefore, two artificial 8 viscosity approaches were presented by van der van der > Du > min 1; r if Du 40; Vegt and van der Ven [222], a subsonic and transonic <> umax À u r i i! flowmodel, and a supersonic flowmodel. Full details of ¼ f >1 if Du 0; these models are given in [222]. > Dur ro :> min 1; min ui À ui otherwise: 5.8. DG space discretization of the NS equations

The compressible Navier–Stokes (NS) equations can 5.7. DG stabilization operator be written in compact vector form as follows qu High-order accurate DG finite-element solutions þrÁfiðuÞþrfvðu; ruÞ¼0, (5.36) qt (with polynomial bases of degree one or higher) do not guarantee monotonicity around discontinuities and where u is the vector of the conservative variables and fi, sharp gradients. The slope limiter of Cockburn et al. fv denote the inviscid and viscous flux functions. The [200] of the previous section guarantees monotone viscous flux fv is a linear function of the gradient ru and solutions for multidimensional scalar conservation laws Eq. (5.36) can be also written as and its extension to the Euler equations. This slope qu þrÁf ðuÞþrÁ½AðuÞru¼0. (5.37) limiter results in a robust numerical discretization and qt i has become quite popular. However, despite its robust- The discretization of the viscous, diffusive part of the ness the slope limiter of Cockburn et al. [200] has the NS equations with the DG method is less well known following serious disadvantages. It may result in an and different than the method described previously for unnecessary reduction in accuracy in smooth parts of the convective, inviscid part. the flow field and slows down or prevents convergence to A simple way to extend the scheme of Eq. (5.5), which steady state. Furthermore, its implementation for multi- was developed for convective problems of the form dimensional cases and high-order discretizations is very u þ u ¼ 0, for the diffusion equation u þ u ¼ 0isto intensive computationally. t x t xx simply replace u by ux and find u 2 V h such that Recently, van der Vegt and van der Ven [222] Z Z suggested that a better alternative for stabilization of d b À uðx; tÞfðxÞ dx ¼ uxfx dx ÀðuxÞjþ1=2fjþ1=2 the DG method is addition of artificial dissipation. This dt K K approach was also followed in the past by Cockburn þ þðubxÞ v , ð5:38Þ and Grenaud [223] and Jaffre et al. [224]. The stabiliza- jÀ1=2 jÀ1=2 tion operators make optimal use of the information where for the lack of an upwind mechanism for the contained in a DG discretization and preserve the diffusive term the numerical flux is the centered flux b 1 L R compactness of the DG method because they use ðuxÞjþ1=2 ¼ 2½ðuxÞjþ1=2 þðuxÞjþ1=2. the jump in the polynomial representation only at the Unfortunately, this simple but naive formulation element faces. leads to numerically stable but inconsistent solutions The stabilization operator D 2 R4Â4 of van der Vegt [201,202]. The numerical solutions seem to converge and van der Ven [222] is for quadrilateral elements and is with mesh refinement but have Oð1Þ errors compared to ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 265 exact solutions of the heat equation [202]. This is a described using a simple purely elliptic problem the pitfall of the DG method when it is applied directly to Helmholtz equation with Dirichlet and Neumann diffusion-type equations. The scheme of Eq. (5.38) is boundary conditions. The extension to the full NS therefore inconsistent because it produces stable but equations was carried out first. Next, an alternative completely incorrect solutions. scheme for ‘‘compact’’ DG approximation of elliptic A formulation of the DG method that is convergent problems was proposed [221]. This alternative scheme and consistent was used by Bassi and Rebay [203] for the overcomes limitations of the method associated with not compressible Navier–Stokes equations. A second suc- optimal accuracy for approximations with odd order cessful method that avoids the inconsistencies of the polynomials and the additional computing cost occur- simple formulation of Eq. (5.38) was presented by ring form the introduction of the auxiliary variable S, Baumann and Oden [204] who added extra penalty The auxiliary variable S is obtained in terms of uh at the terms to the inner boundaries. cost of a block diagonal mass matrix inversion from the The consistent formulation of Bassi and Rebay [203] first equation. However, the formulation of the second for the spatial discretization of the viscous term in the equation, which is obtained in this manner, and contains NS equations was constructed by resorting to a mixed the primal variable alone involves an ‘‘enlarged stencil’’. finite-element formulation. The second-order derivatives This enlarged stencil occurs because the primal un- of the conservative variables required for the viscous known uh for any internal element e is coupled not only terms were obtained by using the gradient of the with unknowns of the neighboring elements but also conservative variables, ru ¼ SðuÞ, as auxiliary un- with unknowns associated to the neighbors of the knowns of the NS equations. The NS equations were neighbors (because the second derivative is evaluated therefore reformulated as the following coupled system as the first derivative of a first derivative). The enlarged to the unknowns S and u. stencil implies additional computing cost. Bassi et al. [221] showed that only the jump contribution to the S Àru ¼ 0, auxiliary variable S is responsible for the non-compact qtu þrÁfiðuÞþrÁf vðu; SÞ¼0. ð5:39Þ support of the scheme described and they propose The weak formulation of the first equation of the modifications that one can use to arrive at a scheme system of Eq. (5.39) is with compact support for elliptic problems and for Z I Z DG discretizations of the viscous terms in the NS equations. Shf dx À uhnf dG þ uhrf dx ¼ 0, (5.40) K e K The modification of the newscheme for the discretization of the elliptic operators follows more closely the where the term uhn in the second (contour) integral of À þ numerical flux function ideas usually employed in Eq. (5.40) is replaced by a numerical flux Hsðu ; u ; nÞ. This numerical flux is a centered flux given as the finite-volume (FV) schemes. A typical treatment of an average between the two interface states elliptic operator in FV schemes is often based on the definition of auxiliary staggered control volumes enclos- À þ 1 À þ Hsðu ; u ; nÞ¼2ðu þ u Þn. (5.41) ing the boundaries of the primal control volumes that are used to construct the diffusive terms, which are The computed auxiliary variables Sh are used to form the second equation of the system in Eq. (5.39) as the analogue to the vector flux S used in the DG follows formulation. Z I d 5.9. The DG variational multiscale (VMS) method uhf dx þ fiðuhÞÁnf dG dt KZ e I The VMS method is a newapproach for LES À f ðu Þrf dx þ f ðu ; S ÞÁn dG i h v h h proposed by Hughes et al. in a series of papers ZK e [225–227]. The VMS method was subsequently further À fvðuh; ShÞrf dx. ð5:42Þ clarified by Collis [228] and applied with the DG method K [229,230] for LES of compressible turbulence. More In Eq. (5.42), the term fvðuh; ShÞÁn is replaced with the recently the VMS method was further developed [231] following centered numerical flux for mixed-type, Fourier-spectral/finite-volume formula- À À þ þ tion. The ideas behind the VMS method and its hvðu ; S ; u ; S ; nÞ implementation with the DG space discretization ¼ 1½f ðuÀ; SÀÞþf ðuþ; SþÞ Á n. 2 v v method are briefly described. A more detailed presentation and analysis of the DG The dynamics of turbulent shear flows are dominated discretization of the viscous part was recently presented by the motions of a small number of relatively large- by Bassi et al. [221]. In Ref. [221], the main points of the scale structures. The separation between the large, DG discretizations of second-order derivatives were energy containing scales and the smallest turbulent ARTICLE IN PRESS

266 J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 scales increases as the Reynolds number increases. The The weak form of Eq. (5.43) is increase in range of scales prevents DNS from being a Z Z T T u i viable tool beyond very simple, lowReynolds number W U t dx þ Wj ðFj À FjÞ dx flows. In contrast, LES attempts to exploit the scale Oe I Oe separation in turbulent shear flows by representing the þ W Tð i À v Þ ¼ ð : Þ Fne Fne dS 0, 5 44 largest scales as accurately as possible on the computa- qOe tional mesh and using a model to account for the where F ¼ F n , n is the outward unit normal vector influence of the unresolved smaller scales. Important n j j e on qO , and W is a continuous weighting function in the progress in turbulent flow simulations with LES was e element O . made, primarily with the application of the so-called e Following the VMS approach of [228] the following dynamic model (see [4–6,233]). It was, however, soon three-level multiscale framework is used to allow direct recognized that the Reynolds numbers for which LES monitoring of unresolved scales. can be applied using this approach are still far too low. As a result, LES is not still an economically feasible U ¼ U þ Ue þ Ub. (5.45) alternative for the simulation of the vast majority of In the partition of Eq. (5.45), of the exact solution U, U engineering flows. The deficiencies of RANS or even e b DES to accurately predict complex separated flows and represent the large scales, U are the small scales, and U the high computing cost of LES for flows of practical are the unresolved scales. This partition into scales is interest lead to the quest of newapproaches [225,232]. depicted in Fig. 92 as a range of Fourier modes in These approaches overcome the weaknesses of LES and wavespace. The large scale equations have no modeling RANS while providing consistency with DNS. The terms while the small scale equations have modeling Discontinuous Galerkin/Variational Multi-Scale (DG/ terms that can range from Smagorinsky closure to a full VMS) method [229] which merges VMS turbulence Reynolds stress model. The large and small scales are modeling [225] with the high-order accurate DG referred to as resolved scales. discretization is a particular synergistic combination Assuming that the basis is orthogonal and substitut- that offers a number of advantages over traditional ing the partition of Eq. (5.45) in the weak form of methods. Eq. (5.44), which in condensed form is BðW; UÞ¼0, The advantages of the VMS/DG approach for LES obtain for the large-scale equations are: BðW; UÞÀCðW; U; UeÞÀRðW; UeÞ ¼ CðW; U; UbÞþRðW; UbÞþCðW; Ue; UeÞð5:46Þ (1) Variational projection with a priori scale separation [225] avoids problems associated with spatial filters. and for the small-scale equations obtain (2) The method converges to the exact DNS. B0ðWe ; U; UeÞÀRðWe ; UeÞ (3) The method is high-order with the potential for e e b e b e e b exponential (spectral) convergence. ¼ RðW; UÞþCðW; U; UÞþRðW; UÞþCðW; U; UÞ. (4) VMS/DG is insensitive to grid quality and suitable ð5:47Þ for complex domains. (5) The method allows for different models to be used in ~ ~ different regions of the flowwhilestill retains formal UU= + U convergence to the exact solution. (6) The VMS/DG approach for turbulence simulations unifies traditional DNS, LES and RANS approaches in a single computational tool.

~ ˆ The VMS/DG method is brieﬂy described next. Large, U Small, U Unresolved, U Consider the compressible Navier–Stokes equations

in strong conservation lawform Energy( k )

i v NðUÞ¼Ut þ F j;j À F j;j ¼ 0, Resolved Unresolved Uðx; 0Þ¼U0ðxÞ, ð5:43Þ where U ¼½r; ru; rET is the conservative variable i k vector F jðUÞ is the inviscid ﬂux vector in the j coordinate ~ ˆ v k k direction, and F j ðUÞ is viscous ﬂux vector in the j coordinate direction. Fig. 92. Wave space decomposition for the VMS method. ARTICLE IN PRESS

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The analysis for the general nonorthogonal basis can The large-scale equation is an approximation because be found in [228]. the effect of the unresolved scales has been ignored. The In Eqs. (5.46) and (5.47) RðW; UeÞ is the generalized modeled large-scale equation takes the form of the exact Reynolds stress, CðW; Ue; UbÞ is the generalized cross- equation only when all scales of motion are contained stress, B0ðW; U; UeÞ is the operator BðW; UÞ linearized within the resolved scales. In this case, the VMS method about U for the linear perturbation Ue. is a DNS. However, by neglecting the influence of the The effect of the unresolved scales on the large scales unresolved scales on the large scales, the modeled large- is seen in Eq. (5.46) which contains the unresolved scale equation has no direct modeling terms. The model Reynolds stress projection onto the large scales simulta- in the small scales indirectly influences the large scales. neously with the large- and small-unresolved generalized This is achieved through the small-scale Reynolds and cross-stresses projections onto the large scales. The cross-stresses. Thus, if the exact solution is fully small-scale equation also contains the unresolved represented by the large scales, then the solution to the Reynolds stresses and cross-stresses. For truncated or modeled equations (5.49) is exact. This consistency is an discrete approximations, the combined small and large important advantage over classical methods [225]. The scales are identified as the resolved scales (see Fig. 92) model applied to the large-scale equation is nothing e and denoted as U ¼ U þ Ue. Therefore, the resolved more than the standard approach used in a Galerkin scale equation (Eq. (5.46)) is written more compactly as method where the projection of the residual of the unresolved scales onto the large scales is zero. Con- f e f b f e b BðW; UÞ¼RðW; UÞþCðW; U; UÞ. (5.48) sidering the simplified case where the bases are The equations for small- (Eq. (5.47)) and large-scales orthogonal, this amounts to weak enforcement of zero (Eq. (5.46) or (5.48)) indicate the need to model unresol- unresolved Reynolds/cross-stresses on the large scales. ved Reynolds and cross-terms appearing on the right- This indicates the particular discretizations play a role in hand side. The modeling assumptions introduced are: altering the model and therefore the results. In the small-scale equation, it is the projection of the unresolved Reynolds and cross-stresses onto the small b (1) The unresolved scales U have negligible direct scales that is modeled using a weak implementation of a influence on the dynamic evolution of the large subgrid-scale model. Again, use of different methods for scales U. Therefore, for sufficient a priori scale the small-scale discretizations is expected to alter the separation, the right-hand side of Eq. (5.46) is small, model. This is an advantage of the VMS framework. b b e b e.g. CðW; U; UÞþRðW; UÞþCðW; U; UÞ0. Although the model is specified without regard to the b (2) The unresolved scales U, however, are expected to specific discretization, the influence of discretization is e significantly influence the small scales U. As a result, obvious in the modeled equations. This fact has not been an appropriate model is needed for the right-hand side completely appreciated in the traditional turbulence e b terms of Eq. (5.47). Therefore we set CðW; U; UÞþ modeling community until recently when it was realized e b e e b e e RðW; UÞþCðW; U; UÞ¼MðW; U h; UhÞ. that differences between the discretization with the ‘‘same model’’ might be as large or larger than The VMS formulation can support a wide range of differences in ‘‘models’’ using the same discretization. models. In addition, a key feature of the VMS method is In the VMS framework, the influence of different that different models can be used in different spatial discretizations is evident and the choice of discretization locations. As a result, the model used depends on the clearly plays an important role in the success of particular flowcharacteristics, the desired fidelity, particular models. Therefore an important area for and the location of the particular subdomain under future research is to explore different bases for use in consideration. defining large and small scales. The modeled large and Using modeling assumptions (1) and (2) the equations small scales in Eqs. (5.49) and (5.50) are combined as for the large and small scales become: follows: e e BðW; U hÞÀCðW; U; UÞÀRðW; UÞ f e f BðW; U Þ¼MðWe ; U ; Ue ÞþðW; SÞ on O . (5.51) e h h h e ¼ RðW; U hÞ, ð5:49Þ The model of Eq. (5.51) can be implemented with VMS 0 e e e e B ðW; U h; UhÞÀRðW; UhÞ by incorporating the additional model term into the small-scale equations. This model term can depend on ¼ RðWe ; UÞþMðWe ; U ; Ue Þ, ð5:50Þ h h both the large and small scales. Therefore it can take where the subscript h in Eqs. (5.49) and (5.50) is forms ranging from classical Smagorinsky model to introduced to denote that these equations contain models used in DES. modeling errors and that the approximate numerical In the proceeding paragraphs, the variational multi- solution introduces discretization errors. scale method was presented for a typical subdomain Oe. ARTICLE IN PRESS

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Summing Eq. (5.51) over all the subdomains in the as an extended first-order system of equations [206]. partition Ph of O and introducing typical continuous This mixed approach has been demonstrated by Lomtev smooth finite-dimensional spaces for U h and the test et al. [235] for two- and three-dimensional unsteady function Wh then, subject to appropriate boundary flows. Unfortunately, the mixed approach in three- conditions and the particular choice of model, obtain the dimensions requires 6 additional unknowns and equa- standard variational multiscale method for classical tions for three-dimensional Navier–Stokes flows. finite elements, spectral elements, or global spectral Furthermore, in applications, the model terms, methods (see e.g., Refs. [225,227–234]). Results from MðWe ; U; UeÞ, often take the form of diffusive terms these methods are quite new, they have shown tremen- (eddy diffusivity models) that may require the addition dous potential including the ability to accurately of even more unknowns. Over the past few years, there simulate wall-bounded and non-equilibrium turbulence has been extensive research on the use of DG methods using a very simple constant coefficient Smagorinsky for elliptic and mixed hyperbolic/elliptic problems (see model on the small scales [227]. However, most of the e.g. [203,204,206,236]). Furthermore, Arnold et al. [237] results presented to date use global spectral methods, showed that the flux formulation, commonly used in which have a rich function space but are only feasible for DG methods, can be readily converted to the primal very simple geometries [226,227]. The preliminary work formulation. of Jansen [234] and co-workers offsets this by using a With this background, the variational multiscale low-order (cubic and lower) hierarchical basis within a method described above can be merged with a DG C finite-element method. While this method can be method. Denoting the boundary of the domain O as applied to complex geometries, the relatively low-order qO ¼ GD [ GN where GD is the portion of the boundary function spaces may not have a sufficient scale separa- where Dirichlet conditions are specified and GN is the tion for effective turbulence simulation. Perhaps of even portion of the boundary where Neumann conditions are greater importance however, is that the reliance of prior set. The element boundary is denoted as G ¼ approaches on C or smoother function spaces limits fGD; GN; G0g where G0 are the inter-element boundaries. ones ability to alter the large/small partition or change Let O1 and O2 be two adjacent elements. Furthermore, ð1Þ ð2Þ the form of the model as a function of space. For let G12 ¼ qO1 \ qO2; and n and n be the correspond- example, in laminar regions of a flowno model should ing outward unit normal vectors at that point. Denoting ðeÞ ðeÞ be used, while in boundary layers a RANS type model U and Fi be the state vector U and flux vectors Fi, may be appropriate (if you only are concerned with the respectively, on G12. Then, define the average h:i and mean flow) while in a wake region an LES type model jump ½: operators on G12 as may be used to capture the large-scale unsteadiness. ð1Þ ð1Þ ð2Þ ð2Þ To address these limitations, the DG/VMS method ðaÞ½Uni¼U ni þ U ni , introduced in [229] combines VMS turbulence modeling ð1Þ ð1Þ ð2Þ ð2Þ ðbÞ½Fn¼Fi ni þ Fi ni , with a DG method in space. The combination of ðcÞhUi¼1 ðUð1Þ þ Uð2ÞÞ, these two methods is particularly synergistic and the 2 1 ð1Þ ð2Þ combined DG/VMS method possesses the following ðdÞhFii¼2 ðFi þ Fi Þ, ð5:52Þ characteristics: where Fn ¼ Fini. With this notation and Eq. (5.51) the discontinuous high-order (even exponential) convergence on highly Galerkin formulation of the BðW; UÞ term, defined in irregular unstructured meshes; Eq. (5.46), is applied to obtain the DG formulation for discretization, large/small partitioning, and model the Navier–Stokes equations as Z equations can be changed on each subdomain, Oe; X T T v all boundary conditions are set weakly through BDGðW; UÞ¼ ðW U;t þ W ;iðFi À FiÞÞ dx O boundary fluxes including fluxes of turbulence stress. OeZ e This enables one to directly enforce zero turbulent T bv b À ð½W nihFi À Fii stress at solid walls—a feature not present in prior G approaches; T b b ÀhðDiWÞ i½ðU À UÞniÞ ds the method is highly localized leading to a great Z T bv b degree of parallelism which is required for the large- À ðhW i½Fn À Fn scale turbulence simulations we are targeting. G0 T b À½ðDnWÞ hðU À UÞniiÞ ds, ð5:53Þ v v The additional complication with the VMS/DG where FnðUÞ¼FiðUÞni; and FnðUÞ¼Fi ðUÞni ¼ DnU: method is that the typical approach for diffusive Quantities with a hat in Eq. (5.53) are numerical problems has been to introduce auxiliary variables for fluxes that must be appropriately defined. For example, À þ the viscous fluxes and re-write the equations of motion the term FiðU ; U Þ is an appropriate approximate ARTICLE IN PRESS

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Riemann flux (see previous sections for various options). Performing linearization of the fluxes F and Fnum The particular choice of Riemann flux plays an obtain important role in determining the dispersion/dissipation 2 characteristics of the method [210]. For example, using FðUi þ DUiÞ¼FðUiÞþAðUiÞDUi þ qðDt Þ the Steger–Warming flux-vector splitting where FnðUÞ is FnumðUi þ DUi; Uj þ DUj; nÞ À þ split into inflowand outflowcomponents F and F as 1 2 2 n n ¼ FnumðUi; Uj; nÞþA DUi þ A DUj þ qðDt Þ, ð5:58Þ follows: ij ij

Æ Æ Æ 1 where Fn ðUÞ¼Rl LU; K ¼ 2ðK ÆjKjÞ (5.54) 1 qFnumðu;v;nÞ 2 qFnumðu;v;nÞ and the approximate Riemann flux in this case is simply qFðUÞ A ¼ qu ; A ¼ qv ; Fb ðUÀ; UþÞ¼FþðUÀÞþFÀðUþÞ where UÆ ¼ lim AðUÞ¼ ; n n n !0 qU A1 ¼ A1ðu ; v ; nÞ; A2 ¼ A2ðu ; u ; nÞ þUðx Æ nÞ. Similarly, various options are available ij i j ij i j for the numerical viscous fluxes [237] and a particularly simple approach is the interior penalty method Ub ¼ with these definitions dropping terms of higher than bv v hUi; F ¼hFi iÀm½Uni where m40 is a stabilization second order in Eq. (5.56) obtain parameter [238,239]. Thus, the DG method is: Z n Z qU n n i Given U0 ¼ U0ðxÞ, for t 2ð0; TÞ, find Uðx; tÞ2 i V i dO À AðU ÞDU ÁrV dO 1 l i i l VðPhÞÂH ð0; TÞ such that Uðx; 0Þ¼U0ðxÞ and Oi qt Oi X Z e e 1 n 2 i BDGðW; UÞ¼MDGðW; U; UÞþðW; SÞ, (5.55) þ ðAijDU i þ AijDU jÞV l dS j qOi 8W 2 VðPhÞ where VðPhÞ is the broken space defined in n ¼ Ri;lðU Þ, ð5:59Þ [204].IfVðPhÞ is restricted to a space of continuous functions, then one recovers the classical Galerkin Z approximation. Further details about turbulence mod- R ðUnÞ¼ FðunÞÁrV i dO eling with the VMS approach and additional implemen- i;l i l Oi I tation issues can be found in the original references X n n 2 [225–229]. À Fnumðui ; uj ; nÞV l dS. ð5:60Þ j qOij b b b 5.10. Implicit time marching of DG discretizations Let U ¼ðU1; ...; UN Þ be the expansion coefficient of e b b b the approximate solution where U i ¼ðUi;1; ...; Ui;N Þ. An implicit time marching algorithm based on the XN backward Euler integration scheme was presented in b i UiðxÞ¼ Ui;lV l ðxÞ; i ¼ 1; ...; Ne. (5.61) [240] for DG discretizations. This algorithm is uncondi- l¼0 tionally stable in two dimensions and provides time accuracy only when its time step resolves the temporal The implicit scheme of Eq. (5.59) is written in matrix scales of the problem. The advantage of the uncondi- form as tional stability is lost in the case of DNS of transitional n MðUnÞDUb ¼ RðUnÞ, and turbulent flows where the time scales that need to n be resolved are often comparable to the time step n D qRðU Þ MðU Þ¼ À , ð5:62Þ imposed by explicit schemes. Unconditional stability is Dt qUb exploited to obtain steady-state solutions with the DG where D denotes the block diagonal mass matrix discretization. D ¼ diagðd1; ...; dNe Þ. Application of backward Euler time integration to the Z weak form of the Euler or Navier–Stokes equations with i i the DG discretizations yields ½dikl ¼ VkVl dO. (5.63) Z Z qUn i vi dO À FðUn þ DUnÞÁrvi dO The spatial accuracy of the solution depends on the qt l i i l Oi I Oi discretization of the residual as for the implicit schemes X presented in Section 2 in the finite difference context. þ F ðUn þ DUn; Un þ DUn; nÞVi dS ¼ 0, num i i j j l Quadratic convergence and large-time steps are possible j qOij only when the spatial discretization of the left-hand l ¼ 1; ...; N, ð5:56Þ side of Eq. (5.62) is consistent with the discretization of where the superscript for the state variable indicates the the right-hand side. The linear system of Eq. (5.62) time level and can be solved with the generalized minimum residual (GMRES) Krylov method [241] with single- or two-level n nþ1 n DUi ¼ Ui À U i . (5.57) preconditioning. ARTICLE IN PRESS

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A second-order accurate in time implicit Runge–Kut- 5.12. Results with the DG method ta method can also be used for time integration of DG discretizations [221] as follows: Second- fourth- and sixth-order accurate numerical Considering the system of the DG discretization solutions for wave propagation on triangular meshes were computed in [244] using polynomial base functions du D À RðuÞ¼0, (5.64) P1, P3, and P5, respectively. The accuracy of the dt numerical solutions was evaluated by comparing the where D is the mass matrix the second order implicit computed results with exact solutions. The first test Runge–Kutta method involves two backward Euler problem with an exact solution [245] was propagation steps and can be written as and reflection from a solid wall of a Gaussian pressure 2 2 nþ1 n pulse given by pðx; yÞ¼expfÀ ln 2½x Àðy À y0Þ =Wg, uj ¼ ui þ g1K1 þ g2K2, (5.65) where W is the width of the pulse and y0 is the distance from the wall. The second problem with n D qRðu Þ n an exact solution is scattering of a similar þ a ; K1 þ Rðu Þ¼0, (5.66) Dt qu Gaussian pressure pulse from the surface of a cylinder [246]. D qRðunÞ The solutions of the linearized Euler equations [244] þ a ; K þ Rðun þ bK Þ¼0 (5.67) Dt qu 2 1 were computed with third-order polynomial basis on a relatively coarse, fully unstructured mesh. The computa- the constants a; b; g and g corresponding to an optimal 1 2 pffiffiffi tional mesh is shown in Fig. 93. At the far-field second-order accurate scheme [242] are a ¼ð2 À 2Þ=2, boundaries, the radiation boundary condition [86] was b ¼ 8að0:5 À aÞ, g ¼ 1 À 1=8a, g ¼ 1 À g . 1 2 1 used. It can be seen that the pressure waves exit the The backward Euler scheme of Eq. (5.61) is recovered computational domain undistorted and there are no by g ¼ 1, g ¼ 0 and a ¼ 1. The Crank-Nicolson 1 2 1 reflections in the interior from the computational algorithm is obtained for g ¼ 1, g ¼ 0, a ¼ 1. 1 2 2 boundaries. Acoustic disturbance propagation is isotropic and does not require use of meshes with pattern 5.11. p-type multigrid for DG [210]. It appears that the unstructured mesh of Fig. 93 is more appropriate for acoustic wave propagation. For A p-type multigrid acceleration method was devel- convenience, evaluation of the numerical method is oped [207] for convergence acceleration with the G performed with solutions computed on meshes obtained method. A simple backward Euler discretization in time from triangulation of structured Cartesian-type grids. was used [207] so that the discrete equation for time The elements of the mesh for the solution of Fig. 94 advancement is followa uniform triangular-mesh generating pattern. It was shown in [210] that the accuracy of the computed 1 M ðunþ1 À unÞþRðunþ1Þ¼0, (5.68) solution depends on the triangular-mesh generating Dt pattern. Comparing the computed solutions of Figs. 93 where M is the mass matrix, R is the residual vector, and and 94 it appears that the coarse, canonical grid solution u are the degrees of freedom to be evolved in time from ðDx ¼ 2Þ obtained with the third order P3 polynomial time level n to n þ 1. basis shows more distortion than the fully unstructured For steady-state solutions the nonlinear system RðUÞ¼0 is solved using a p-multigrid scheme with a linear Jacobi smoother. A generic iterative scheme for Eq. (5.68) can be written as unþ1 ¼ un À PÀ1RðunÞ, (5.69) where the preconditioner, P, is an approximation to qR=qU. In p-multigrid the low-frequency error modes can be effectively corrected by smoothing with lower order expansion of the approximate solution that serve as the worse grids [243]. p-multigrid fits naturally with the frame work of high-order DG discretizations. There is no need to store additional grid information since the same spatial grid is used by all levels. The transfer operators between the different p expansions, prolonga- tion and restriction, are local and are stored for the Fig. 93. Solution computed with an unstructured, triangular reference element. mesh and third-order polynomial basis. ARTICLE IN PRESS

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Fig. 94. Comparison of the pressure field computed with Dx ¼ 2 triangular elements and third-order polynomial basis (right) with the exact solution (left). mesh of Fig. 93 that has comparable resolution. The P1,2nd order upper part of Fig. 94 shows, however, that for fine 3 th 0.1 P ,4 order grids and high-order accuracy the bias introduced P5,6th order by the triangular-mesh generating pattern is very 0.08 Exact small. This is again consistent with the theoretical 0.06 analysis of [210]. Solutions shown in the following paragraphs were 0.04 computed [244] until final time T ¼ 25 on triangular 34 36 38 canonical meshes with mesh generating pattern (see 0.02 Fig. 95). Comparisons are carried out with the exact 0 solution given in [245]. A comparison of the solutions Pressure, p computed with grid spacing Dx ¼ 2:0 is only shown in -0.02 ¼ Fig. 95 because within plotting accuracy for Dx 1:0 -0.04 the differences between the fourth- and sixth order- accurate solutions are not visible. The same time step -0.06 Dt ¼ 0:01, which is below the stability limit of the 10 20 30 40 Distance from the wall, y Runge–Kutta method was used for all solutions. The comparison with the exact result of Ref. [245] is shown Fig. 95. Comparison of the pressure computed with P1, P3, and along the symmetry line, which is normal to the wall at P5 polynomial bases with the exact solution; in all cases the same canonical triangular mesh with Dx ¼ 2:0 is used. x ¼ 0. For the same location, the error er ¼ðpcom À pexÞ of the computed solutions is shown in Fig. 96. Clearly, only for Dxo1:0 the results computed with In Fig. 97, the grid convergence of the second-, the first-order polynomial basis (second-order accurate fourth-, and sixth-order accurate solutions is shown. solution) provide the accuracy level needed in aero- The error norm in Fig. 97 was computed on the acoustic computations. symmetry line, where the error norm is expected to be ARTICLE IN PRESS

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P1 error, ∆ x = 1 -1 10 3 0.03 P1 error, ∆ x = 2 P P3 (error x 100), ∆ x = 2 P5 P5 (error x 1000), ∆ x = 2 P1 0.02

10-2 0.01 comp -P

0 exact

-3 Log(||e||L2) 10 -0.01 Error, P

-0.02 10-4 -0.03 10 20 30 40 50 0123 Distance from the wall ComputingTime

Fig. 96. Error of the solutions computed with P1, P3, and P5 Fig. 98. Computing time required in order to achieve certain polynomial bases and canonical triangular meshes with Dx ¼ 2. error level (L2 error norm of the computed pressure) with second- (P1), fourth- (P3), and sixth-order (P5) accurate solutions.

10-1 P3 Exact 5 P P1, 2nd order P1 0.1 P3, 4th order slope = 1.4 0.08 -2

10 slope = 4.1 0.06

0.04

10-3 0.02 norm of the error 2 Pressure, p L 0

-0.02 slope = 6.4 10-4 -0.04 012345678 -0.06 Grid spacing log(Dx) 10 20 30 40 50 Distance from the wall Fig. 97. Grid convergence (in the L2 norm of the computed 1 3 pressure) of the second- (P ), the fourth- (P ), and sixth-order Fig. 99. Comparison of pressure ﬁeld computed with quad- 5 (P ) accurate solutions computed with triangular elements. rilateral canonical meshes and P1, P3 polynomial bases with the exact solution. the largest, and not for the entire domain. Norms of the order methods. Furthermore, it was found that for this error computed in the full domain do not differ much simple problem the solution computed with fourth- or from the error norm obtained from the deviation of the sixth-order accuracy and single precision practically computed solution from the exact result only on the converges when Dx ’ 1:0 and the remaining errors symmetry line. The grid convergence plot of Fig. 97 are mainly due to time integration. In Fig. 98, the shows that all solutions achieve the expected order of reduction of the average error is plotted versus the accuracy. Once again it is evident that the desired order required computing time for the solution. It can be seen of accuracy for CAA can only be achieved with higher- that use of higher-order accuracy yields savings in ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 273 computing time when the solution must reach certain basis, for example, the unit area (for Dx ¼ dy ¼ 1) is error level. spanned by 16 polynomials in quadrilateral elements Solutions computed with quadrilateral meshes are while for triangular element discretization the same area presented next. A comparison of the pressure ﬁeld is covered by two elements and spanned by twenty 3 computed with third-order polynomial basis with the (2 Â Pj¼1;...;10 ¼ 20) polynomials. In terms of computing exact result is shown in Fig. 99. It was found [244] that efﬁciency, the solution obtained with quadrilateral the solutions computed with quadrilateral elements do elements requires slightly less computing time. not achieve the ðk þ 1Þth order of accuracy. This is expected because in a computation with third-order

Fig. 100. Comparison of the computed pressure ﬁeld obtained Fig. 102. Computed pressure ﬁeld with triangular mesh and P5 with P3 quadrilateral elements with the exact solution for polynomial basis for scattering of a pressure pulse from the scattering from a cylinder surface. surfaces of two cylinders.

Fig. 101. Comparison of the computed pressure at r ¼ 5:0 and 0pTp8 for f ¼ 0, and f ¼ 90 with the exact solution of Ref. [33] for pulse scattering. ARTICLE IN PRESS

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Solutions for more complex domains were also computed. First, scattering of sound waves from the surface of a cylinder, which is one of the benchmark problems of [246], is considered. The computed solution with third-order polynomial basis and quadrilateral elements at T ¼ 8 is compared qualitatively with the exact result in Fig. 100. Very good agreement with exact result is achieved for the solutions computed with both triangular and quadrilateral elements. The computational domain for the solution obtained with triangular elements contains 5437 elements or 2810 vertices. Discretization of the same domain with isoparametric quadrilaterals includes 2000 elements or 2091 vertices. In both cases the cylinder was represented by 40 elements. The time variation of the computed solution for 0pTp8 is shown in Fig. 101. Again good agreement with the exact result [246] is achieved. Next, scattering of a Gaussian pulse from the surfaces of two cylinders with different diameters is considered. The large cylinder has diameter D1 ¼ 2 and the small cylinder has diameter D2 ¼ 1. The distance between the centers is 8:5D2, the large cylinder center is at ðÀ5; 0Þ, the small cylinder center is at ð3:5; 0Þ and the source is located at ð0; 0Þ. The solution is computed using triangular elements because the meshing of the domain with triangular elements is easier and the resulting mesh is more isotropic. The surface of the large cylinder is represented by 40 elements and the surface of the small cylinder is represented by 20 elements. The entire domain contains 7671 elements or 3929 vertices and the solution is obtained with the p5 polynomial basis. The computed pressure field at time T ¼ 10 is shown in Fig. 102. Numerical solutions obtained with the DG finite- element method for benchmark aeroacoustic problems demonstrated that only p3 (fourth-order) or higher- order accurate discretizations provide the required resolution for CAA. In terms of computing time, solutions with equivalent order of accuracy obtained on triangular meshes require about 10% more resources than the solutions obtained with quadrilateral meshes. In both cases, the CFL stability limitations become more stringent, cf. [208] with the increase of the order of spatial accuracy. For isotropic meshes, the resolving ability of both triangular and quadrilateral space discretizations is approximately equivalent. For two- dimensional problems, it is feasible to achieve spatial accuracy up to sixth order. Sixth- or higher-order of accuracy is possibly advantageous for long time propagation of complex waveforms. Numerical tests for pure convection ðut þ ux ¼ 0Þ of complex one- dimensional waves of the form uðx; tÞ¼0 ¼ ½2 þ cosðaxÞ exp½À ln 2ðx=10Þ2, a ¼ 1:7, using up to 10th-order accurate DG discretizations showed that for long time integration ðT4400Þ at least sixth-order accuracy is needed. Application of the DG finite-element Fig. 103. Multigrid levels in an adapted mesh about the NACA 0012 airfoil. ARTICLE IN PRESS

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Fig. 104. Adapted mesh around an oscillating NACA0012 airfoil, contours of density and surface pressure coefﬁcient Cp for a ¼ 3:77 (pitching upward) and a ¼ 4:0 (pitching downward); M ¼ 0:8, o ¼ p=10.

method for three-dimensional CAA applications is shown in Figs. 103 and 104. Very good resolution of straightforward using the available finite-element frame- transonic shocks was obtained with the stabilization work [42] but beyond the scope of the present paper. It operators. Adaptive grid refinement with ‘‘hanging appears, however, that three-dimensional CAA applica- nodes’’ that is possible for DG method yielded very tions with the DG method are feasible in terms of the good resolution of the shocks. required computing resources but discretizations with Computations of viscous turbulent flows with the DG polynomial bases of order higher than three are possibly method were recently presented by Bassi et al. [221]. too intensive computationally. Examples from these computations are shown in Computations of compressible flows over realistic Figs. 105–107. Fig. 105 shows the computational mesh aerodynamic configurations with the DG method were over turbomachinery blades. An implicit Runge–Kutta carried out by Van der Vegt and Van der Ven [247] and method was used for time marching in order to avoid the more recently in [222]. Examples of the recent computa- stringent stability limitations imposed by the small grid tions of transonic steady and unsteady airfoil flows of spacing required for the resolution of the near wall [222] using the space–time DG method and stabilization viscous phenomena. The unsteady flowfield is shown operators of Section 5.10 instead of slope limiters are with snapshots in Figs. 106 and 107. Additional results ARTICLE IN PRESS

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Fig. 105. Global viewand detail of the grid around the blade.

Fig. 106. Snapshot of Mach number and turbulence intensity computed with P2 elements. with the DG method can be found in other recent publications [248–256].

for high-order solutions occurs because of the large 6. The spectral volume (SV) method number of the polynomial coefficients that need to be determined and the evaluation of high-order surface and The discontinuous Galerkin method of Section 5 is volume integrals that are expensive to compute. A kth based on the finite-element framework and assumes a order DG method (k À 1 polynomial basis) requires high-order data distribution for each element. As a 2kth accurate quadratures for the line or surface result, the computed state variables are not continuous integrals and ð2k À 1Þth order accurate quadrature across the element boundaries and a Riemann solver is formula for the volume integrals. required to compute the fluxes through the element The spectral volume (SV) method is a newconserva- boundaries. The DG method is fully conservative and tive high-order accurate numerical method developed by does not depend on grid smoothness. It was mentioned, Wang in a series of papers [46,257,258]. The SV method however, in Section 5 that the main disadvantages of the is a finite-volume method for unstructured grids that DG method are the high computing cost and the unlike other FV methods does not require information deterioration of the CFL stability with the increase of from neighboring cells to perform reconstruction. The the order of approximation. The high computing cost SV method like other FV methods and the DG method ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 277

Fig. 107. Shedding from the blade trailing edge shown with density contours.

is suitable for discontinuity capturing with the applica- The domain O is discretized into N nonoverlapping tion of TVD or TVB limiters. cells Si called spectral volumes, which are the same as the usual finite volumes. The integral form of the scalar 6.1. Fundamentals of the SV method conservation lawis Z I qu The basic idea of the spectral volume method [46] is dV þ ðF Á nÞ dA ¼ 0, (6.2) qt presented for the following multidimensional conserva- Si qSi tion lawfor the scalar uðx; y; tÞ where n is the outward unit normal on qSi. The cell- averaged state variable for the spectral volume S is qu qf ðuÞ qf ðuÞ i þ þ ¼ 0 (6.1) defined in the usual way as qt qx qy Z 1 in the domain O with flux F ¼ðf ; gÞ, initial condition ui ¼ u dV, (6.3) V i Si uðx; y; 0Þ¼u0ðx; yÞ, and appropriate boundary conditions. where V i is the area or volume of the spectral volume Si. ARTICLE IN PRESS

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The discrete formula of the integral conservation law, Eq. (6.2), for a spectral volume with M faces is expressed as Z du 1 XM i þ ðF Á nÞ dA. (6.4) t V d i m¼1 Am The surface or line integral on each face or edge, respectively can be performed with a kth order accurate Gauss quadrature as follows Z Fig. 108. (a) A possible reconstruction stencil for a quadratic XJ reconstruction in high-order k-exact finite-volume scheme. (b) ðF Á nÞ dA ¼ omjFðuðxmj; y ÞÞ Á nmAm mj The partition of a spectral volume into six control volumes Am j¼1 supporting a quadratic reconstruction. þ OðhkÞ, ð6:5Þ where omj are the Gauss quadrature weights and xmj; ymj are the Gauss quadrature points. element while in the SV method is determined by A kth order accurate approximation of the state the number of subdivisions in control volumes. Possi- ble minimum CVs subdivisions of a SV for linear, variable in the spectral volume (SV) Si can be obtained with a ðk À 1Þth order polynomial in x and y as quadratic, and cubic data reconstructions are shown in Figs. 109–111, respectively. k piðx; yÞ¼uðx; yÞþOðh Þ; ðx; yÞ2Si. (6.6) The cell-averaged state variables ui for the spectral volumes S are updated from the time level n to the time This approximation of the state variable is in general i level n þ 1 with Eq. (6.8). The crucial step in the SV discontinuous across the SV boundaries unless the state method is that the update Du ¼ unþ1 À un must be variable is a ðk À 1Þth order polynomial or less. Similar i i i scattered back to the cell-averaged variables in the to the FV or DG methods, the flux integration on the control volumes c ; j in S . Then the same high-order two sides of the SV faces involves two disconti- i i reconstruction can be used again at the time level n þ 1 nuous state variables. Therefore, this flux integration is to the time level n þ 2, etc. The relationship between the carried out using an exact or an approximate Riemann cell-averaged variables for S and the cell-averaged solver i variables for the CV in Si is ð ð ÞÞ Á P F u xmj; ymj nm K k j¼1ui; jV i; j ffi F Riemannðpiðxmj; ymjÞ; pi;mðxmj; ymjÞ; nmÞþOðh Þ, ui ¼ , (6.9) V i ð6:7Þ where ui; j denotes the cell-averaged state variable of the where p is the kth order approximation of the exact i;m jth control volume ci; j of Si and V i; j is its volume.P The state variable in the neighboring cell. K total number of CV in Si is K and V i ¼ j¼1 V i; j. The kth order accurate scheme of the SV, Si is Furthermore, in order to ensure conservation the CVs XM XJ updates must satisfy dui 1 k P þ omjF R ðpi; pi;m; nmÞAm ¼ Oðh Þ. ð6:8Þ K dt V i Dui; jV i; j m¼1 j¼1 Du ¼ j¼1 . (6.10) i V For a high-order FV method, including FV ENO and i WENO schemes, the high-order polynomial approxima- A scattering scheme for high-order ðk42Þ reconstruction of the state variable on the cell under consideration tion is obtained as follows. Each CV inside a SV is (see Fig. 108a) is obtained using a stencil that contains treated separately to update the cell-averaged state the cell and sufficient number of neighboring cells. For a variable for the CV. Inside a particular SV, however, DG method, on the other hand, the polynomial form is the state variables are continuous across the interior assumed and the expansion coefficients are the variables. boundaries of the CV. Therefore, it is not necessary to In the SV method, instead of using a large number of use a Riemann flux or flux splitting for the boundaries of neighboring cells, the reconstruction is performed by the CVs. At the CVs boundaries analytic fluxes are used partitioning the SV into subcells called control volumes and Riemann fluxes are only necessary at the SV (see Fig. 108b). The order of accuracy of the SV method boundaries. High-order accuracy is preserved by is determined by the number of CVs in which each SV is using high-order quadratures not only for the Riemann subdivided. The weak analogy between the DG method fluxes at the SV boundaries, but also for the ana- and the SV method is that for the DG method the order lytical fluxes through the interior CV boundaries within of accuracy is related to the number of nodes on the the SV. ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 279

(2) Use of a kth order quadrature (exact for k À 1 polynomial) and a Riemann solver to compute the surface fluxes at the SV boundaries. (3) Use of a kth order quadrature and analytic fluxes to compute the surface fluxes at the CV boundaries. (4) Use of a suitable time marching scheme.

It appears that the SV method shares many of the DG method desirable features, e.g. it is compact, therefore suitable for parallel implementation, high-order accu- Fig. 109. Control volumes in a triangular linear spectral rate, conservative and capable of handling complex volume. (a) Type 1, n1 ¼ 0, n3 ¼ 1, n6 ¼ 0. (b) Type 2, n1 ¼ 0, geometries. In fact, Zhang and Shu [259] performed an n3 ¼ 1, n6 ¼ 0. analysis of the SV method considering the SV method as a Petrov–Galerkin ﬁnite-element method where the solution space and the test function space are different. The SV method requires larger number of ﬂux computations at the CV boundaries, while the DG method requires evaluation of volume integrals. The implementation of the SV method is presented in detail starting from the one-dimensional scalar conservation law.

6.2. SV method in one dimension Fig. 110. Possible triangular quadratic spectral volume parti- 1 1 tions. (a) Type 1 with d ¼ and Type 2 with d ¼ ; n1 ¼ 0, 3 4 Consider the one-dimensional conservation lawin n3 ¼ 2, n6 ¼ 0. (b) A singular partition; n1 ¼ 0, n3 ¼ 0, n6 ¼ 1. ½a; b quðx; tÞ qf ðuðx; tÞÞ þ ¼ 0. qt qx

uðx; 0Þ¼u0ðxÞ. (6.11) The domain ½a; b is divided into N nonoverlapping SVs

[N ½a; b¼ Si; Si ¼½xiÀ1=2; xiþ1=2, (6.12) n¼1

where xiþ1=2 À xiÀ1=2 ¼ hi. Given the desired order of accuracy k, for the numerical solution of Eq. (6.11), each spectral volume k Si is subdivided into k control volumes fxi; jþ1=2gj¼0 denoted ci; j with xi; jþ1=2 ¼ xiÀ1=2 and xi;kþ1=2 ¼ xiþ1=2 (see Fig. 112). The cell-averaged state variables for ci; j are R x i; jþ1=2 uðx; tÞ dx i ¼ 1; ...; N; xi; jÀ1=2 ui; j ¼ ; ð6:13Þ hi; j j ¼ 1; ...; k; Fig. 111. Possible cubic triangular spectral volumes. (a) Type 1, n ¼ 1, n ¼ 1, n ¼ 1. (b) Type 2 with d ¼ 1 and Type 3 with 1 3 6 6 where hi; j ¼ x þ À x À . 1 i; j 1=2 i; j 1=2 d ¼ 15; n1 ¼ 1, n3 ¼ 1, n6 ¼ 1. (c) n1 ¼ 1, n3 ¼ 1, n6 ¼ 1. (d) n ¼ 1, n ¼ 3, n ¼ 0. 1 3 6 6.2.1. Reconstruction for SV in one dimension The reconstruction problem is: Given the cell-aver- In summary, the main steps for a kth order (k À 1 aged values ui; j for all CVs in Si construct a polynomial polynomial reconstruction) SV method include: piðxÞ for Si of degree k À 1 the most such that piðxÞ is a kth order accurate approximation of uðxÞ inside Si, e.g. (1) Computation of the state variables at the quadrature k points. piðxÞ¼uðxÞþOðh Þ; x 2 Si; i ¼ 1; ...; N. ð6:14Þ ARTICLE IN PRESS

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= = Control Volumes Ci, j (xi, j-1/2, xi, j+1/2) , j 1,....,k At the control volume boundaries, Uðxi; jþ1=2Þ is exactly known from the cell averages u . with{ui,j} cell averaged values in each Ci,j i; j j Z X xi;rþ1=2 Ci-1, k Ci,1 Ci, j Ci, k Ci+1,1 Uðxi; jþ1=2Þ¼ uðxÞ dx r¼1 xi;rÀ1=2 h i,j Xj x + ≡ x x − x + x + ≡ x + i-1, k 1/2 i,1/2 i, j 1/2 i, j 1/2 i, k 1/2 i 1,1/2 ¼ u h ; j ¼ 1; ...; k i;r i;r r¼1 Si−1 Si+1 Si subdivision of each spectral volume Si Uðxi;1=2Þ¼0 for j ¼ 0: ð6:20Þ into no-equal size CVs, Ci,j The unique polynomial that interpolates Uðxi; jþ1=2Þ, 0 Fig. 112. Subdivision of the spectral volume Si into control j ¼ 0; ...; k is PiðxÞ. Then for piðxÞ¼PiðxÞ have volumes for the one-dimensional problem. Z Z 1 xi; jþ1=2 1 xi; jþ1=2 p ðxÞ dx ¼ uðxÞ dx h i h i; j xi; jÀ1=2 i; j xi; jÀ1=2

¼ ui; j; j ¼ 1; ...; k. ð6:21Þ In particular, piðxÞ gives the following approximation of 0 uðxÞ on the CV boundaries Therefore, piðxÞPiðxÞ is the polynomial of Eq. (6.14) that satisfies the accuracy requirement of ui; jþ1=2 piðxi; jþ1=2Þ Eq. (6.15). i ¼ 1; ...; N; The constants crj are obtained considering the k ¼ uðxi; jþ1=2ÞþOðh Þ; ð6:15Þ Lagrange interpolation polynomial PiðxÞ j ¼ 0; ...; k: Xk Yk x À x P ðxÞ¼ U i;mþ1=2 ; x 2 S , This reconstruction problem is solved with the same i i;rþ1=2 x À x i r¼0 m¼0 i;rþ1=2 i;rÀ1=2 method presented in Section 4 for the ENO reconstruc- mar tion. For completeness, the procedure of Section 4 is ð6:22Þ repeated here in SV context. 8 There is a unique polynomial of degree k À 1 at most > Xk < Xk whose cell average in each of the CVs in Si (see Fig. 112) 1 p ðxÞ¼ h satisfies i > i;r ½ðx À x Þ ...ðx À x Þ0 R r¼1 : q¼r i i;1=2 i;qþ1=2 xi; jþ1=2 piðxÞ dx 9 xi; jÀ1=2 > ¼ ui; j; j ¼ 1; ...; k. (6.16) => hi; j X Yk Xk Â ðx À xi;pþ1=2Þ>ui;r frðxÞui;r. This is the polynomial we are looking for in Eqs. m¼0 p¼0 ;> r¼1 (6.14) and (6.15). The mappings from given cell averages maq paq;m ui; j to the CV boundary values of the state variables are ð6:23Þ linear. Therefore, coefficients crj can be found, which Considering that p ðxÞ¼P0ðxÞ and substituting depend on the order of accuracy k and the CV size hi; j in i i Eq. (6.17) in the expression for p ðxÞ obtain Si but not on u itself, such that i Xk Xk 1 crj ¼ hi;r ui; jþ1=2 ¼ crj ui;r; j ¼ 0; ...; k. (6.17) q¼r ðx À xi;1=2Þðx À xi;3=2Þ ...ðx À xi;qþ1=2Þ r¼1 Xk Yk k u ; ; u As a result, given the cell averages i;1 ... i;k for Â ðxi; jþ1=2 À xi;pþ1=2Þ. ð6:24Þ the CVs in Si the reconstructed point values at the CV m¼0 p¼0 maq paq;m boundaries xi; jþ1=2, computed from Eq. (6.17), are kth order accurate For CV of equal size Eq. (6.24) simplifies to

k Xk Xk Yk ui; jþ1=2 ¼ uðxi; jþ1=2ÞþOðh Þ; j ¼ 0; ...; k. (6.18) 1 Q crj ¼ k ðj À qÞ. (6.25) The primitive function UðxÞ of uðxÞ is used to obtain q¼r p¼0 ðr À pÞ m¼0 p¼0 paq maq paq;m the constants crj in Eq. (6.17). The primitive function of uðxÞ is At the interior CV boundaries in Si the state variables Z x are continuous and it is not necessary to use a numerical UðxÞ¼ uðxÞ dx; x 2 Si. (6.19) ﬂux to update the reconstructed variables at the CV xi;1=2 boundaries. In other words, for the update at the ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 281 interior in [46] that it is necessary to cluster ‘‘grid points’’ near the boundaries of the SV interval to make the du 1 i; j ¼À ðfb À fb Þ (6.26) reconstruction less oscillatory. dt h i; jþ1=2 i; jÀ1=2 i; j One way to perform CV subdivision is to use the the numerical flux is the analytical flux function Gauss-Lobatto points for CV subdivision. For the b standard interval ½À1; 1 the Gauss-Lobatto points are f i; jþ1=2 ¼ f ðui; jþ1=2Þ; j ¼ 1; ...; k À 1. defined by At the SV boundaries xiÀ1=2 and xiþ1=2 or xi;1=2 and jp x ¼Àcos ; j ¼ 0; ...; k. (6.29) xi;kþ1=2 (see Fig. 112) the state variables are discontin- i; jþ1=2 k uous because at xiÀ1=2 for example there is one value due to the reconstruction in Si and another due to the Another clustering function is the hyperbolic tangent reconstruction in SiÀ1. Therefore, a numerical flux is function defined by computed at xiÀ1=2 with a Riemann solver 2bj b b b tanh À m f i;1=2 ¼ f RiemannðuiÀ1;kþ1=2; ui;1=2Þ¼f iÀ1;kþ1=2. (6.27) k xi; jþ1=2 ¼ ; j ¼ 0; ...; k, (6.30) The numerical flux must satisfy the following condi- tanhðbÞ tion to ensure monotonicity of the first-order scheme where b is a constant controlling the degree of clustering near the endpoints. Large values of b result in stronger (1) Locally Lipschitz and consistent fbðu; uÞ¼f ðuÞ, grid clustering. The value b ¼ 0:6 in Eq. (6.30) yields (2) fb is a nondecreasing function of its first argument, quite even distribution and the polynomials are highly (3) fb is a non-increasing function of the second oscillatory at the end points. For b ¼ 1:6 a grid similar argument. to the Gauss-Lobatto grid is obtained and the oscillations of the polynomials diminishes significantly. The Lax–Friedrich’s or the Roe flux can be used as It is well known that the Gibbs phenomenon numerical fluxes. Time marching of the semidiscrete associates with high-order schemes in the presence of form discontinuities causes loss of monotonicity in the du solution of hyperbolic conservation laws. The SV ¼ LðuÞ, (6.28) dt method is not excluded for Gibbs phenomenon. In 2 3 2 3 particular, since the lowest order reconstruction for the u1;1 L1;1ðuÞ SV method is linear (second-order accurate scheme), and 6 7 6 7 6 . 7 6 . 7 there are no linear second- or higher-order schemes 6 . 7 6 . 7 6 . 7 6 . 7 which guarantee monotonicity, some limiting approach 6 7 6 7 6 7 6 7 must be applied to the SV method to achieve mono- u ¼ 6 ui; j 7; LðuÞ¼6 Li; jðuÞ 7, 6 7 6 7 tonicity. In Ref. [46] the limiting approach originally 6 . 7 6 . 7 developed by van Leer [127,128], is applied. The van 6 . 7 6 . 7 4 5 4 5 Leer’s approach limits the reconstruction so that the uN;k LN;kðuÞ reconstructed solution is monotonic. Details about 1 limiting for the SV method in one dimension can be Li; j ¼À ½f i; jþ1=2ðuÞÀf i; jÀ1=2ðuÞ found in [46]. hi; j can be obtained with TVD RK schemes of Section 2. 6.3. Polynomial reconstruction in triangular SVs The reconstruction quality strongly depends on the choice for CV subdivision. The obvious choice of equal Further development of the SV method for two- size CVs subdivision of a spectral volume is not optimal dimensional scalar equations was presented by Wang for high-order ðk43Þ reconstruction. It was found in and Liu in [257]. The two-dimensional formulation of [46] that high-order reconstructions on uniform CV the SV method is becomes subdivisions are highly oscillatory near the two SV Z XK boundary grid points xi;1=2 and xi;kþ1=2. The source of du 1 i; j þ ðF Á nÞ dS ¼ 0, (6.31) this oscillatory behavior for high-order reconstructions t V d i; j r¼1 Sr was identified to be the oscillatory behavior of Lagrange R polynomials that interpolate non-polynomial functions where ui; j ¼ 1=V i; j u dV are the cell-averaged ci; j on equidistant points. Taking into account that La- variables for ci; j, K is the total number of faces (or grange polynomials are used for the interpolations of edges in 2D) in ci; j and Sr denotes the rth face Eqs. (6.22) and (6.23) it is expected that the subdivision of ci; j. Using a kth order accurate Gauss quadrature of a SV into equal size CVs is not optimal. It was found ðk ¼ m þ 1Þ for the evaluation of the surface integral ARTICLE IN PRESS

282 J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 in Eq. (6.31) imation space Pm of degree m polynomials in two Z dimensions is XJ ðF Á nÞ dS ¼ orqFðuðxqr; yrqÞÞ Á nrSr m þ 2 ðm þ 1Þðm þ 2Þ Sr q¼1 N ¼ ¼ . (6.38) m 2 2 k þ OðArh Þ, ð6:32Þ The reconstruction of u in Pm is achieved by where J ¼ðk þ 1Þ=2 is the total number of quadrature partitioning Si into a Nm nonoverlapping CVs, ci; j, points, orq are the Gaussian quadrature weights, and h is j ¼ 1; ...; Nm. The reconstruction problem is to find a the maximum length of all CVs obtain polynomial p 2 Pm such that Z m Z XK XJ dui; j 1 k p ðx; yÞ dV ¼ uðx; yÞ dV; j ¼ 1; ...; Nm. þ orq Fðuðxrq; yrqÞÞ Á nSr þ Oðh Þ. m t V c c d i; j r¼1 q¼1 i; j i; j (6.33) (6.39) e ðx; yÞ2P Assuming that a multidimensional polynomial in x Introducing the complete polynomial basis l m, he polynomial pm can be expressed as and y of degree k À 1 the most exists on Si which is the kth order approximation of the state variable in Si XNm p ¼ clel ðx; yÞ, (6.40) k m piðx; yÞ¼uðx; yÞþOðh Þðx; yÞ2Si. (6.34) l¼1 T This polynomial approximation yields discontinuous where e ¼½e1; ...; eN is the basis function vector and state variables across the SV boundaries (unless the cl, l ¼ 1; ...; N are the reconstruction coefficients, and variables are polynomials of k À 1 degree or less) and a substituting Eq. (6.40) in Eq. (6.39) obtain flux integration with an exact or approximate Riemann Z 1 XNm solver is needed, i.e. c e ðx; yÞ dV ¼ u ; j ¼ 1; ...; N . (6.41) V l l i; j m i; j l¼1 ci; j Fðuðxrq; yrqÞÞ Á nr ¼ F Riemannðpiðxrq; yrqÞ; pi;rðxrq; yrqÞ; nrÞ This equation is written in matrix form as þ Oðpi À pi;rÞ, ð6:35Þ ½Rc ¼ u, (6.42) where p denotes the reconstruction polynomial of a i;r T r where u ¼½ui;1 ...ui;N and ½R is the reconstruction neighboring CV ci; j which shares the face Ar with the ci; j and since both p and p are kth order accurate matrix given by i i;r 2 Z Z 3 approximations of the exact state variables the semi- 1 1 6 e1ðx; yÞ dV eN ðx; yÞ dV 7 discrete kth order accurate scheme for ci; j is 6 V V 7 6 1 Zci;1 1 Zci;1 7 XK XJ ½R¼6 7 dui; j 1 4 1 1 5 þ orq e1ðx; yÞ dV eN ðx; yÞ dV t V V N V N d i; j r¼1 q¼1 ci;N ci;N k ÂF Riemannðpiðxrq; yrqÞ; pi;rðxrq; yrqÞ; nrÞSr þ Oðh Þ (6.43) ð6:36Þ and the reconstruction coefficients in Eq. (6.40) are obtained by or in vector form c ¼½RÀ1u. (6.44) du ¼ R ðuÞ, (6.37) dt h Substituting of Eq. (6.44) in Eq. (6.39) yield an 2 3 2 3 expression of pm in terms of the basis functions as u1;1 R1;1ðuÞ 6 7 6 7 XNm u ¼ 4 ui; j 5; RhðuÞ¼4 Ri; jðuÞ 5, pm ¼ Ljðx; yÞui; j ¼½Lu, (6.45) j¼1 uI;N RI;N ðuÞ

½L¼e½RÀ1. (6.46) 1 XK XJ Ri; j ¼À orqF Riemannðp ðxrq; y Þ, The values of u at the quadrature points x , y within V i rq rq rq i; j r¼1 q¼1 the SV are given according to Eq. (6.45) by p ðxrq; y Þ; nrÞSr. i;r rq XNm Approximation of the solution can be obtained with pmðxrq; yrqÞ¼ Ljðxrq; yrqÞui; j. (6.47) any linearly independent functions. Reconstruction in j¼1 [257] is obtained with polynomial basis functions as in This equation expresses an interpolation of a functional Eq. (6.34). The dimension of the polynomial approx- value at a point using the cell-averaged values with ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 283 weights equal to the basis functional value evaluated at the corresponding point. Spectral volumes of various types, e.g. triangular, quadrilateral, or polygons of other types are possible choices. The special case where all SVs are triangular and all CVs are polygons with straight edges is considered. For this case, SVs with different shapes, which are partitioned in CVs of similar geometrical shape, have the same reconstruction, e.g. the functional values of the bases at corresponding mesh points are the same. For triangular shape SVs, consider the map c : s ! D,ofFig. 113a. This ‘‘map’’ transforms an arbitrary triangle s to a right triangle D. Another map (see Fig. 113b) often used transforms s to an equilateral triangle E. Assume that one node of s is at the origin r0 ¼ð0; 0Þ and the other two at r1 ¼ðx1; y1Þ and r2 ¼ðx2; y2Þ. The nodes corresponding to r0, r1 and r2 in D are ð0; 0Þ, ð1; 0Þ and ð0; 1Þ, respectively. The map c can be written as Fig. 113. The schematic of the mapping from the physical triangle to the standard triangle. c : r ¼ r1x þ r2Z; xX0; ZX0; x þ Zp1. (6.48) The map of Eq. (6.48) is linear therefore for a complete set of basis functions eðx; yÞ2Pm have The reconstruction formula of Eq. (6.47) with Eq. (6.52) are used to evaluate the state variables at 1 dV ¼ dx dy ¼ 2V dx dZ; V ¼ 2jr1 Â r2j, (6.49) the quadrature points. Triangles of different shape have identical bases Ljðx; ZÞ in the transformed space D if eðx; yÞ¼eðx; ZÞT, (6.50) their partition into polygonal CVs are similar. The where T is the transformation matrix containing only reconstruction is carried out only once for an arbitrary the geometry information. For example, if eðx; yÞ¼ shape triangle. The matrix R in Eq. (6.43) is inverted ½1; x; y; x2; xy; y2 is the hierarchical basis then T becomes analytically with Mathematica using exact arithmetic to 2 3 derive the reconstruction coefficients, which are identical 10 0 0 0 0 for all triangles. Exact integration formulas of poly- 6 7 6 7 nomials over polygons with arbitrary shape can be 6 0 x1 y1 0007 6 7 found in [261]. 6 0 x2 y 0007 6 2 7 All the CVs in a SV use the same data reconstruction T ¼ 6 2 2 7. 6 00 0 x1 x1y1 y1 7 therefore, it is not necessary to use a Riemann solver for 6 7 4 00 02x1x2 x1y2 þ x2y1 2y1y2 5 the interior boundaries between the CVs of a particular 2 2 SV. For the interior CV boundaries, exact fluxes are 00 0 x2 x2y2 y2 used. Use of Riemann fluxes is only necessary at the (6.51) spectral volume boundaries. the Eq. (6.46) becomes

L ¼½e1ðx; ZÞ; ...; eN ðx; ZÞ 6.4. Linear, quadratic, and cubic SV reconstructions R R 2 3À1 e1ðx; ZÞ dx dZ ÁÁÁ eN ðx; ZÞ dx dZ ci;1 ci;1 In the previous section, it was shown that the 6 7 6 7 reconstruction problem is equivalent for all triangles. 6 . . 7 ¼ 6 . . 7 Different order reconstructions are carried out for the 4 R R 5 transformed space using the equilateral triangle E of e1ðx; ZÞ dx dZ ÁÁÁ eN ðx; ZÞ dx dZ ci;N ci;N 2 3 Fig. 113b. The triangle E is partitioned into N V 1 nonoverlapping CVs that satisfy the conditions: 6 7 6 2V 7 6 7 6 . 7 Â6 . 7. ð6:52Þ The CVs are symmetric with respect to all symmetries 6 . 7 4 5 of the triangle. V N The CVs are convex. 2V The CVs are polygons with straight lines. ARTICLE IN PRESS

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The last condition is not absolutely required but degree m, i.e. simplifies the formulation. In general, however, iso- ðm þ 1Þðm þ 2Þ parametric SV can be used to handle curved boun- n1 þ 3n3 þ 6n6 ¼ . (6.55) daries. Possible partitions of the standard equi- 2 lateral triangle for cubic reconstruction were shown in Linear SV ðm ¼ 1Þ: The solution of Eq. (6.55) for Fig. 111. m ¼ 1 yields two partitions. These partitions, Type 1 Given a partition a necessary condition for valid and Type 2 are shown in Fig. 109a and b, respectively. reconstruction is that the matrix R of Eq. (6.43) is 13 The Lebesgue constant for Type 1 is 3 ¼ 4:3333 and for nonsingular. Furthermore, in the one-dimensional SV it 43 Type 2 15 ¼ 2:8667, therefore, the L1 error with Type 2 was mentioned that not all nonsingular reconstructions should be smaller than the error with Type 1 SV located are convergent. High-order polynomial reconstruction at the middle of the edge. based on equidistant CVs, for example, are not Linear reconstruction requires 1 G quadrature convergent in one dimension although the reconstruc- point, for the evaluation of the line integral. Due to tions are nonsingular. symmetry, functional values of the cardinal bases are A criterion for valid, convergent partitions [257] is required at two quadrature points, i.e. total of six based on the value of coefficients which are the same for all triangles with similar partitions. XN Quadratic SV ðm ¼ 2Þ: The solution of Eq. (6.55) for kGPk¼max jLjðx; ZÞj, (6.53) x2E j¼1 m ¼ 2 yields two partitions. These partitions are shown in Figs. 110a and b, respectively. The partition of where Fig. 110b is singular. For the partition of Fig. 110a all partitions with 0 d 0:5 are admissible. The partitions XN o o d ¼ 1 and d ¼ 1 (corresponding to the Gauss–Lobatto lðx; ZÞ¼ jLjðx; ZÞj (6.54) 3 4 j¼1 points of the edge) were considered in [257]. The 1 partition with d ¼ 3 is called Type 1 and the partition is referred to as Lebesgue function of the interpolation 1 with d ¼ 4 is called Type 2. The Type 1 partition and kGPk is the Lebesgue constant. The smaller the has Lebesgue constant 9.3333 and for the Type 2 Lebesgue constant the better the interpolation poly- partition the Lebesgue constant is 8, and this partition is nomial. Therefore the criterion for the best partition is expected to yield more accurate results. Quadratic to obtain small kGPk. Partitions of the standard reconstruction requires 2 G quadrature points for the equilateral triangle E which support linear, quadratic, evaluation of the line integral. Due to symmetry, 30 and cubic reconstructions with small Lebesgue constants coefficients corresponding to functional values of were given in [257]. the bases at five quadrature points must be computed and stored. Cubic SV ðm ¼ 3Þ: The solution of Eq. (6.55) for m ¼ 6.5. Spectral volume partitions 3 yields two different types of partitions. Among possible partitions shown in Fig. 111, only the partitions Consider the standard equilateral triangle E and of Fig. 111a and c are admissible while the partition of partitions of E in convex, symmetric CVs with straight Fig. 111d is singular. Variation of the partition distance edges. Partitions in CVs that contain the centroid of E d in Fig. 111b affects the value of Lebesgue constant. must be symmetric with respect to the three edges and The smallest value of the Lebesgue constant 3.44485 is vertices. Simple inspection shows that at most one type 1 obtained for d ¼ 15. Cubic reconstruction requires two of CV can exist, and this CV is said to possess degree 1 Gauss quadrature points for the evaluation of the line symmetry. Similarly, a CV is said to possess degree 3 integral. Due to symmetry, 100 coefficients correspond- symmetry when two other symmetric CVs exist in the ing to functional values of the bases at 10 quadrature same partition, and 6 symmetry when five other points must be computed and stored. symmetric CVs exist in the same partition. The degrees of 1, 3, and 6 symmetry groups in a partition are denoted as n1, n3, and n6. Partitions with different 6.6. Multidimensional limiters degrees of symmetry are shown in Fig. 111 for cubic spectral volumes. Different partitions for linear and The TVB idea [262] is used in [257] for the quadratic spectral volumes are shown in Figs. 109 and construction of limiters in order to obtain uni- 108, respectively. form accuracy away from discontinuities. Consider a The total number of CVs in the partition n1 þ 3n3 þ SV with a partition into N control volumes. Given 6n6 is equal to the dimension of the approximation space the cell-averaged values ui; j of the variables for that supports the unique reconstruction polynomial of all CVs and the k À 1 reconstruction polynomial ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 285 R p ðx; yÞ dV ¼ V i; jui; j j ¼ 1; ...; N and denoting 6.7. The SV method for two-dimensional systems ci; j i r ¼ 1; ...; K; Following the basic formulation for the development Durq ¼ piðxrq; yrqÞÀui; j; (6.56) q ¼ 1; ...; J of the SV method of [257,258], which was presented in it is not necessary to limit the data if the previous section, Wang et al. [260] continued the development of the SV method for hyperbolic conserva- r ¼ 1; ...; K; 2 tion laws. The two-dimensional Euler equations in jDurqjp4Mh ; (6.57) rq q ¼ 1; ...J: conservative form U ¼½r; ru; rv; rE were considered [260]. The cell-averaged conservative variable u of any Similar to the one-dimensional case in Eq. (6.57) M conservative variable in U is represents some measure of the second derivative of R the solution and hrq is the distance from ðxrq; yrqÞ to the c uðx; y; tÞ dx dy j ¼ 1; ...; m; u ðtÞ¼ i; j ; (6.61) centroid of ci; j. In Ref. [257], M was selected close to the i; j V i; j i ¼ 1; ...; N; maximum absolute value of the second derivative over the domain. where V i; j is the volume of ci; j. Given the cell averages If Eq. (6.57) is violated for any r and q, then it is from Eq. (6.61) a polynomial piðx; yÞ of degree k À 1at kÀ1 assumed that ui; j is near a steep gradient and limiting is most ðpiðx; yÞ2P Þ can be obtained such that piðx; yÞ applied. Therefore, instead of the polynomial piðx; yÞ in is a kth order accurate approximation of uðx; yÞ inside ci; j where Eq. (6.57) is violated it is assumed that the Si, i.e. data is linear, i.e. ðx; yÞ2Si; p ðx; yÞ¼uðx; yÞþOðhkÞ; (6.62) ui; jðx; yÞÀui; j þrui; j Áðr À ri; jÞ8r 2 ci; j, (6.58) i i ¼ 1; ...; N: where r is the position vector of the centroid of c .At i; j i; j The reconstruction polynomial the same time, it is required that the reconstructed R solutions at the quadrature points of ci; j satisfy the p ðx; yÞ dx dy ci; j i monotonicity constraint. ¼ ui; j; j ¼ 1; ...; m (6.63) V i; j r ¼ 1; ...; K; min max can be found analytically as for the scalar equation case ui; j pui; jðxrq; yrqÞpui; j ; (6.59) q ¼ 1; ...; J; and is expressed as

min max Xm where ui; j and ui; j are piðx; yÞ¼ Ljðx; yÞui; j, (6.64) max j¼1 ui; j ¼ max ui; j max ui; j;r , 1prpK kÀ1 where Liðx; yÞ2P are the shape functions satisfying R min Lmðx; yÞ dx dy ui; j ¼ min ui; j min ui; j;r , (6.60) ci; j 1prpK ¼ djm. (6.65) V i; j where ui; j;r denotes cell averages at neighboring CV of The shape functions can be obtained analytically. The ci; j sharing the face r. For highest resolution, the values of the shape functions at the Gauss quadrature magnitude of the solution gradient rui; j in Eq. (6.58) must be maximized. Several different approaches were points are given by Wang et al. [260]. The integral form of the Euler equations for CV mean suggested in [257] for the estimation of rui; j. In the first approach, all gradients are reconstructed values is I using cell-averaged data at ci; j and its neighbors. If any XK dUi; j 1 of the reconstructed variable at the quadrature points is þ ðf Á nÞ dS ¼ 0, (6.66) t V min max d i; j r¼1 Sr out of the range ½ui; j ; ui; j then the gradient is limited. In the second approach, only one gradient is computed where U i; j is the CV-averaged vector of the conservative and limited using so that the reconstructed solution at variables in ci; j, f ¼ðE; FÞ, is the flux vector K is the all quadrature points satisfy the monotonicity con- number of faces of the control volume ci; j, and Sr straint. This limiter is called ‘‘minmod’’ limiter in [257]. denotes the rth face (edge) of ci; j. The surface (line) In addition to these approaches where the gradients of integrals are evaluated numerically using a kth order the solution are reconstructed using data from neighbor- accurate Gauss quadrature formula as follows ing CVs, a third approach was suggested in Ref. [257] I XJ that uses the polynomials, which are available in the SV ðf Á nÞ dS ¼ o fðUðx ; y ÞÞ Á n S for the reconstruction. The limiter of this approach, rq rq rq r r Sr q¼1 which is named ‘‘CV’’ limiter in Ref. [257], is more k efficient than the limiters in the other two approaches. þ OðArhÞ , ð6:67Þ ARTICLE IN PRESS

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min max where J ¼ integer½ðk þ 1Þ=2 is the number of quad- where ui; j and ui; j are the minimum and the maximum rature points on the rth edge, and orq are the Gauss cell-averaged solutions among all neighboring CVs. The quadrature weights. The resulting conservative variables neighboring CVs either share a face (face neighbors) or a obtained with the polynomial distribution of Eq. (6.64) node (node neighbors) with ci; j, which is the control on each SV are discontinuous across the SV boundaries volume under consideration. For the second-order and ﬂux integration is carried out using approximate scheme, the face neighbors (see Fig. 109a) were used to min max Riemann solvers as deﬁne ui; j and ui; j for the limiter. This choice was made in order to reduce the number of cells that are fðUðx ; y ÞÞ Á n rq rq r limited in the higher-order schemes. The TVB idea is f RiemannðU Lðxrq; yrqÞ; U Rðxrq; yrqÞ; nrÞ. ð6:68Þ employed for limiting in [260] and small oscillations are allowed in the solution. Expressing the reconstruction Using Eqs. (6.65)–(6.67) obtain the following semi- for the quadrature points as discrete scheme for the CV ci; j Durq ¼ p ðxrq; y ÞÀui; j (6.75) dU 1 XK XJ i rq i; j þ o t V rq no limiting is required if d i; j r¼1 q¼1 2 Âf RiemannðU Lðxrq; yrqÞ; U Rðxrq; yrqÞ; nrÞSr ¼ 0. ð6:69Þ jDurqjp4Mqhrq, (6.76)

The Roe’s and Rusanov approximate Riemann where hrq ¼jri; j À rrqj is the distance of the CV centroid solvers were used in [260]. The ﬂux evaluation with to the quadrature point. Similar to the scalar case, Mq these Riemann solvers is obtained as can be chosen to be the maximum second derivative of Roe’s ﬂux: the solution. In [260], Mq was scaled as

max min f RoeðUL; UR; nÞ Mq ¼ Mðu À u Þ, (6.77) 1 min max ¼ 2½f ðU LÞþf ðU RÞÀjAjðUR À U LÞ, ð6:70Þ where u and u are computed over the entire qE qF computational domain. No data limiting is applied when where A ¼ nx þ ny and qU qU Eq. (6.76) is satisfied, even if the condition of Eq. (6.75) À1 jAj¼RjLjR (6.71) is not fulfilled. A CV is assumed to be close to with R and RÀ1 the right and left eigenvector matrices, a discontinuity of Eq. (6.76) and (6.75) are violated. and L the diagonal eigenvalue matrix all evaluated at the For limiting, the solution in the CV is assumed locally Roe-averages. linear, i.e. Rusanov flux: ui; jðx; yÞ¼ui; j þrui; j Áðr À ri; jÞ8r 2 ci; j. (6.78) f RusanovðU L; U R; nÞ Furthermore, the solution is assumed linear for all 1 ¼ 2½f ðU LÞþf ðU RÞÀaðU R À ULÞ, ð6:72Þ other CVs inside an SV if any of the CVs in the SV is limited. The gradient computed with cell-averaged where a is the maximum absolute eigenvalue solution that satisfies Eq. (6.76) is limited (replaced) by a ¼jvnjþc, (6.73) multiplying with a scalar f 2½0; 1 so that Eq. (6.78) becomes where vn and c are the average normal velocity and sound speed computed from U L and UR. ui; jðx; yÞ¼ui; j þ frui; j Áðr À ri; jÞ, (6.79) where the scalar f is computed as 6.8. Multidimensional TVD and TVB limiters 8 ! > Du > rq Stability is maintained for the nonlinear Euler > min 1; max if Durq40; > u À ui; j equations by performing data limiting with TVD or < i; j ! f ¼ (6.80) TVB limiters. TVD limiters enforce strict monotonicity > Durq > min 1; if Durqo0; by degrading solution accuracy at local extrema. TVB > min À > ui; j ui; j limiters relax monotonicity requirements in order to : 1 otherwise: achieve uniform accuracy away from discontinuities. The limiters in [260] the component-wise limiting For M ¼ 0, this limiter becomes TVD. approach was chosen because of its efficiency compared At the interior CV boundaries inside a SV the with the characteristic-wise approach of limiting. The reconstructed conservative variables are continuous, following numerical monotonicity criterion was estab- since the analytical flux is used if no limiters are lished in [260] for each CV. imposed. The monotonicity condition of Eq. (6.75) does

min max not guarantee positive pressure for problems with strong ui; j pui; jðxrq; yrqÞpui; j , (6.74) discontinuities because it strictly enforces monotonicity ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 287 only for the conserved variables. To ensure positive du 1 i; j þ pressure near strong discontinuities it is suggested in dt V i; j I I ! [260] to limit pressure instead of total energy. The total XK XK energy is computed at the quadrature points using the Â bue Á n dS À mqb Á n dS ¼ 0. limited reconstructed density, momentum and pressure. r¼1 Sr r¼1 Sr This limiting that guarantees positivity in both density ð6:83Þ and pressure was found very robust [260]. Using the Roe approximate Riemann solver, which degenerates in the scalar case to an upwind flux, obtain for the inviscid flux 6.9. The SV method for the Navier–Stokes equations ( uL b Á n40; ue ¼ (6.84) The SV method was recently extended to solve the uR b Á no0: Navier–Stokes equations [263]. The viscous terms were treated with a mixed formulation named in [263] local The other two numerical fluxes are defined as discontinuous Galerkin approach. It was shown [263] ub ¼ uL; ub ¼ uR that the desired order of accuracy is achieved with this or formulation for the scalar advection diffusion equation b b and for the Navier–Stokes with reconstruction poly- q ¼ qR; q ¼ qL. (6.85) nomials of all orders. The extension of the SV method to The numerical solution of Eqs. (6.83) follows the SV the Navier–Stokes equations is based on developments method described in the previous sections. for the SV method based on the one-dimensional pure diffusion equation [264]. The local SV (LSV) and the 6.9.2. Extension of the SV method to the NS equations penalty SV (PSV) approaches were found consistent, Consider the two-dimensional Navier–Stokes equa- stable and convergent. Furthermore, it was shown [264] tions written in conservation form that the LSV approach achieved the optimal order of accuracy, i.e. ðk þ 1Þth order for degree k poly- qu þrÁFiðUÞÀrÁFvðU; ruÞ¼0, (6.86) nomial reconstruction, while the PSV approach qt achieved only kth order accuracy for even k.Asa where U is the conservative variables vector and Fi, Fv result, the LSV approach was selected in [263] for the are the inviscid and viscous flux vectors, respectively. extension to the Navier–Stokes equations. The LSV Following the LDG approach for the convection–dif- formulation is briefly presented for one-dimensional fusion equation define the following auxiliary variable: linear and nonlinear and two-dimensional convection– G ¼rU. (6.87) diffusion equations. Then Eq. (6.86) becomes qU 6.9.1. SV formulation for 2D convection–diffusion þrÁF ðUÞÀrÁF ðU; GÞ¼0. (6.88) qt i v equation The convection–diffusion equation in the domain O is Considering the integral form of Eq. (6.88) for the CV ci; j and integrating by parts obtain qu I þrÁðbuÞÀrÁðmruÞ¼0, (6.81) 1 XK qt Gi; j ¼ Un dS, V i; j S where b is the convective speed and m is the diffusion r¼1 r " coefficient. The domain O is discretized with the spectral I dU 1 XK volume method. Following the local DG (LDG) i; j þ F ðUÞÁn dS dt V i approach an auxiliary variable q ¼ru was defined in i; j r¼1 Sr I # [263] to obtain the following system: XK À FvðU; GÞÁn dA ¼ 0. ð6:89Þ q ¼ru, r¼1 Sr Both U and G are discontinuous at SV boundaries. qu Therefore, the auxiliary flux, the inviscid fluxes and the þrÁðbuÞÀrÁðmqÞ¼0. (6.82) e e qt viscous fluxes are replaced by numerical fluxes U, F i,and e F v. The auxiliary and viscous fluxes are defined [263] as Integrating Eq. (6.82) by parts in ci; j obtain b I U U L, 1 XK q ¼ ubn dS, i; j V e R i; j r¼1 Sr F v FvðU L; G Þ.(6.90) ARTICLE IN PRESS

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For the inviscid fluxes Roe or any other consistent flux The order of accuracy of the SV method is one order splitting can be used. higher than the reconstruction polynomial degree. The Numerical tests were carried out in [263] for the linear optimum choice of parameters for the partitions of the convection–diffusion equation and for the viscous SV in CVs required for certain degree of accuracy is Burger’s equation. It was found that for the linear determined by minimizing the Lebesgue constant of one-dimensional convection–diffusion equation ux þ the reconstruction matrix (see [265] for details). The ux þ uxx the LSV approach achieved the optimum order partitioning of tetrahedra makes use of the partitioning of accuracy. For the one-dimensional viscous Burger’s of the triangular faces. Comments on the partition of equation ut þ uux À muxx ¼ 0, m ¼ 0:1, x 2ð0; 1Þ with triangular faces can be found in previous section. initial condition uðx; 0Þ¼Àtanhðx=2mÞ, boundary con- Further information on triangle partitions is given in dition uð0; tÞ¼0, uð1; tÞ¼Àtanhð1=2mÞ, and exact [265]. Tetrahedra SV partitions, which yield degree of solution uðx; tÞ¼Àtanhðx=2mÞ it was found again that accuracy up to three, all CVs have at least one face the LSV approach achieved the optimum orders of of the SV boundary and there are no CVs in the interior accuracy in all cases, e.g. second-, third- and fourth- of the SV. All the CVs consist of vertices of the 2D CVs order of accuracy. The LSV approach achieved also for each face of the SV connected to the SV centroid optimum orders of accuracy for the two-dimensional with straight edges. Fig. 114a shows the partition of a convection–diffusion equation ut þ cðux þ uyÞÀ SV face into CVs. For linear partition the four CVs are mðuxx þ uyyÞ¼0. Furthermore, the LSV method was hexahedra with all faces being planar quadrilaterals. tested for laminar flowover a flat plate at Re ¼ 500 and There are 12 CV faces on the boundary of the SV and 6 M ¼ 0:3. The computed results were in good agreement interior CV faces. The CVs for quadratic partitioning with the Blasius solution. (see Fig. 114b) are members of two symmetry groups. For the one-parameter partition, one consists of the four 6.10. The SV method in three dimensions hexahedra at the corners of the SV and includes three interior no planar faces. The other group consists of the The high-order SV method was extended to three six mid-edge polyhedra. Each polyhedron consists of dimensions in Ref. [265]. Similar to the two-dimensional two exterior triangular faces and the two quadrilateral case presented in the previous section it was shown [265] interior faces it shares with the corner CVs. As in the that if all grid cell are partitioned into structured sub- two-dimensional case, an analytic minimum value of the cells, the discretizations become universal and are Lebesgue constant does not exist this constant. How- reduced to the same weighted sum of unknowns ever, the Lebesgue constant asymptotes to a value of involving just a fewsimple adds and multiplies. approximately 11. A quadratic partition is shown in The development of the SV method in three dimen- Fig. 114b. The six mid-edge CVs are nowoctahedra with sions starts with the integral form for a control volume. two pentagonal faces on the SV boundary and six Similar to the two-dimensional case the partitions of interior planar quadrilateral faces. each SV into CVs depends on the choice of the basis functions for the reconstruction. For a complete 6.11. Results with the SV method polynomial basis, the reconstruction of degree of accuracy k requires a partition into at least N CVs where Application of the spectral volume method showed 8 very good performance for test problems used to > k þ 11D; > demonstrate the accuracy of other high-order accurate <> ðk þ 1Þðk þ 2Þ 2D; methods presented in the previous sections. N ¼ 2 (6.91) > The first problem considered in [260] is the isentropic > ðk þ 1Þðk þ 2Þðk þ 3Þ : 3D: vortex convection (see Section 3 for details on problem 6 definition). Numerical simulations were carried out until

Fig. 114. Two-dimensional faces for three-dimensional SV partitions. ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 289 t ¼ 2 on both regular and irregular meshes. The leads to a spurious Mach stem at the downstream Rusanov flux was used and no limiting was necessary bottom wall. In the numerical test of [260] no special for this problem that has smooth solution at all times. It treatment was used for the singularity. The effect of the was found that the second-order SV is less accurate than the FV method with the same number of degrees of freedom. However, the third-order SV method was found 3.81 times more accurate than the second-order FV scheme and the fourth-order SV method 435 times more accurate than the second-order FV scheme with the same degrees of freedom. The high-order SV schemes were shown to be [260] more efficient than the second-order FV scheme in achieving the same solution quality. For example, the third-order SV scheme had a relative cost 0.44, and the fourth-order SV scheme had a 0.023 relative cost compared to the FV scheme for the same solution accuracy. The performance of the spectral volume method was tested extensively in [258] for two problems, suggested by Woodward and Colella [179]. Both problems were used widely to assess the performance of shock- capturing methods. The first problem is the Mach 3 wind tunnel with a step and the second problem is the double Mach reflection. Results for these problems from Ref. [261] that demonstrate the effectiveness and resolution characteristics of the SV method variants, including numerical flux evaluation and effect of limit- Fig. 116. Density contours computed on the coarse mesh using ers, are shown next. the Rusanov flux and TVD limiter. Thirty even contour lines For the Mach 3 wind tunnel problem it is well known between 0.09 and 4.53; (a) second-order SV scheme; (b) third- that the corner of the step is a singularity that often order SV scheme; (c) fourth-order SV scheme.

Fig. 115. Density contours computed on the coarse mesh using Fig. 117. Density contours computed on the coarse mesh using the Roe ﬂux and TVD limiter. Thirty even contour lines the Rusanov ﬂux and TVB limiter with M ¼ 10. Thirty even between 0.09 and 4.53; (a) second-order SV scheme; (b) third- contour lines between 0.09 and 4.53; (a) second-order SV order SV scheme; (c) fourth-order SV scheme. scheme; (b) third-order SV scheme; (c) fourth-order SV scheme. ARTICLE IN PRESS

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Riemann solver and order of accuracy was tested using a limiting and the non-uniform subcell mesh produced 1 coarse mesh with 1763 triangular elements and h ¼ 20. large errors. The Roe’s flux was found less dissipative All simulations for the wind tunnel were carried out and produced weaker Mach stems than the Rusanov until t ¼ 4. The computed density contours obtained flux. For the TVB limiter test of Fig. 117 the Mach from numerical solutions with the second-, third-, and number was increased to M ¼ 10. The performance of fourth-order accurate SV methods, using the Roe flux the TVB limiter was still very similar with that of the and the TVD limiter are shown in Fig. 115. The same TVD limiter except that the TVB limiter produced more computations with the Rusanov flux are shown in oscillations. Fig. 116.InFig. 117 the effect of the limiter (TVD versus TVB) is demonstrated using again the Rusanov flux. It can be seen that the fourth-order SV performed worse than the second- and third-order SV scheme because many reconstructions were linear due to data

Fig. 118. Density contours computed using a third-order SV scheme with the Rusanov ﬂux and TVD limiter. Thirty even 1 Fig. 120. Computed density contours with the second-order SV contour lines between 0.09 and 4.53; (a) h ¼ 40 (52,476 DOFs); 1 scheme using the Rusanov ﬂux and TVD limiter. (b) h ¼ 80 (222,876 DOFs).

Fig. 119. Three computational grids with different degrees of Fig. 121. Computed density contours with the third-order SV reﬁnement near the singular corner point. scheme using the Rusanov ﬂux and TVD limiter. ARTICLE IN PRESS

J.A. Ekaterinaris / Progress in Aerospace Sciences 41 (2005) 192–300 291

Grid refinement studies performed in [260] demon- increases significantly. Even though the third-order strated (see Fig. 118) that resolution of the third-order results were more oscillatory than the second-order 1 1 scheme for the medium h ¼ 40 and fine h ¼ 80 grids results the spurious Mach stems were weaker. Successive grid refinement, with the meshes shown in Fig. 119, was used to demonstrate that the singular corner is the cause of the spurious Mach stem. Computed density contours from the second- and third-order accurate schemes with the Rusanov flux and the TVD limiter are shown in Figs. 120 and 121, respectively. It can be seen that increased grid density weakened the spurious Mach stem and the entropy layer

Fig. 122. Density contours computed using a second-order SV scheme with the Rusanov ﬂux and TVD limiter. Thirty even 1 contours between 1.25 and 21.5. (a) h ¼ 30 (24,600 DOFs), 1 1 (b) h ¼ 60 (98,808 DOFs), (c) h ¼ 120 (392, 484 DOFs).

Fig. 123. Density contours computed using a third-order SV Fig. 124. Close-up viewof the density contours near the double 1 scheme with the Rusanov ﬂux and TVD limiter. Thirty even Mach stem: (a) second-order SV scheme, h ¼ 120 (392,384 1 1 contours between 1.25 and 21.5. (a) h ¼ 30 (49,200 DOFs), DOFs); (b) third-order SV scheme, h ¼ 60 (197,616 DOFs); 1 1 1 (b) h ¼ 60 (197,616 DOFs), (c) h ¼ 120 (784,968 DOFs). (c) third-order SV scheme, h ¼ 120 (784,968 DOFs). ARTICLE IN PRESS

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mþ1=2 downstream the shock reflection. Higher resolution of case, Oðh Þ accuracy is obtained L2 norm and mþ1 shock waves and contact was obtained with the third- Oðh Þ accuracy is L1 norm. For k4m, or for regions order accurate scheme. away from the discontinuities when kXm, Oðhkþ1Þ The double Mach reflection problem (see Section 4 for accuracy is obtained in L1, L2, and L1 norms. There- the description of the problem) tests were carried out for fore, there is no advantage when a polynomial of degree the Rusanov flux, the TVD limiter, and approximate k4m is used in the vicinity of the discontinuity and for 1 1 1 mesh sizes 30, 60 and 120. The density contours at t ¼ 0:2 maximal efficiency a polynomial of degree k ¼ m should for the second- and third-order SV schemes are shown in be used near the discontinuity. For a better capturing of Figs. 122 and 123, respectively. The third-order SV the discontinuity itself, locally h-refinement must be scheme has much higher resolution than the second- employed. For smooth analytic solutions ðm !1Þ,on order scheme for complex flowstructures near the the other hand, high degree numerical approximations double Mach stem. Fig. 124 shows that the density should be used with no or little reliance on local contours computed with the third-order scheme and half h-refinement similar to spectral methods. 1 the mesh size ðh ¼ 60Þ display finer structures than the Another aspect of high-order accurate methods is computation performed with the second-order scheme robustness. When the solution has a discontinuity in the 1 on a fine mesh with mesh size h ¼ 120. The third-order mth derivative, then oscillations in the vicinity of the accurate computations of [260] were found to have discontinuity appear because of the Gibbs’ phenomen- better resolution than the DG scheme on the same mesh on. Oscillations could lead to unphysical states, such as even for computations obtained with DG methods of regions with negative pressure or density, during the the same order on finer meshes. course of the solution process unless care it taken with slope limiters [38] or shock switches [60,63]. The oscillations, which could appear in the commutated 7. Conclusions solution for degree of accuracy kom on coarse or purely resolved meshes, would disappear with grid refinement. For vortical flows without discontinuities, many In CFD (also in many CAA) applications the governing different numerical schemes can achieve high accuracy. equations and in many cases the numerical scheme as Centered schemes with spectral-type filters work well for well is nonlinear. In these cases, inadequate resolution of these problems. So do all other shock-capturing features could present serious problems, primarily methods such as schemes with explicit, characteristic- having to do with the solvability of the nonlinear system type filters [28,36], essentially non-oscillatory (ENO) and secondary, possibly with the unphysical nature of schemes [32], weighted WENO schemes [33,34], and the the convergent solution. discontinuous Galerkin method. However, only the The sensitivity to grid quality is another major high-order accurate shock-capturing methods, such as problem that is common even for second-order accurate WENO schemes, the DG method, and centered schemes methods. Grids are generated based on a priori knowl- with characteristic-type filters, are suitable for computa- edge of the solution. For example, the number of points tions of high-speed flows with shocks. Finite-difference and stretching ratio required to resolve boundary layers, centered compact differentiation formulas and WENO vortical structures, and detached layers is usually schemes are quite efficient computationally. In addition, predetermined. The real power of unstructured grid when combined with implicit time integration methods methods is their ability to adapt to complex flow [149,29] are suitable for the numerical solution of time- features without being constrained by considerations depended high Reynolds number viscous flows. Numer- such as grid structure and topology. As a result, once a ous applications of Sections 3 and 4 demonstrate the valid solution is obtained on any initial grid, it is at least potential of these methods. conceptually possible to adapt the grid, either through In Ref. [266], additional important observations were grid refinement/coarsening and/or by moving grid made regarding the efficiency with high-order accurate points. To date there have been a fewnotable efforts schemes by examining the local errors achieved when [267] for relatively simple CFD problems. The real employing polynomial base functions or high-order challenge is application of anisotropic grid adaptation accurate stencils. These observations stem from the for high Reynolds number complex flows. For example, study of a steady advection problem with source in one in rotorcraft and high lift applications the flowfeatures direction. If the exact solution has a discontinuity in the of interest are attached and separated boundary layers, mth derivative (m ¼ 0 denoting discontinuity in the wake roll-ups and tip vortices, possible shocks on slats solution), the local error in L1 norm is no better than or blade tips and interactions among all these features. It OðhmÞ, where h is the mesh size, regardless of the is very likely that the grids produced by adaptation polynomial basis or the stencil order being used. The would be highly stretched, deformed, and anisotropic. It m Oðh Þ accuracy in L1 norm is achieved by using kth is a challenge for current flowsolvers to achieve solution degree polynomial basis functions with kXm. In this reliability and accuracy on anisotropic adapted grids. ARTICLE IN PRESS

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The ability to retain high-order accuracy on any grid is Table A.1 even more difficult, but nevertheless remains a key requirement. Several of the recently proposed high-order kr j¼ 0 j ¼ 1 j ¼ 2 j ¼ 3 methods, such as the discontinuous Galerkin finite- 1 À11 element method and the spectral volume methods have 01 the potential to overcome problems associated with anisotropic meshes. However, demonstration was shown 2 À1 3 À1 only for simple cases. 2 2 0 1 1 The discontinuous Galerkin (DG) and the spectral 2 2 1 À1 3 volume (SV), which can be interpreted as a Petrov– 2 2 Galerkin method where the test and basis functions are different, methods have the potential for grid adaptation. 3 À1 11 7 1 6 À6 3 In addition the discontinuous Galerkin method is well 0 1 5 1 3 6 À6 suited for hp-type refinement and anisotropic adaptation 1 À1 5 1 6 6 3 with hanging nodes [222,247]. Second-order accurate 2 1 À7 11 discretization with the DG and SV methods are 3 6 6 comparable in computing cost to the finite-volume 4 À1 25 23 13 1 methods. Application of TVB limiters with high-order 12 À12 12 À4 0 1 13 5 1 accurate DG discretizations is computationally intensive. 4 12 À12 12 1 1 7 7 1 Recently, progress was achieved for efficient high-order À12 12 12 À12 2 1 5 13 1 computation of flows with discontinuities with the 12 À12 12 4 3 1 13 23 25 introduction of stabilization operators [222]. A major À4 12 À12 1 obstacle for the application of the DG and SV methods in large scale, high-Reynolds number aerodynamic simulations is time step limitation that deteriorates with the increase of the order of the method. Recent that simplifies for uniform grid to P Q progress in implicit time marching algorithms [210,221], k k Xk l¼0 q¼0 ðr À q þ 1Þ space/time DG discretizations [222], and p-type multigrid lam a c ¼ Q q m;l . (A.3) [207] is expected to make possible high-order accurate rj ðl À mÞ m¼jþ1 l¼0 computations of practical aerodynamic flows. These lam developments combined with the DG/VMS method, The values of the constants crj for uniform grid and which combines VMS modeling with the ability of the order of accuracy between k ¼ 1 and k ¼ 6 are given in DG method to obtain high accuracy in complex domains, Table A.1. offer tremendous potential for the accurate simulation of From Table A.1 and Eq. (4.5) obtain for example wall-bounded and non-equilibrium turbulence. 1 7 7 1 4 uiþ1=2 ¼À12 uiÀ1 þ 12 ui þ 12 uiþ1 À 12 uiþ2 þ OðDx Þ, 49 71 79 uiþ1=2 ¼ 20 uiþ1 À 20 uiþ2 þ 20 uiþ3 Appendix 163 31 1 6 À 60 uiþ4 þ 30 uiþ5 À 6 uiþ6 þ OðDx Þ. The Lagrange interpolation polynomials are given by

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