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A Comparison of High-Resolution, Finite-Volume, Adaptive±Stencil

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A Comparison of High-Resolution, Finite-Volume, Adaptive±Stencil Advances in Water Resources 24 (2001) 29±48 www.elsevier.com/locate/advwatres A comparison of high-resolution, ®nite-volume, adaptive±stencil schemes for simulating advective±dispersive transport Matthew W. Farthing, Cass T. Miller * Department of Environmental Sciences and Engineering, Center for the Advanced Study of the Environment, University of North Carolina, Chapel Hill, NC 27599-7400, USA Received 1 February 2000; received in revised form 8 May 2000; accepted 16 May 2000 Abstract We investigate a set of adaptive±stencil, ®nite-volume schemes used to capture sharp fronts and shocks in a wide range of ®elds. Our objective is to determine the most promising methods available from this set for solving sharp-front advective±dispersive transport problems. Schemes are evaluated for a range of initial conditions, and for Peclet and Courant numbers. Based upon results from this work, we identify the most promising schemes based on eciency and robustness. Ó 2000 Elsevier Science Ltd. All rights reserved. 1. Introduction ecient for solving relatively simple advective-domi- nated problems. These approaches, however, can pose The processes of solute and energy transport in nat- mass conservation problems, are not well suited for ural systems are most often described using an advec- problems with multiple sources and non-linear mass tive±dispersive equation (ADE) with additional terms transfer terms, and can be diusive and oscillatory for sources, reactions, and interphase exchange of mass [3,19,38,44]. or energy [34]. For many problems, especially those in- Over the last decade, simulators in the water re- volving multiple species, ADE solutions can be a sub- sources ®eld have employed so-called high-resolution stantial part of the computational eort involved for a methods more and more for discretizing the advective given ¯ow and transport simulator. Economical solution portion of the ADE [6,9,20,25,30,35,40,51,52,61,67]. By of this equation is still elusive for cases in which sharp high-resolution methods, we mean a class of adaptive± fronts in space and/or time develop. Dozens of ap- stencil methods, usually explicit in time, for resolving proaches have appeared in the literature in a wide sharp fronts. This class has been actively investigated in variety of ®elds, including water resources and envi- the ®elds of gas dynamics and applied mathematics over ronmental engineering [2,29,66], chemical and petro- the last two decades [14,15,18,22,24,28,32,42,45,48,49, leum engineering [10,54], atmospheric science [43], 53,57,59,64]. applied mathematics, and gas dynamics [11,37]. Some issues should be considered before directly Most current subsurface transport codes use ®xed- applying high-resolution schemes from the gas dynamics grid Eulerian methods, method of characteristics, or and applied math literature to the ADE. Much of the particle tracking approaches ± each of which has its own work in these ®elds is primarily concerned with how well limitations. For example, conventional low-order ®nite- the schemes resolve solutions arising from non-linear dierence or ®nite-element methods are widely used, but equations or systems of equations (e.g., Burgers' equa- known to be overly diusive, subject to phase error, tion or the Euler equations) whose fronts are usually undershoot, peak depression, and oscillations [37]. combinations of self-sharpening shocks, rarefaction Combining the advantages of Eulerian and Lagrangian waves, and contact discontinuities in the presence of approaches, method of characteristic approaches can be negligible physical dispersion [11,28,49]. In many re- spects, the ADE is much simpler and often serves as a useful initial test problem. However, the ability to re- * Corresponding author. E-mail addresses: [email protected] (M.W. Farthing), solve sharp fronts for linear advective±dispersive prob- [email protected] (C.T. Miller). lems is not the chief criterion for determining the overall 0309-1708/01/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0309-1708(00)00027-0 30 M.W. Farthing, C.T. Miller / Advances in Water Resources 24 (2001) 29±48 Notation df limited higher-order ¯ux correction (FCT), Eq. (32) à C concentration gj term (gj Cj r =2) (PPM) ~ j1=2 C higher-order approximation (ULT), Eq. (45) hk ideal weight for kth ENO stencil Cb extrapolated constraint bound (ULT), Eq. (46) (WENO) CL cell reconstruction approached from left k Courant±Friedrichs±Lewy number side of cell face (PPM) k uDt=Dx CR cell reconstruction approached from right n normalized spatial coordinate side of cell face (PPM) n x xj1=2=Dx 2xj1=2; xj1=2 p à à L R C higher-order approximation (PPM), Eq. (49) rj1=2 sum rj1=2 Cj1=2 Cj1=2 (PPM) D dispersion coecient rj sum rj Cj1 Cj Fl lower-order numerical ¯ux, Eq. (18) ss local spatial truncation error Fh higher-order numerical ¯ux, Eq. (18) u ¯ux-limiter function, Eq. (20) M minmod operator, Eq. (24) x weighting parameter (COX) Nx number of cells along x-axis xk ®nal weighting applied to kth ENO stencil Pe mesh Peclet number Pe uDx=D (WENO), Eq. (67) S numerical slope, Eq. (17) Subscripts and superscripts Sf sign of higher-order ¯ux correction (FCT), Eq. (33) j spatial grid cell identi®er (subscript) a starting spatial coordinate for semi-ellipse n time level identi®er (superscript) pro®le, Table 4 b ®nal spatial coordinate for semi-ellipse Abbreviations pro®le, Table 4 ADE advective±dispersive equation ASO alternating split operator ic center cell number for semi-ellipse pro®le, Table 4 CFL Courant±Friedrichs±Lewy number COX Cox and Nishikawa transport scheme if ®nal cell number for semi-ellipse pro®le, Table 4 CPU total elapsed CPU time ENO essentially non-oscillatory is initial cell number for semi-ellipse pro®le, Table 4 FCT ¯ux-corrected transport i half cell-width for semi-ellipse pro®le, Table 4 HARM harmonic transport scheme w LIU Liu et al. transport scheme qk kth ENO stencil (WENO), Eqs. (64)±(66) t time coordinate MMOD minmod transport scheme u interstitial velocity MUSCL monotone upstream-centered scheme for x spatial coordinate conservation laws PPM piecewise parabolic method Greek symbols Ref primary literature source(s) for method, Dj second-order dierence, Eq. (13) Table 1 DC general higher-order ¯ux correction, Eq. (19) SBEE superbee transport scheme Df higher-order ¯ux correction (FCT), Eq. (34) TV total variation, Eq. (1) X spatial domain under consideration TVD total variation diminishing, Eq. (2) ak weighting fraction for kth ENO stencil ULT ultimate transport scheme (WENO), Eq. (68) UNO uniformly non-oscillatory bk smoothness monitor for kth ENO stencil UNO2 second-order uniformly non-oscillatory (WENO), Eqs. (69)±(71) transport scheme bc compression coecient (COX), Eq. (38) WENO weighted essentially non-oscillatory bu compression coecient (COX), Eq. (38) ZDL Zaidel transport scheme dj ®rst-order dierence dj Cj Cj1 1-PT one-point upstream weighting dj centered ®rst-order dierence, Eq. (27) 2-PT two-point upstream weighting à à L R dj1=2 ®rst-order dierence dj1=2Cj1=2Cj1=2 flop ¯oating point operation (PPM) hpm hardware performance monitor M.W. Farthing, C.T. Miller / Advances in Water Resources 24 (2001) 29±48 31 X quality of a scheme in the gas dynamics or applied TV C jCj1 Cjj 1 mathematics ®elds. j The objectives of this work are: 1. to review evolving classes of high-resolution ®nite- of the solution is non-increasing [27,48,64], i.e. volume methods that hold promise for the water re- TV Cn1 6 TV Cn: 2 sources ®eld; 2. to summarize a signi®cant set of high-resolution Appropriately, these schemes are known as total varia- methods; tion diminishing (TVD). 3. to compare the eciency and robustness of a broad The notion of total variation leads to another classi- set of methods for solving the ADE; ®cation of high-resolution methods that is closely tied to 4. to recommend approaches for applications and their accuracy at smooth local extrema. TVD schemes extension to other cases. necessarily degrade to ®rst-order at local extrema [11]. In [18], the TVD criterion was replaced by the uniformly non-oscillatory (UNO) property. UNO schemes insure 2. Background that the number of local extrema does not increase rather than enforcing the stricter TVD condition, which The challenge for modeling sharp fronts in advective- requires that local extrema are damped at each time step. dominated problems is to resolve the front over rela- As a result, the second-order UNO scheme presented in tively few computational cells without introducing [18] maintains its formal accuracy at smooth local ex- spurious oscillations in the solution. The scheme should trema. The notion of essentially non-oscillatory (ENO) provide higher-order accuracy in smooth regions of the methods presented in [17] represents a further general- solution as well. Godunov's original scheme (simple ization. ENO schemes allow local extrema to be in- upwinding) [12] produces a monotone solution from creased at times but guarantee that the total variation monotone initial data but introduces excessive numeri- increase is on the order of the grid spacing. cal diusion, losing the sharp nature of the physical TV Cn1 6 TV CnO D xp1 for p > 0: 3 solution. Godunov also showed that any linear scheme of second or higher-order necessarily introduces oscil- As a result, ENO schemes also maintain higher-order lations in the numerical solution. Modern high-resolu- accuracy around smooth extrema. tion schemes have circumvented this barrier because Schemes can also be categorized according to the they are non-linear. That is, they use an approximation order of interpolation and extrapolation used to ap- that is at least second-order in smooth regions but then proximate the intercell face values.
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