Perfectly Matched Layers and High Order Difference Methods for Wave

Dedicated to my father Mr. Ambrose Duru (1939 – 2001)

List of papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I K. Duru and G. Kreiss, (2012). A Well–posed and discretely stable perfectly matched layer for elastic wave equations in second order formulation, Commun. Comput. Phys., 11, 1643–1672 (DOI:10.4208/cicp.120210.240511a). contributions: The author of this thesis initiated this project and performed all numerical experiments. The manuscript was prepared in close collaboration between the authors

II G. Kreiss and K. Duru, (2012). Discrete stability of perfectly matched layers for anisotropic wave equations in ﬁrst and second order formulation, (Submitted). contributions: The author of this thesis initiated this project and performed all numerical experiments. The manuscript was prepared in close collaboration between the authors.

III K. Duru and G. Kreiss, (2012). On the accuracy and stability of the perfectly matched layers in transient waveguides, J. of Sc. Comput., DOI: 10.1007/s10915-012-9594-7. In press. contributions: The author of this thesis initiated this project performed all numerical experiments and had the responsibility of writing the paper. The remaining time was spent between the author and his advisor correcting misconceptions, improving the texts and the theory.

IV K. Duru and G. Kreiss, (2012). Boundary waves and stability of the perfectly matched layer. Technical report 2012-007, Department of Information Technology, Uppsala University, (Submitted) contributions: The author of this thesis initiated this project and had the responsibility of writing the paper. The remaining time was spent between the author and his advisor correcting misconceptions, improving the texts and the theory.

V K. Duru and G. Kreiss, (2012). Numerical interaction of boundary waves with perfectly matched layers in elastic waveguides. Technical report 2012-008, Department of Information Technology, Uppsala University, (Submitted). contributions: The author of this thesis initiated this project, performed all numerical experiments and had the responsibility of writing the paper. The remaining time was spent between the author and his advisors correcting misconceptions, improving the texts and the theory.

VI K. Duru, G. Kreiss and K. Mattsson, (2012). Accurate and stable boundary treatments for elastic wave equations in second order formulation, (Submitted). contributions: The author of this thesis initiated this project, performed all numerical experiments and had the responsibility of writing the paper. The remaining time was spent between the author and his advisors correcting misconceptions, improving the texts and the theory.

Reprints were made with permission from the publishers. Contents

1 Introduction ...... 9

2 Non-reﬂecting boundary conditions (NRBC) ...... 12 2.1 Exact NRBC ...... 12 2.2 Local NRBC ...... 14

3 Absorbing layers ...... 16 3.1 Model problem ...... 18 3.2 Construction of the PML equations ...... 18 3.2.1 First order formulation ...... 19 3.2.2 Perfect matching ...... 20 3.3 Well–posedness of the PML ...... 21 3.4 Stability of the Cauchy PML ...... 22 3.5 Stability of the PML for IBVPs ...... 23 3.6 Stability of the discrete PML ...... 26

4 Finite difference methods for second order systems ...... 29 4.1 The wave equation ...... 30 4.2 Integration–by–parts ...... 31 4.3 Summation–by–parts ...... 31 4.4 Weak boundary treatment with SAT ...... 32

5 Summary of papers ...... 34 5.1 Paper I ...... 34 5.2 Paper II ...... 35 5.3 Paper III ...... 35 5.4 Paper IV ...... 36 5.5 Paper V ...... 36 5.6 Paper VI ...... 37

6 Summary in Swedish ...... 39

References ...... 42

1. Introduction

Wave motion is a dominant feature for problems in many branches of engineering and applied sciences. For instance, both industrial materials such as steel pipes and plates [78, 93], and natural structures like the Earth’s surface can support propagating waves that can lead to failure or disaster. Numerical simulations of propagating waves can serve as a complement to theoretical and experimental investigations, and can possibly treat many more important sce- narios that can not be investigated by theory or experiments. Thus, providing valuable information which can be useful in developing modern technologies, exploring natural minerals from the subsurface, and understanding natural dis- asters such as earthquakes and tsunamis. A defining feature of propagating waves is that they can propagate long dis- tances relative to the characteristic dimension, the wavelength. As an example, strong ground shaking resulting from an earthquake or a nuclear explosion can be recorded far away in another continent sometime after it has occurred. In practice, because of limited resources, computer simulations of such problems are restricted only to areas of interest, where the effects of the strong ground motion is significant. This is typical of numerical simulations of many wave propagation problems including seismic imaging, wireless communication and ground penetrating radar (GPR) technologies. In numerical simulations, large spatial domains must be truncated to fit into the finite memory of the computer, by introducing artificial boundaries. One can immediately pose the question:

Which boundary conditions ensure that the numerical solution of the initial boundary value problem inside the truncated domain converge to the solution of the original problem in the unbounded domain?

Provided the numerical method used in the interior is consistent and stable, the numerical solution will converge to the solution of the original problem in the unbounded domain only if artificial boundaries are closed with ‘accurate’ and reliable boundary conditions. Otherwise, waves traveling out of the computational domain generate spurious reflections at the boundaries, which will then travel back into the computational domain and pollute the solution everywhere. It becomes apparent that the most important feature of artificial boundary conditions is that, all out–going waves disappear (or are absorbed) without reflections. Therefore, an effective numerical wave simulator should be able to treat both physical and artificial boundaries efficiently.

9 One objective of this thesis is to design effective artificial boundary conditions suitable for numerical solutions of partial differential equations (PDE) describing wave phenomena. The effort to design efficient artificial boundary conditions began over thirty years ago [35] and has evolved over time to become an entire area of research. Artificial boundary conditions in general can be divided into two main classes: absorbing or non-reflecting boundary conditions (NRBC) [35] and absorbing layers [20]. We also note that for time-harmonic problems there are accurate and efficient artificial boundary procedures, see [107, 47]. Problems of this class are not considered in this thesis. Most domain truncation schemes for numerical time-dependent wave propagation have been developed for constant coefficient wave propagation problems. Many of these methods such as high order local NRBCs and the perfectly matched layer (PML) which we will discuss in Chapter 2 and Chap- ter 3, respectively, are efficient for certain problems, particularly the scalar wave equation and the Maxwell’s equations in isotropic homogeneous media. For many other problems in this class, such as the linearized magneto– hydrodynamic (MHD) equations and the equations of linear elasticity, there are yet unresolved problems, see for example [17, 11]. Artificial boundary conditions for more difficult problems such as variable coefficient and non-linear wave propagation problems are less developed. In practice, ad hoc methods are still in use. However, a better understanding of the mathematical properties of the corresponding linear problems will enable the development of efficient boundary conditions for problems of this class. Second order hyperbolic systems often describe problems where wave propagation is dominant. In the numerical treatment of second order systems, the equations are usually rewritten and solved as a first order system, introducing additional degrees of freedom. This is in part due to the maturity of the theory and numerical techniques developed in the computational fluid dynamics (CFD) community. Rewriting second order systems to first order system can lead to a loss of computational efficiency [72, 50]. The second objective of this thesis is to derive high order accurate and strictly stable (or time–stable) finite difference approximations for systems of second order hyperbolic partial differential equations on bounded domains. Numerical methods for second order hyperbolic systems are briefly presented in Chapter 4. In this thesis, we consider linear time–dependent constant coefficient wave propagation problems in second order formulation. Note that many linear wave equations that appear as first order systems can be rewritten in second order formulations. Our focus is on the scalar wave equation, curl–curl Maxwell’s equations and the equations of linear elasticity. However, with limited modifications the results obtained in this thesis can be applied to other wave equations. We begin Chapter 2 by illustrating the fundamental ideas underlying the derivation of NRBCs and reviewing some well known results of NRBCs. In

10 Chapter 3, a model problem for second order anisotropic wave equations is considered, and the PML equations in second order and ﬁrst order formulations, respectively, are derived. Some of the mathematical properties of the models are also discussed. Chapter 4 is devoted to a short presentation of numerical methods for second order hyperbolic systems. A summary of the included papers is presented in Chapter 5.

11 2. Non-reﬂecting boundary conditions (NRBC)

The amount of literature on non-reflecting boundary conditions is enormous and constantly increasing. In this section we attempt a brief review of NRBCs for the wave equation. Elaborate discussions can be found in the review papers [107, 41, 46]. A NRBC, as the name implies, is a boundary condition imposed on an artificial boundary to ensure that no (or little) spurious reflections occur from the boundary [41]. These boundary conditions can be classified into exact (or global) NRBCs, and approximate (or local) NRBCs1. If a boundary condition is such that the artificial boundary appears perfectly transparent, it is called exact. Otherwise it will correspond to a local (NRBC) approximation and generate some spurious reflections.

2.1 Exact NRBC To begin, we consider the second order scalar wave equation in Cartesian coordinates, 1 u = u + u , for t > 0, (x,y) ∈ 2, c2 tt xx yy R (2.1) u = u0, ut = v0, at t = 0.

Here, u is the wave field, c is the wave speed, t denote time, (x,y) are the spatial coordinates, and u0,v0 are the initial data. The wave equation (2.1) can describe pressure waves in a homogeneous isotropic media. After some as- sumptions, the wave equation (2.1) can also describe shear waves propagation in elastic solids or wave propagation in an electromagnetic media. Assume we want to compute the solution of the wave equation (2.1) in the half plane x < 0. In order to close the statement of the problem and make accurate computations, a NRBC is needed at the artificial boundary at x = 0. Exact boundary conditions for (2.1) were first derived in the pioneering work by Engquist and Majda [35] and have recently been reviewed [41, 46] from a modern perspective. The construction of the absorbing boundary conditions

1We note that all exact NRBCs are global, but all global conditions are not exact. Similarly all local NRBCs are approximate but all approximate NRBCs are not local.

12 [35] uses the fact that any right–going solution u(x,y,t) to (2.1) can be represented by a superposition of plane waves travelling to the right. Such solutions are described by q ω 2 2 −i ( ) −ky x uˆ(x,ky,ω) = a0e c . (2.2)

Here, a0 is the amplitude of the wave, and (ω,ky) are the duals of (t,y), sat- 2 2 2 isfying ω /c − ky > 0 with ω > 0, and the solutionu ˆ(x,ky,ω) is the wave function in the Fourier space. By Engquist and Majda [35], the correct boundary condition which annihilates all right–going waves at x = 0 is s 2 ∂ ω kyc uˆ(x,ky,ω) = −i 1 − uˆ(x,ky,ω), x = 0. (2.3) ∂x c ω

Notice that the boundary condition (2.3) is prescribed foru ˆ(x,ky,ω) and not for the time–space dependent wave funtion u(x,y,t). Butu ˆ(x,ky,ω) is related to u(x,y,t) via the inverse Fourier transform. Therefore, in order to obtain a boundary condition for u(x,y,t) we need to invert the the Fourier transform in (2.3). In theory we can always compute the inverse Fourier transform to determine ∂u(x,y,t)/∂x. Unfortunately, the inverse Fourier transform of (2.3) yields the operator  s  2 ∂ −1 ω kyc u(x,y,t) = −F i 1 − uˆ(x,ky,ω), x = 0, ∂x c ω (2.4) Z ∞ Z ∞ −1 i(ωt+kyy) with F uˆ(x,ky,ω) = e uˆ(x,ky,ω)dkydω, −∞ −∞ which is nonlocal in both time and space. This√ is manifested in the difficulty 2 in inverting the pseudo–differential operator 1 − s , (with s = kyc/ω), which does not have an explicit local representation. Engquist and Majda acknowl-√ edged this difficulty and instead resorted to some approximations of 1 − s2 which yield a local differential operator upon inversion.√ The accuracy of the boundary condition then depends on how well 1 − s2 is approximated. As we will see later, this is the basic idea behind local NRBCs. The contribution [43, 44, 45] by Grote and Keller is another pioneering work in NRBC. Using the Dirichlet to Neumann (DtN) map, Grote and Keller derived the first exact NRBC on a spherical boundary. We note that the Grote and Keller NRBC is inherently three dimensional, since it is based on special properties of the spherical harmonics. It is believed that no such boundary conditions can be constructed in two space dimensions. This is related to the fact that the Green’s function associated with the wave operator in two space dimensions has an infinitely long tail. Similarly, starting from the Helmhlotz equation,

2 ξ uˆ = uˆxx + uˆyy, (2.5)

13 Hagstrom [52] employed the DtN technique to derive the exact boundary condition

∂u 1 ∂u −1 2 + + F |ky| K (|cky|t) ∗ F u = 0, x = 0, ∂x c ∂t (2.6) J (t) 1 Z 1 q K(t) ≡ 1 = 1 − ρ2 cosρtdρ, t π −1 which only involves a Fourier transform in the tangential direction and a con- volution in time. It has been reported that by the use of fast algorithms, together with the Fast Fourier Transform, the boundary condition (2.6) can be imposed directly [9].

2.2 Local NRBC Due to the non-locality of the exact boundary condition (2.3), Engquist√ and Majda [35] proposed the first hierarchy of local NRBC by replacing 1 − s2 with some rational√ approximations before inverting the Fourier transform. Ap- proximating 1 − s2 by Taylor expansions and including only the first term yields 1 u + u = 0, x = 0, t ≥ 0. (2.7) c t x This is the so–called first order Engquist–Majda boundary condition. The boundary condition (2.7) remains exact for a two-dimensional wave propagating normal to the boundary. Including the second term of the expansion yields the second order Engquist–Majda boundary condition 1 1 1 u + u − u = 0, x = 0, t ≥ 0. (2.8) c2 tt c tx 2 yy We see that including higher order terms of the Taylor expansion to increase the accuracy of the boundary condition in turn introduces higher order derivatives at the boundary. However, the inclusion of higher order terms of the Taylor expansion to improve accuracy of the approximation ceases to yield a well-posed problem. This can be cured by the use of Padé approximations, though. The Padé expansion is not the only possible choice. Other expansions like Chebyshev approximations have been studied [106]. Higdon [57] has a more general representation of these boundary conditions cosα ∂ ∂ cosα ∂ ∂ ( m + )···( 1 + )u = 0, x = 0, t ≥ 0, (2.9) c ∂t ∂x c ∂t ∂x where α1 ···αm, are arbitrary parameters. The second order Engquist–Majda boundary condition (2.8) corresponds to α1 = α2 = 0 radians. The higher order derivatives appearing in these boundary conditions greatly complicate

14 their use in any numerical scheme. As a result, first and second order boundary conditions are most commonly used in practice. In 1993, Collino [27] introduced a smart idea to remove higher order derivatives while retaining high order accuracy√ at the boundary. The fundamental idea lies in the approximations of 1 − s2 by Padé expansions, and conse- quently introducing a sequence of auxiliary variables φ j. Collino’s idea gave the opportunity to improve the work of Engquist and Majda. Local NBRCs sharing this structure are often referred to as high order local NBRCs, see [41]. Many different high order local NRBCs have been proposed in the past fifteen years, see [41, 53, 48, 107, 46]. The most important property of these high order local NRBCs is that arbitrary order of accuracy can be achieved by introducing more auxiliary variables. However, the convergence of the boundary conditions depends on the convergence of the Padé expansion. Starting from a complete plane wave representation of the time-dependent wave field incorporating both the propagating and evanescent modes, Hagstr- om et al. derived a new local NRBC [55] that annihilates outgoing waves in both propagative mode and evanescent mode regimes. They also presented strong numerical results indicating the long time efficiency of their model. The general structure shared by all high order local NRBC is that no derivatives beyond second order appear, and there are no normal derivatives of any of the auxiliary variables φ j. We point out that on a boundary which has corners, special corner conditions must be used in order for these boundary conditions to yield a well-posed problem. The major difference between exact NRBCs and approximate NBRCs is that exact NRBCs are non-local in both time and space. Storage requirement remains a drawback for exact NBRCs. Another practical challenge in the implementation of exact NBRCs is the shape of the boundary. The boundary can be very complex such that it becomes extremely difficult to have an explicit representation of the boundary conditions. We also note that while the discussion here is formulated in two space dimensions, the ideas carry over immediately to three space dimensions.

15 3. Absorbing layers

Another approach to truncate an unbounded spatial domain is to surround the computational (truncated) domain with an artificial absorbing layer of finite thickness, see figure 3.1. All absorbing layers are constructed by modifying the underlying equations such that solutions in the layer decay rapidly. For

Figure 3.1. A computational domain surrounded by the PML. this method to be effective, it is important that all waves traveling into the layer, independent of angle of incidence and frequency are absorbed without reflections. This approach is analogous to the physical treatment of the walls of anechoic chambers. Absorbing layers with these desirable features are called perfectly matched layers (PML). By perfect matching we mean that the interface between the computational domain and the layer exhibits a zero reflection coefficient1. In practice one takes advantage of this property by reducing the width of the layer dramatically, then choosing the damping coefficient as strongly as possible to minimize the reflections from the edge of the layer. The accuracy of the entire scheme is then determined by the numerical method used in the interior. The PML was first introduced for the Maxwell’s equations in the seminal paper [20] by J-P. Bérenger . In its original form [20], the PML is derived by

1A more general interpretation of perfect matching is that the restriction of the solution to the PML problem in the interior coincides with the solution to the original problem, see [10].

16 splitting the wave functions (magnetic and electric fields) before special lower order terms that simulate the absorption of waves are added. This makes it possible for the PML to absorb all waves exiting the computational domain without reflections, independent of angle of incidence and frequency. There are other formulations of the PML which do not require the unphysical splitting of the field variables, see for instance [24, 52, 10]. These are the so-called unsplit PML. The unsplit PML is more elegant mathematically. It is also more straightforward to be implemented in a computer program. The PML, both split [20] and unsplit [24, 52, 10], have also been extended to many application areas like acoustics, elasticity and quantum dynamics. In general, the PML can be interpreted as a complex change of spatial coordinates for the wave equation in the Fourier (or Laplace) space. The main focus of this thesis is on the PML. However, we also mention that there are other absorbing layers that have been developed, which do not have the perfect matching property. Before the emergence of the PML, Israeli and Orzag [60], Kosloff and Kosloff [66] had begun the construction of absorbing layers for wave equations. In [28, 29], Colonius and Ran proposed an absorbing layer (the super-grid scale model) for compressible flows, by stretching the grid and filtering high frequency components. The paper [12] by Appelö and Colonius extended the work of Colonius and Ran to linear hyperbolic systems, in first and second order formulations. In [34], Efraimsson and Kreiss performed a semi-dicrete analysis of a scalar linear model of an absorbing layer (buffer zone). Using their theoretical results Efraimsson and Kreiss suggested how to choose the layer parameters in order to enhance performance, then they applied the result to a fully non-linear problem (the Euler equations). Unlike the PML, these absorbing layers do not require the use of auxiliary variables. By construction the super-grid scale models [28, 29, 12] are linearly stable. They can be used for non-linear problems where the notion of perfect matching is obscure. However, for linear problems, it is doubtful whether these layers can compete with the PML. In this chapter, we review the PML. To begin, we introduce a simple model problem for second order strongly hyperbolic systems in two space dimensions. We will then derive a PML model using a plane wave decomposition that leads to stretching the physical spatial coordinates onto a carefully chosen complex contour, where spatially oscillating solutions are turned into exponentially decaying solutions. The second order model problem will be rewritten as a first order system. We will also derive a PML model corresponding to this first order system. Finally, we will comment on perfect matching, well-posedness and stability of the models.

17 3.1 Model problem Consider the model problem, the so-called anisotropic scalar wave equation,

utt = uxx + uyy + (αuy)x + (αux)y , α,x,y ∈ R, |α| < 1, (3.1) describing electromagnetic waves propagating in an anisotropic dielectric media. If |α| < 1, it is easy to show that (3.1) is strongly hyperbolic, see Paper II.

3.2 Construction of the PML equations The idea is to introduce new coordinates deﬁned by special complex metrics, see [102, 24]. To begin with, we take the Fourier transform in time Z ∞ uˆ(x,y,ω) = u(x,y,t)e−iωtdt, −∞ and consider the time-harmonic problem 2 − ω uˆ = uˆxx + uˆyy + (αuˆy)x + (αuˆx)y . (3.2) The wave equation (3.2) admits plane wave solutions,

−iω(k1x+k2y) uˆ(x,y,ω) = u0e .

Here, (k1,k2) = (kx/ω,ky/ω) are real, u0 is a constant amplitude, and (kx,ky) is the wave vector. Let us consider the case where we want to compute the solution in the half– plane x ≤ 0, and the PML is introduced outside that half–plane, that is, in x > 0. We now look for the modiﬁcation of the wave equation such that it has the exponentially decaying solutions

−k1Γ(x) −iω(k1x+k2y) vˆ(x,y,ω) = u0e e , in the PML. Here, Γ(x) is a real valued non–negative increasing smooth function, which is zero for x ≤ 0. The decaying solution can be rewritten as

Γ(x) −iω k1 x+ iω +k2y vˆ(x,y,ω) = u0e . This is a plane wave solution to the wave equation in the transformed coordinates (x˜,y), where Γ(x) x˜ = x + . iω Next, we analytically continue the wave equation (3.2) onto the complex con- tourx ˜ where spatially oscillating solutions are turned into exponentially decaying solutions. We have the complex coordinates transformation

∂ 1 ∂ dx ˜ σ1 (x) dΓ(x) = , s1 := = 1 + , where σ1 (x) = . (3.3) ∂x˜ s1 ∂x dx iω dx 18 In (3.3), σ1 (x) ≥ 0 is called the damping function. The PML is then derived by ﬁrst replacingu ˆ withv ˆ in (3.2), and applying this complex change of variables (3.3) to the wave equation (3.2) (deﬁned forv ˆ), thus yielding 2 1 1 1 1 − ω vˆ = vˆx + vˆyy + (αvˆy)x + α vˆx . (3.4) s1 s1 x s1 s1 y We note that in order to enhance the absorption and stability properties of the layer more complicated complex metrics s1 have been proposed in literature, see [11] and Paper I. Notice that the plane wave satisfying (3.4) is

−iω(k1(s1x)+k2y) vˆ(x,y,ω) = e u0. (3.5) Also note thatu ˆ solves the wave equation (3.2) in the half–plane x < 0 andv ˆ is the solution of the PML equation (3.4) in x > 0. The wave equation (3.2) is perfectly coupled to (3.4) with the coupling condition

uˆ(0,y,s) = vˆ(0,y,s), uˆx (0,y,s) = vˆx (0,y,s). (3.6)

Note that the coupling condition (3.6) is achieved if σ1 (0) = 0. We localize the PML in time by ﬁrst introducing the auxiliary variables,

1 vˆ vˆy φˆ = x , ψˆ = . s1 iω iω Inverting the Fourier transforms yields the time–dependent PML equations

vtt + σ1vt = vxx + vyy + (αvy)x + (αvx)y − (σ1φ)x + (σ1ψ)y, (3.7) φt = vx − σ1φ, ψt = vy.

We note in passing that when σ1 (x) ≡ 0 we recover the wave equation (3.1).

3.2.1 First order formulation In order to demonstrate the construction of the PML for a ﬁrst order system, we introduce extra variables and rewrite (3.1) as a ﬁrst order system in time and space. T ut = Aux + Buy, with u = (u1,u2,u3) , (3.8) 0 1 0 0 0 1 where A = 1 0 0, B = α 0 0, and α 0 0 1 0 0

u1 = ut, u2 = ux + αuy, u3 = uy + αux.

19 To derive the corresponding PML model for the ﬁrst order system (3.8) we take the Fourier transform in time and apply the complex change of variables (3.3) in the x-direction. Then choosing the auxiliary variable w and inverting the Fourier transform, we have

vt = Avx + Bvy − σ1Aw, (3.9) wt = vx − σ1w. Also we comment that the auxiliary variables in (3.7) and (3.9) will inﬂuence the solutions, v and v, only in the layers where σ1 6= 0.

3.2.2 Perfect matching The most important property of the PMLs (3.7) and (3.9) is the perfect matching. This means that the restriction of the solutions to (3.7) and (3.9) in the half–plane x < 0 coincides with the solutions to (3.1) and (3.8), respectively. There are two standard methods [10, 102] that have been used to study the perfect matching property of the PML. The approach [102] uses plane wave analysis and only accounts for propagating modes. Here, we use the technique [10] which is rooted in the construction of the general solution to the wave equation in the Laplace–Fourier space. The technique [10] is more general since it includes both the propagating mode regime and the evanescent mode regime. To start with, we take the Laplace tranform in time t → s and the Fourier transform in the tangential direction y → iky. The equations (3.7) and (3.9) are perfectly matched if

u¯(x,iky,s) = v¯(x,iky,s), andu ¯x (x,iky,s) = v¯x (x,iky,s), x ≤ 0, (3.10) u¯ (x,iky,s) = v¯ (x,iky,s) x ≤ 0. The variables u,v,u,v are related tou ¯,v¯,u¯,v¯ via the inverse Laplace–Fourier transformation. Observe that the perfect matching of the second order PML (3.7) requires continuity of the solution and its normal derivative across the interface, while the ﬁrst order PML (3.9) is perfectly matched if the solutions are continuous across the interface. We can construct modal solutions

λx λx u¯ = e u¯0 (iky,s), u¯ = e u¯ 0 (iky,s), (3.11) for the problem in the half–plane x ≤ 0, and

1 R x 1 R x λ(x+ σ1(z)dz) λ(x+ σ1(z)dz) v¯ = e s 0 u¯0 (iky,s), v¯ = e s 0 u¯ 0 (iky,s), (3.12) for the PML problem in the half–plane x ≥ 0. Direct calculations show that the ﬁrst order PML (3.9) is perfectly matched by construction for arbitrary σ1, while perfect matching is achieved if σ1(0) = 0 in the second order case (3.7). In computations however, additional smoothness at the interface is often beneﬁcial.

20 Remark 1 The modal PMLs [52, 11, 10] are derived by ﬁrst assuming a modal ansatz of the form (3.12). From the modal ansatz the complex metric (3.3) is derived. The PML is then constructed by performing the complex change of variables.

3.3 Well–posedness of the PML An important mathematical property of a partial differential equation is well– posedness. By a well–posed problem, we mean that there is a unique solution which depends continuously on the data of the problem. To be precise, consider the Cauchy problem ∂ ∂ ut = P , u, t ≥ 0, ∂x ∂y (3.13) u(x,y,0) = u0(x,y).

Here, (x,y) ∈ R2 are the spatial coordinates and the symbol P(∂/∂x,∂/∂y) denotes the spatial operator. The Cauchy problem (3.13) is weakly (resp. strongly) well–posed if for every t0 ≥ 0,

2 κ(t−t0) 2 ||u(t)|| ≤ Ke ||u0||Hs , (3.14) s for u0 given in the Sobolev space H , with s > 0 (resp. s = 0). Here κ and K are independent of u0 and t0. We demonstrate the well–posedness of the second order and ﬁrst order PML models (3.7) and (3.9). By introducing auxiliary variables we can rewrite (3.7) as a ﬁrst order system in time and space.

Ut = A1Ux + A2Uy + σ1A3U, (3.15) 0 1 2α 0 0 0 0 1 −1 0 0 0 1 0 0 0 0 0 0 0  0 −1 0 0 A1 =  , A2 =  , A3 =  . 0 0 0 0 1 0 0 0  0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 2 It is easy to show that ∀S = (Sx,Sy) ∈ R normalized to satisfy 2 2 Sx + Sy + 2αSxSy = 1, (3.16) the matrix Aˆ = SxA1 + SyA2 has real eigenvalues and a complete system of eigenvectors. It follows that the PML model (3.7) is strongly hyperbolic, thus strongly well–posed, see [49]. However, because of the lower order term σ1A3U the system (3.15) can have solutions that grow in time. Now consider the PML (3.9) for the ﬁrst order system (3.8). We can rewrite the PML model (3.9) as follows

Vt = B1Vx + B2Vy − σ1B3V, (3.17)

21 A 0 B 0 A 0 B = , B = , B = . 1 I 0 2 0 0 3 0 0

Since the matrix A in (3.8) has a zero eigenvalue, the matrix Bˆ = SxB1 + SyB2 is not diagonalizable. Therefore, the PML model (3.9) is weakly hyperbolic, and weakly well-posed. This is also true for Bérenger’s PML. If we can rewrite (3.1) as a first order system such that the coefficient matri- ces A and B are invertible it is possible to construct a strongly hyperbolic PML. This is possible for the acoustic (and advective acoustic) wave equations. But for more complex problems such as the elastic wave equations, rewriting the second order system as a first order system will introduce a non propagating mode which will lead to a weakly well-posed PML, see [17, 11]. However, many examples and computations in literature [19, 26, 11, 17] indicate that, for linear constant coefficient problems, loss of strong well- posedness may not be ‘disastrous’. We note that if PML models derived for linear constant coefficient problems are to be used for variable coefficient or non-linear problems, it is essential that PML equations are strongly well- posed. It is also important to note that the discussions here are for Cauchy problems. When physical boundary conditions are important the analysis must be adjusted by considering the theory for IBVPs [74, 75], see also Papers IV and V.

3.4 Stability of the Cauchy PML Here, we are interested in the temporal behavior of the solution of the PML in the absence of physical boundaries. A typical set–up is a computational domain completely surrounded by the PML as in figure 3.1. For time–dependent problems, it is not sufficient that the PML is well–posed, it must also be stable. By definition well–posed problems support exponentially growing solutions. This is of course undesirable of an absorbing model. The Cauchy problem (3.13) is weakly (resp. strongly) stable if for every t0 ≥ 0, 2 s 2 ||u(t)|| ≤ K (1 + γt) ||u0||Hs , (3.18) s for u0 giving in the Sobolev space H , with s > 0 (resp. s = 0). Here K,γ are independent of u0 and t0. A necessary condition for weak stability is that all eigenvalues λ j of the symbol P(ikx,iky) have non–positive real parts Perhaps, the earliest (numerical) instability of the PML are reported in [58, 4]. When using the (split-field) PML for the linearized Euler equation, Hu [58] employed a filter to ensure that all fields in the layer decay with time. In [4], Abarbanel and Gottlieb performed a mathematical analysis of Bérenger’s PML for Maxwell’s equation and found that the PML is only weakly well– posed, and under certain perturbations the solutions of the PML can grow

22 exponentially in time. Because of the result [4] several unsplit PML were developed [1]. In the paper [3], Abarbanel et al. studied the longtime stability of these unsplit PML. The result is that these layers suffer from long time growth due to the Jordan block present in the lower order damping term. Bécache and Joly [19] used standard Fourier techniques and energy methods to establish the well–posedness and stability of Bérenger’s PML for Maxwell’s equations. They reported that though Bérenger’s PML is weakly well–posed, the damping term appearing in the layer can at the worst lead to a linear growth. In a subsequent paper [18], Bécache et al. showed that the introduction of the complex frequency shift [77] eliminates the long time growth in the unsplit PML. While the growth found in [4] is not fatal (since it appear after a very long time), the growth observed in [58] can be destructive. It is inherent in the underlying physical problem. Similar growths were also reported for the elastic wave equation in [17]. In [17], Bécache et. al. established an important but negative stability result. They found that the shape of the slowness curve for an arbitrary first order hyperbolic system determines whether a stable split–field PML can be constructed. This is related to the existence of propagating modes with oppositely directed phase and group velocities, that is the so–called backward propagating modes. It turns out that this is also true for the modal PML, see [11, 10]. In Paper I, we have also shown that the second order PML for elastic wave equations can support growing solutions when this geometric stability condition is violated. However, for some (simple) problems such as (3.1), there are linear transformations which can modify the slowness diagrams such that a stable PML can be constructed, see [14, 30]. For more complex systems like the elastic wave equations such linear transformations may not be possible. We conclude that the stability properties of the continuous first order PML model (3.9) and the second order PML model (3.7) are linearly equivalent. The PMLs (3.7) and (3.9) can support growing solutions if α 6= 0. As we commented earlier this problem can be cured by introducing extra terms in the PML which modifies the slowness diagrams, see [14] or introducing such a linear transformation first before the PML is derived [30].

3.5 Stability of the PML for IBVPs For time–dependent PDEs, the introduction of boundary conditions often leads to a more complicated mathematical problem to be analyzed. In the discrete setting, more challenging numerical questions can also arise. A PML which is Cauchy stable can exhibit growth if boundary conditions are included. For instance in isotropic elastic waveguides, not only that backward propagating modes can be supported [97]. In the discrete setting, if the boundary condi-

23 tions are not well treated, the PML can pollute the numerical solutions everywhere [97, 12]. For ﬁrst order hyperbolic PDEs, the theory of IBVPs is rather well–developed. The Friedrichs theory [37, 38, 39] for symmetric systems with maximally dis- sipative boundary conditions was available, but not sufﬁcient for non-symmetric systems or more general boundary conditions. A more general theory is the Kreiss theory [70, 49, 68] based on the normal mode analysis [7]. The Kreiss technique can be summarized by the following. Consider the problem (3.13) on the upper half–plane (∞ < x < ∞, 0 < y < ∞), with boundary conditions at y = 0, Lu = g(x,t), y = 0. (3.19)

Here L is a linear boundary operator and g(x,t) is the boundary data. To begin with, • localize the problem by freezing all coefﬁcients • split the problem into a Cauchy problem and a half–plane problem with boundary conditions at y = 0. The corresponding Cauchy problem can be analyzed using standard Fourier methods. The half–plane problem is analyzed by a Laplace–Fourier technique. That is, we take the Laplace transform in time t → s, and the Fourier transform in the tangential direction x → ikx, thus yielding an ODE

d suˆ (ikx,y,s) = P ikx, uˆ (ikx,y,s) + fˆ(ikx,y,s), ℜs > 0, dy (3.20) Luˆ (ikx,0,s) = gˆ (ikx,s), to be solved on the half–line 0 < y < ∞. The complex function fˆ(ikx,y,s) is given by the initial data. The ODE (3.20) with homogeneous initial fˆ(ikx,y,s) ≡ 0 and boundary gˆ (ikx,s) ≡ 0 data, corresponds to a non–linear eigenvalue problem with s as an eigenvalue. If there are non–trivial solutions of the eigenvalue problem with ℜs > 0, the IBVP (3.13) with (3.19) is ill–posed. The boundary condition (3.19) is not useful in a practical sense, and is therefore discarded. Solving the eigenvalue problem (3.20) with homogeneous data we arrive at the linear system

ϒ(kx,s)δ = 0, ℜs > 0, (3.21) where ϒ(kx,s) is a square matrix determined by the boundary operator L, δ is a set of parameters to be determined. To ensure well–posedness, we require δ ≡ 0 for all ℜs > 0, thus obtaining the determinant condition

F(kx,s) ≡ detϒ(kx,s) 6= 0, ℜs > 0. (3.22)

24 The IBVP (3.13) with (3.19) is not well-posed if there are roots s with ℜs > 0 satisfying the characteristic equation

F(kx,s) ≡ detϒ(kx,s) = 0. (3.23)

The Kreiss theory has recently been extended to second order hyperbolic systems [75]. However, the PML is a more complicated system. For second order systems, the PML is a combination of a second order system of PDEs for the transformed wave equation and a ﬁrst order system of PDEs for the auxiliary differential equations. Unfortunately no stability theory exists for the corresponding IBVP including the PML. Skelton et al. [97], investigated the stability of the PML for elastic waveguides, in the frequency domain. They found that an elastic waveguide bounded by free–surface boundary conditions can support backward propagating modes, and these modes deteriorate the performance of the PML. However, they suggested that the problem could be cured by constructing a frequency–dependent PML with a damping whose sign depends on the frequency. It is non–trivial to extend their results to the time–domain. In Papers III, IV and V we study the stability of the PML for IBVPs. In order to demonstrate the stability of the PML for IBVPs, we will again consider the model problem (3.1) on the upper half–plane 0 < y < ∞. At y = 0 we set the ﬂux to zero, having

uy + αux = 0, y = 0. (3.24)

This is a model for the free–surface boundary condition in linear elastodynamics. Note that the wave equation (3.1) with the boundary condition (3.24) satisfies a strict energy estimate. However, if we use the physical boundary condition (3.24) with the PML (3.7) the solution explode immediately2, see figures 3.2 and 3.3. In fact, the system (3.7) with (3.24) is unstable in the sense of Kreiss. Note also that when the physical boundary condition (3.24) only is used in the PML, the derivation of the PML is not consistent on the boundaries. We know that the PML is derived by the complex change of variables ∂/∂x → 1/s1∂/∂x, defined in equation (3.3). Since the boundary condition (3.24) con- tains ux, in order to ensure a consistent derivation of the PML on the boundary we must modify the boundary condition (3.24) by applying the complex change of variables (3.3) to ux. Firstly, we take Fourier transform in time and apply the complex change of variables (3.3) to the physical boundary condition (3.24). Secondly, we chose auxiliary variables and invert the Fourier transforms. Depending on the choice of our auxiliary variables we obtain two

2Note that when α 6= 0 the Cauchy PML can support growth independent of the boundary conditions. However, we have used |α| = 0.5 and 10 grid points in the PML such that the discrete Cauchy PML is stable, see Paper II for more details.

25 possible mathematically equivalent boundary conditions

vy + αvx + σ1ψ = 0, y = 0, (3.25) or vy + αvx − σ1αφ = 0, y = 0, (3.26) in the PML. When σ1 (x) ≡ 0 we recover the physical boundary condition (3.24). Note that both (3.25) and (3.26) are mathematically equivalent. Using the Kreiss technique, we can show that the problem (3.1) with the boundary condition (3.25) or (3.26) does not support temporally growing modes. How- ever, an SBP–SAT, described in the next chapter, approximation of the PML (3.7) with the boundary conditions (3.25), (3.26) results in two different discrete systems of equations. Using (3.26) results in temporally growing discrete solution while (3.25) supports no growing solution, see ﬁgure 3.2. For more details and results we refer the reader to Papers IV and V.

2 10 BC1 BC2 Physical BC 0 10 energy −2 10

−4 10 0 50 100 150 200 time

Figure 3.2. Time history of the L2 norm of the energy kuk with various boundary conditions in the PML. The curve BC1 corresponds to using the transforemd boundary condition (3.25), the curve BC2 corresponds to using the transformed boundary condition (3.26) and the curve Physical BC corresponds to using the physical boundary condition (3.24) without modiﬁcation.

3.6 Stability of the discrete PML In literature, very little attention has been paid to the study of stability of discrete PML models. This is in part due to the fact that if a continuous PML model supports growth, it may be very difﬁcult to construct accurate and stable discrete approximations. However, in the discrete setting, derivatives are

26 time = 1.5 time = 3

0.25 0.2

0.2 0.15 0.15

0.1 0.1

0.05 0.05

time = 1.5 time = 3

0.45 0.45 0.4 0.4 0.35 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05

Figure 3.3. Snapshots of the wave ﬁeld |u| (with |α| = 0.5) in a rectangular waveguide bounded on the top and the bottom by the boundary conditions (3.24) in the y–direction. In the x–direction the domain is truncated by a PML containing 10 grid points. The upper panel corresponds to using the transformed boundary condition (3.25) in the PML. The lower panel corresponds to using the physical boundary condition (3.24), without the PML transformation. Initially, both solutions are the same. As time passes the physical boundary condition (3.24) in the PML initiates an explo- sive growth, which propagates ﬁrst inside the PML, and then spreads everywhere.

approximated, for example by ﬁnite differences. In addition, the discrete PML is a ﬁnite width layer containing only a few grid points. For a given discretization the instability or stability in the continuous model can be strengthened or weakened. Therefore a continuous analysis, still very useful, might be incomplete or too restrictive for the discrete PML. A relevant article to this study is the recent work [2]. In this paper, Abar- banel et al. performed a systematic experimental study of the long-time behavior of discrete unsplit PML models for the Maxwell’s equations. They found that the long-time stability of a given unsplit PML also depends on the chosen discrete approximation. The study of discrete stability of the PML is also a topic of Papers I, II and V. In Paper II, we determined the number of grid points needed to ensure a stable discrete PML for several anisotropic elastic materials violating the geometric stability condition. In Paper V, we formulate transformed free–

27 surface boundary conditions in the PML, for elastic wave equations suitable for the SBP–SAT schemes.

28 4. Finite difference methods for second order systems

Second order hyperbolic systems often describe problems where wave propagation is dominant.Typical examples are the scalar wave equation, Maxwell’s equations, the elastic wave equations and Einstein’s equations of general rel- ativity. In the numerical treatment of second order systems, the equations are often rewritten and are solved as a first order system. This is in part due to the maturity of the theory and numerical techniques developed for CFD. There are several benefits with solving the equations in second order formulation, though. One advantage is that less degrees of freedom are required, and for the same accuracy we need fewer number of grid points per wavelength. A second advantage is that we can avoid the spurious high frequency modes which can be poisonous when using central difference schemes for first order systems. However, while the theory and numerical methods for first order hyperbolic systems are well developed, numerical techniques to solve second order hyperbolic systems are less developed. For wave propagation problems, it has been demonstrated that high-order accurate time marching methods as well as high-order spatially accurate schemes are more efficient for problems on smooth domains [50, 69]. A major difficulty for high order finite difference schemes is the numerical treatment of boundary conditions. For other methods such as finite element methods [61] and spectral element methods [63, 65], numerical enforcement of boundary conditions can be more straightforward. In addition, the flexibility of numerical computations on unstructured meshes allows for the resolution of complicated geometries. However, computational efficiency has continued to make computations on structured meshes attractive. For instance, numerical algorithms formulated with finite difference approximations are more intuitive, simple to analyze, and can be readily implemented. For high order finite difference methods, a methodology that ensures stable boundary treatment is the SBP-SAT scheme [84, 85], see also Papers III and VI. The SBP-SAT scheme uses high order finite difference operators that satisfy the so-called summation-by-parts (SBP) rule. Boundary conditions are imposed weakly using the Simultaneous Approximation Term (SAT) method [23]. The scheme leads to a strictly stable approximation. In this chapter, we will briefly describe the SBP-SAT schemes. For more elaborate discussions we refer the reader to the suggested references above.

29 4.1 The wave equation To begin, we consider the scalar wave equation in one space dimension,

utt = uxx + f (x,t), x ∈ (a,b), t > 0, (4.1) u(x,0) = u0 (x), ut (x,0) = v0 (x).

Here, u0(x), v0(x) and f (x,t) are smooth functions which are compactly supported in (a,b). In order to obtain a well-posed problem, we augment (4.1) with compatible boundary conditions,

α1ut − ux + γ1u = 0, x = a, (4.2) α2ut + ux + γ2u = 0, x = b, where, αi ≥ 0,γi ≥ 0 (i = 1,2) are real parameters. To begin with, we introduce the standard L2 scalar product and the corresponding norm Z b (u,v) = vudx, kuk2 = (u,u). a Let us deﬁned the energy by [32]

2 2 2 2 Eu(t) := kutk + kuxk + γ1|u(a)| + γ2|u(b)| . (4.3)

Multiplying (4.1) with ut, adding the conjugate of the product, and integration– by–parts yield d E (t) ≤ ( f ,u ) + (u , f ) dt u t t ≤ 2kutkk f (t)k. p p p We write Eu(t) = Eu(t) Eu(t). Using the fact Eu(t) ≥ kutk, yields p p Z t Eu(t) ≤ Eu(0) + k f (τ)kdτ. (4.4) 0 The energy (4.3) is bounded. Thus the problem (4.1) with (4.2) is well–posed. If we can derive discrete approximations of (4.1) with (4.2), and a corresponding discrete energy satisfying (4.4) we say that the discrete approximation is strictly stable. This means that the numerical approximations do not allow any growth not called for by the partial differential equations. For time–dependent problems, strict stability is an important property, particularly if long-time calculations are desired. We note that for IBVPs, it can be difﬁcult to derive strictly stable and higher order accurate schemes. In order to derive strictly stable discrete approximations, it is important that the discrete operators approximating the spatial derivatives ∂/∂x, ∂ 2/∂x2 mimic the properties of the continuous operators. Note that the main tool in deriving the estimate (4.4) is the integration–by–parts property.

30 4.2 Integration–by–parts Consider the scalar field u(x,t) defined on the unit interval x ∈ [0,1]. The spatial operators ∂/∂x, ∂ 2/∂x2 satisfy ∂u Z 1 ∂u ,u = u dx {integration–by–parts} ∂x 0 ∂x (4.5) ∂u = − u, + u(1)2 − u(0)2, ∂x ∂ 2u Z 1 ∂ 2u ,u = u dx {integration–by–parts} x2 x2 ∂ 0 ∂ (4.6) ∂u 2 ∂u ∂u = − + u(1) (1) − u(0) (0). ∂x ∂x ∂x It is possible to construct discrete approximations of the spatial derivatives ∂/∂x, ∂ 2/∂x2 mimicking the integration–by–parts properties (4.5), (4.6) in a certain discrete norm. Such operators are called summation–by–parts (SBP) operators, and are topics of several works, see for instance [87, 50, 100, 86, 84, 85, 99]. Below, we will briefly describe SBP operators.

4.3 Summation–by–parts Let u(x,t) be discretized on the uniform grid 1 x = ( j − 1)h, h = , j = 1,··· ,N, j N − 1 in the unit interval [0,1]. The grid function at x j is denoted v j(t) and the semi- T discrete solution vector is v(t) = [v1(t),v2(t),v3(t),...,vN(t)] . We introduce the discrete scalar product T hv,wiH := w Hv, (4.7) where H is symmetric and positive definite. The scalar product (4.7) defines a norm. Note that with H = hI (where I is the identity operator) we get the standard discrete norm. Let D1, D2 be discrete operators approximating the first and the second derivatives respectively, that is ∂ ∂ 2 D ≈ , D ≈ . 1 ∂x 2 ∂x2

The discrete operators D1, D2 are called SBP operators if they satisfy the following properties

−1 T D1 = H Q, Q + Q = BN, T T (4.8) BN = diag(−1,0,0,...,1), H = H , v Hv > 0 ∀v 6= 0,

31 −1 T D2 = H (−M + BNS), M = M , (4.9) vT Mv ≥ 0, H = HT , vT Hv > 0, ∀v 6= 0. Here, H deﬁnes a norm, Q is almost a skew-symmetric operator, M is called the symmetric part of the operator D2 and the operator S is a consistent approximation of the ﬁrst derivative ∂/∂x on the boundaries. Note that D1, D2 satisfy T hD1v,viH = v HD1v = hQv,vi T T = v −Q + BN v 2 2 = −hv,D1viH + v1 − v0, and T T hD2v,viH = v HD2v = v (−M + BNS)v T = −v Mv + vN(Sv)N − v0(Sv)0.

Thus the operators D1, D2 completely mimic the integration by part properties (4.5), (4.6) in the H–norm. Discrete operators satisfying the summation–by–parts properties (4.8) and (4.9) are usually derived using even order standard centered difference schemes in the interior. Close to the boundaries lower order schemes are used such that the operators satisfy (4.8) and (4.9) in a given H–norm. Note that the operator H can be a diagonal norm or a block norm. When using block norms, it can be difﬁcult to prove stability if variable coefﬁcients or curvilinear grids are considered, see [101]. We have used diagonal norms throughout in this thesis.

4.4 Weak boundary treatment with SAT The construction of SBP operators does not include boundary conditions. For IBVPs, the SBP property alone usually does not ensure a strictly stable scheme. To ensure strict stability, special boundary treatments are required. Here, we discuss a particular boundary procedure, the SAT method [23]. The main idea is to discretize the PDE (4.1) separately using SBP operators, and the boundary conditions (4.2) are discretized using special one–sided difference operators. The semi–discrete PDE and the semi–discrete boundary conditions are then patched together using penalties. The penalty strengths are determined by requiring stability. The semi–discrete approximation of the boundary conditions (4.2) is

Bα vt + BNSv + Bγ v = 0, (4.10) where

Bα = diag(α1,0,...,α2), BN = diag(−1,0,...,1), Bγ = diag(γ1,0,...,γ2),

32 and S is the special one–sided approximation of the ﬁrst derivative ∂/∂x, appearing in the second derivative SBP operator D2, deﬁned in (4.9). A SBP– SAT discretization of (4.1) with the boundary condition (4.2) reads

vtt =D2v − SAT + F(t), (4.11) v(0) = u0, vt (0) = v0, where the SAT term is deﬁned by

−1 SAT = τnH Bα vt + BNSv + Bγ v , and τn = 1 is a penalty. We can rewrite (4.11) as

vtt = Bvt + Dv + F(t), (4.12) v(0) = u0, vt(0) = v0. Here, −1 −1 D = −H M + Bγ , B = −H Bα . Let the semi-discrete energy be deﬁned by,

2 T 2 2 Ev(t) := kvtkH + v Mv + γ1v1 + γ2vN. (4.13) T Multiplying (4.12) with vt H, adding the transpose of the product leads to d E (t) ≤ hf,v i + hv ,fi dt v t H t H ≤ 2kvtkH kfkH . p p p As before, we write Ev(t) = Ev(t) Ev(t), and use Ev(t) ≥ kvtkH , to obtain p p Z t Ev(t) ≤ Ev(0) + kf(τ)kH dτ. (4.14) 0 The semi–discrete approximation (4.11) is strictly stable. In Paper VI, we derived SBP–SAT schemes for more general second order hyperbolic systems. For Dirichlet boundary conditions, it is possible to impose the boundary conditions strongly, by injection. The injection method for Dirichlet conditions in the SBP framework, for systems of second order hyperbolic PDEs is also discussed in Paper VI.

33 5. Summary of papers

This chapter presents the summary of the individual papers included in this thesis. The material contained in this thesis discuss the construction, analysis and numerical implementation of the perfectly matched layer for second order wave equations. We investigate the stability of both the continuous and the discrete PML models. We also derive strictly stable ﬁnite difference approximations suitable for systems of second order hyperbolic PDEs. The sections below brieﬂy describe each paper, and highlight the main results in the papers.

5.1 Paper I In this paper, we construct a general PML for second order hyperbolic systems, focusing on the linear anisotropic elastic wave equations in two space dimensions. The layer equations are derived by applying a complex coordinate stretching directly to the second order equations. The resulting system is strongly hyperbolic. When the geometric stability condition is violated, by a standard perturbation argument our PML at constant coefficients suffers from the same high frequency instability affecting the modal PML and the split field PML for first order systems. However, in computations using standard second order finite differences, our PML behaves much better than a standard first order PML. We have found several reasons for this. In a discrete setting the unstable modes may be of higher frequency than can be represented, or well represented, on the grid. The temporal behavior of such modes cannot be expected to be predicted by continuous analysis. We also show that this effect can be enhanced by coordinate compression and the complex frequency shift in the layer. However, coordinate compression increases the stiffness of the problem. Secondly, we observe that the geometric instability gives rise to growing modes, with bulk localized to part of the layer, and propagating tangentially. If a Cartesian domain is surrounded by layers, the bulk of the unstable mode eventually moves into a corner region, and decays. We analyze the stability properties of the corner region, as before. The result is that, the corner region does not support temporally growing solutions. All waves propagating into the corner region will decay immediately.

34 5.2 Paper II In Paper I, numerical experiments using standard second order central finite differences for anisotropic elastodynamics demonstrate that the PML in first order formulation is very sensitive to high frequency instability. The corresponding second order formulation behaves dramatically better. In this paper we investigate this observation from a theoretical point of view. The result is that, if the so–called geometric stability condition is violated, the spurious modes introduced by a straightforward discretization of the first order formulation lead to high frequency instability at most resolutions. In the corresponding second order discretization the PML supports growth only if growing modes are well resolved. In applications, the PML is used only in a few grid points around the boundaries of the computational domain. A continuous analysis is still very useful though, but it can be incomplete or too restrictive for the discrete PML. We analyze a discrete PML and determine the number of grid points needed in the layer to ensure stability, for several anisotropic elastic materials. Numerical experiments are presented to verify the theoretical results.

5.3 Paper III The PML is usually derived by assuming a homogeneous medium and an infinite domain in all directions. In a domain with physical boundaries, in order to avoid unwanted reflection from the interface, the underlying boundary conditions must be accurately extended from the interior into the PML. In addition, once the PML is discretized and truncated to a finite width layer for numerical simulations on the computer, the discrete PML is no longer a perfectly absorbing medium. The PML parameters must be tuned and optimized, by a combination of discrete analysis and numerical experiments in order to achieve optimal performance. This article begins the study of the stability, accuracy and the grid convergence properties of the PML for transient wave propagation problems, in the presence of physical boundaries. As a model, we consider a semi-infinite waveguide governed by the scalar wave equation. Two commonly used physical boundary conditions, homogeneous Neumann and homogeneous Dirichlet conditions, are considered. For this simple case of a uniform isotropic acoustic waveguide, there are no modes with oppositely directed phase and group velocities –the so–called backward propagating modes. However, to ensure the accuracy of the PML, we derive a set of equivalent boundary conditions. At constant coefficients, we prove that the PML subject to homogeneous Neu- mann and homogeneous Dirichlet conditions is stable. At variable coefficients we derive energy estimates. We demonstrate that a straightforward imposition of the Neumann boundary conditions in the PML can pollute the numerical solution in the interior. In the discrete setting, the modified boundary conditions

35 are essential in deriving discrete energy estimates analogous to the continuous energy estimates, thus proving numerical stability and convergence of the discrete PML.

5.4 Paper IV There are many wave propagation problems where boundary phenomena such as surface or glancing waves are dominant. Typical examples are in non- destructive testing, seismology, earthquake engineering, ultrasonics, and ground penetrating radar (GPR) technologies. These problems can be described by symmetric time–dependent partial differential equations (PDE) in semi-bounded domains. For time–dependent partial differential equations, the introduction of boundary conditions often leads to several difﬁculties. For instance, a PML which is Cauchy stable can support growth (or become ill–posed) when boundary conditions are introduced. From a theoretical point of view, IBVPs are much more difﬁcult to analyze than the corresponding Cauchy problems. In this paper, we study the stability of the perfectly matched layer (PML) for symmetric second order hyperbolic partial differential equations on the upper half plane, with boundary conditions at y = 0. First we derive a PML that truncates the boundary in the x–direction. We also modify the x–derivatives (∂/∂x) in the boundary condition with the PML metric s1 (i.e. ∂/∂x → 1/s1∂/∂x). Then by choosing auxiliary variables, we construct new boundary conditions in the PML. Using a mode analysis, we develop a stability theory for the corresponding IBVP. A main result is that if the characteristic function, similar to (3.23), for the undamped problem is a homogeneous function1, then the introduction of the PML will immediately move all non–zero roots of the characteristic equation into the stable half–plane. The zero roots are not per- turbed by the PML. We apply our technique to the PML for the elastic wave equation subject to free surface and homogeneous Dirichlet boundary conditions, and to the PML for the curl–curl Maxwell’s equation subject to insulated walls and perfectly conducting walls boundary conditions. The conclusion is that these half–plane problems do not support temporally growing modes.

5.5 Paper V In this paper, we consider the PML in a time–dependent isotropic elastic waveguide. The waveguide is bounded in the y–direction by free–surface boundary conditions, but extends inﬁnitely in the x–direction. Our ﬁrst objective is to extend the analysis of the half–plane problem in Paper IV to a rectan-

1 k Let f (v) be a function with the vector argument v. If f (αv) = α f (v) with α 6= 0 and k ∈ Z, then f (v) is a homogeneous function of degree k.

36 gular waveguide bounded on the top and the bottom by free–surface boundary conditions. We show that if the waveguide supports backward propagating modes, the continuous PML can support temporally growing solutions. A useful practical point is that the frequency bands where backward propagating modes exist are small and most prominent at lower spatial frequencies (or wave numbers). For a ﬁnite width PML, numerical approximations can immediately exclude backward propagating modes in the PML. In Paper IV, we derive a PML, together with two mathematically equivalent set of boundary conditions in the PML. The second objective of this paper is to derive a stable numerical scheme for the PML, and numerically evaluate the boundary conditions proposed in Paper IV. Note that the resulting boundary conditions are more complicated than the original free–surface boundary conditions. One strength of the SBP–SAT machinery is that boundary conditions can be easily imposed using penalties. We propose to use the SBP framework [85, 50]. The boundary conditions are imposed weakly using the SAT method [23]. However, an SBP–SAT approximation of the PML for the two mathematically equivalent boundary conditions yield two different discrete systems of equations. While the ﬁrst boundary condition supports only non–growing solutions, the second boundary condition is unstable.

5.6 Paper VI Paper VI presents a systematic way to obtain high order accurate and strictly stable finite difference approximations for systems of second order hyperbolic PDEs. The paper focuses on the equations of linear elastodynamics. We consider a class of boundary conditions yielding an energy estimate. For the elastic wave equation in second order formulation, many boundary conditions relevant in applications consist of a combination of normal and tangential derivatives, and possibly time derivatives on the boundaries. A main difficulty with the second order formulation is that, it is not obvious how to discretize both normal and tangential derivatives, and enforce the discrete boundary conditions in an explicit and stable manner. One of such boundary conditions, and probably the most important, in geophysics and structural mechanics is the free–surface or traction–free boundary condition. The free–surface boundary condition corresponds to setting the traction (stresses) to zero on the boundary. Another difficulty with the free–surface boundary condition is that it couples the shear waves and the pressure√ waves on the boundaries. For large values of the velocity ratio2 ν 2, this can lead to coupling slowly propagating modes with high spatial frequencies to fast modes with low spatial frequencies [74]. To ensure efficient numerical treatment higher order accurate schemes (at least 4th order accuracy) are essential [74].

2Here µ,λ are the Lamé parameters and ρ is the density of the media, the P–wave speed is p p cp = (2µ + λ)/ρ, the S–wave speed is cs = µ/ρ and ν = cp/cs.

37 In this paper, we propose to use the SBP framework. Natural and mixed boundary conditions are imposed weakly using SAT [23]. Dirichlet boundary conditions are imposed strongly by injection. By mimicking the continuous energy estimate we show strict stability. The class of boundary conditions our schemes can treat include free–surface boundary conditions, Dirichlet conditions, linear friction laws and energy absorbing boundary conditions.

38 6. Summary in Swedish

Vågutbredningsfenomen beskrivs av andra ordningens system av hyperboliska partiella differentialekvationer (PDE). Då dessa system löses numeriskt skrivs de ofta först om till ett ekvivalent första ordningens system, främst för att det finns välutvecklade numeriska metoder för flödesberäkningar. Dock finns det flera fördelar med att lösa systemet på andra ordningens form. Bland annat krävs färre frihetsgrader; för att uppnå en given noggrannhet krävs ett mindre antal nätpunkter per våglängd, och högfrekventa parasitlösningar kan und- vikas. Medan teorin och de numeriska metoderna för att lösa första ordningens hyperboliska system är välutvecklad så är motsvarande teori för system på andra ordningens form inte lika långt framskriden. Det generella vågutbredningsproblemet är formulerat på en domän som är stor relativt våglängden medan numeriska simuleringar nödvändigtvis måste begränsas till en mindre beräkningsdomän. Denna restriktion hanteras genom att introducera artificiella ränder och tillhörande randvillkor. Det huvudsak- liga arbetet i denna avhandling är att konstruera effektiva artificiella randvillkor för andra ordningens hyperboliska system genom att använda tekniken med perfekt matchande lager (PML) och att härleda strikt stabila differensapproximationer på begränsade domäner lämpliga för dessa system. Till en början konstrueras ett strikt välställt PML för andra ordningens system i två rumsdimensioner, med fokus på den linjära elastiska vågekvationen. En fördel med detta angreppssätt är att hjälpvariabler kan väljas sådana att det perfekt matchande lagret blir strikt hyperboliskt och därmed välställt. En annan fördel är att ett mindre antal hjälpvariabler krävs jämfort med vid an- vändning av existerande första ordningens formuleringar. I det kontinuerliga fallet är stabiliteten för första och andra ordningens formulering linjärt ekvi- valenta. Dock är det inte alltid fallet vid diskretiseringen av ekvationerna. I denna avhandling visas att om det geometriska stabilitetsvillkoret inte är upp- fyllt, så leder de vanliga centrala differensapproximationerna av ett PML till högfrekvent instabilitet vid de flesta nätupplösningar. Diskretiseringen på andra ordningens form beter sig mycket mer stabilt. Vid en andra ordningens formulering framträder instabilitet endast om de instabila moderna är väl up- plösta. En diskret modell av ett PML analyseras och antalet nätpunkter som krävs i lagret för att garantera stabilitet bestäms för flera anisotropiska material och numeriska experiment som verifierar de teoretiska resultaten presenteras. Det finns många vågutbredningsfenomen där randfenomen såsom ytvågor eller reflekterade “glancing” vågor är av intresse. Typiska exempel återfinns i icke-destruktiva tester, seismologi, ultraljud och markpenetrerande radar.

39 Dessa problem kan beskrivas av symmetriska tidsberoende PDE i delvis be- gränsade områden. För dessa problem introducerar PML ränder där lagret interagerar med den fysiska randvillkoret. För tidsberoende PDE ger införandet av randvillkor ofta upphov till svårigheter. Till exempel, ett PML som är Cauchystabilt kan bli icke-välställt då randvillkor ansätts. Från en teoretisk synvinkel är (begynnelse - randvärdesprob- lem mycket svårare att analysera än motsvarande Cauchyproblem. I prak- tiken uppkommer de mest utmanande svårigheterna vid numerisk behandling av randvillkor. PML–problem där fysiska ränder är viktiga har studerats. I ett första skede betraktas den skalära vågekvationen med ett PML i en vågledare. För en vå- gledare begränsad av Neumannvillkor härleds ett antal modifierade randvillkor som garanterar stabilitet och noggranhet. Senare behandlas ett PML för symmetriska andra ordningens hyperboliska system definierade på ett halv- plan. För en familj av välställda (stabila) randvillkor konstrueras modifierade randvillkor, och välställdhet av motsvarande halvplansproblem bevisas. Slut- ligen utvidgas analysen till rektangulära isotropisk - elastiska vågguider, be- gränsade vid topp och botten av “free-surface” randvillkor. Det visas att våg- guiden stödjer bakåt propagerande moder, ett kontinuerligt PML kan stödja lösningar som växer i tiden. Vidare demonstreras hur bakåtpropagerande moder inte skadar ett diskret PML. Genom att använda partialsummations–operatorer av hög ordning härleds strikt stabila finita differensapproximationer av hög ordning för andra ordningens tidsberoende hyperboliska system på begränsade områden. Naturliga och blandade randvilkor ansätts svagt med simultaneousapproximationterm– metoden. Dirichletrandvillkor ansätts starkt med injektion. Genom att konstruera strikta kontinuerliga energinppskattningar och analoga strikta diskreta energinppskattningar visas strikt stabilitet.

40 Acknowlegement

I am highly indebted to my advisor Prof. Gunilla Kreiss for all her support and encouragement, and more importantly for generously sharing her invaluable expertise. I thank you Gunilla for your kindness, patience and the freedom you gave me to develop my own interests. Surely for the past six years, I have studied under an enthusiastic mind, and one of the most generous and benevolent hearts bequeathed to mankind. I thank my secondary advisors Prof. Jan Nordström and Dr. Ken Mattsson. Ken the discussions with you improved my work signiﬁcantly. I am grateful to Prof. Michael Thuné for guiding me during my teach- ing assignments. Thank you Dr. Tom Smedsaas for all the help. I express my gratitude to Carina Lindgren who continually put up with me turning in late forms. I gratefully acknowledge all the help and many useful discussions with Dr. Maya Neytcheva. I thank Emil Kieri and Sven-Erik Ekström for reading this comprehensive summary and providing many useful comments. Kristoffer Virta thank you for contributing to the Swedish summary. I thank Martin Kronbichler for proofreading the early versions of Papers III and VI. I am thankful to Samsidy Goudiaby and Katharina Kormman for proofreading Paper IV. Many thanks to Prof. Eric Dunham, Dr. Jeremy Kozdon and Dr. Brittany Erickson for hosting my visit to Stanford. Thank you Dr. Daniel Appelö for hosting my visit to CALTECH. To Dr. Ernest Eteng, I say thank you for your advice. My sincere gratitude goes to Clara Rogo and John Rogo for your kindness and for always making me feel at home. I thank Ali Dorostkar and Farshid Hassani for the lunch time spent together. I am grateful to my parents and my siblings for all the support and prayers. I also thank auntie Gladys, uncles Clement and Alphonsus and my friends Nicholas Nwogu and Godwin Chukwunta, for all your support and encouragement. The travel grant from Lliljewalchs restipendium is gratefully acknowledged. My research was funded in part by the Swedish Research Council grant: VR 2009–5852.

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