Difference Methods with Boundary and Interface Treatment for Wave Equations

Transcription

1 IT Licentiate theses Difference Methods with Boundary and Interface Treatment for Wave Equations KRISTOFFER VIRTA UPPSALA UNIVERSITY Department of Information Technology

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3 Difference Methods with Boundary and Interface Treatment for Wave Equations Kristoffer Virta October 2013 Division of Scientific Computing Department of Information Technology Uppsala University Box 337 SE Uppsala Sweden Dissertation for the degree of Licentiate of Technology in Scientific Computing c Kristoffer Virta 2013 ISSN Printed by the Department of Information Technology, Uppsala University, Sweden

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5 Abstract Wave motion in acoustic and elastic media is highly influenced by the presence of outer boundaries and media interfaces. The solutions to the equations governing the wave motion at any point in the domain as a function of time can be sought either through analytical or numerical techniques. This thesis proposes provably stable finite difference schemes to accurately simulate wave interaction with boundaries and interfaces. Schemes for the acoustic wave equation in three spatial coordinates, general domains and heterogeneous media and the elastic wave equation in two spatial dimensions and layered media are presented. A study of the Rayleigh surface wave in almost incompressible media is carried through. Extensive numerical experiments designed to verify stability and accuracy as well as applicability to non - trivial boundary and interface phenomena are given.

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7 List of papers This thesis is based on the following three papers referred to as Paper I, Paper II and Paper III. I Kristoffer Virta and Ken Mattsson Acoustic Wave Propagation in Complicated Geometries and Heterogeneous Media. Submitted to J. Sci. Comput, II Kristoffer Virta and Gunilla Kreiss Surface Waves in Almost Incompressible Elastic Materials. Available as preprint arxiv: , III Kristoffer Virta and Kenneth Duru High Order Finite Difference Schemes for the Elastic Wave Equation in Discontinuous Media. Available as preprint arxiv: ,

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9 Contents 1 Introduction 7 2 The wave equation Acoustics Elasticity An elastic half - space Two elastic half - spaces in contact Discretization of the wave equation Previous work SBP - SAT discretization of the wave equation Summary of papers Paper I Paper II Paper III Summary and future work 29 5

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11 Chapter 1 Introduction The spreading over time of an initially localized disturbance in a medium forms the basis for the study of wave propagation. Familiar manifestations of wave propagation include transmission of sound and radio waves, seismic tremors in the earth and the spreading of ripples on a pond of water. This thesis concerns the spreading of acoustic and seismic waves. Both phenomena are governed by an equation commonly referred to as the wave equation. Seismic waves originating from a rupture in the earth soon encounters the surface of the earth. Acoustic signals reflect and refract from the ocean bottom in an ocean acoustic setting. That is, outward propagation of waves from an initial disturbance inevitably encounter and interact with boundaries. While the propagation of solid and acoustic waves are governed by the same equation, the types of boundary interaction phenomena is one factor that makes each application distinct. For example, a single solid wave, be it compressional or shear, will on striking a boundary produce both compressional and shear reflections. A feature that never occurs in a strictly acoustic scenario. Wave boundary interaction is modeled by adding to the wave equation suitable boundary conditions. Now the general acoustic and elastic wave propagation problem can be stated: find the solution to the wave equation with the prescribed initial and boundary conditions as a function of time at any point in the domain of interest. Much work has been devoted to the analytic study of this problem. However, it turns out that exact solutions can only be obtained for a small subset of cases. These cases involve simplifying assumptions such as time harmonic behavior, plane boundaries and homogeneous media. While these solutions provide invaluable information and have served to understand real world observations, the general problem 7

12 can not be solved analytically. One approach is to discretize the wave equation with the boundary conditions and thus find an approximate solution to the general acoustic and elastic wave propagation problem. An approximate solution to the general acoustic and elastic wave propagation problem involves many details. The focus of this thesis is to propose numerical methods that handle the approximation of relevant boundary conditions. These methods will then be used to study phenomena that arise as waves interact with boundaries. It turns out that the analytical studies that has been made now provide inspiration for the construction of non - trivial numerical tests once the numerical codes have been developed. Key ingredients in the numerical approximation of the solution to the wave equation are stability and accuracy. That is, the numerical algorithm should not allow for any growth in the numerical solution not present in the analytical solution and the approximation should be consistent with the equation. The two properties then guarantee convergence of the numerical solution to the true solution as the discretization becomes finer. Most of the workspentinthisthesisistoprovideproofsthattheproposedmethodsare stable. The outline of the thesis is as follows. Chapter 2 serves the purpose of introducing the wave equation in the context of acoustics and elasticity, with particular focus on boundary phenomena. Chapter 3 then discusses the discretization of the wave equation and briefly introduces the framework used in the present work. A summary of the papers that discusses the different numerical algorithms and studies is given in Chapter 4. Future work is mentioned in Section 5. 8

13 Chapter 2 The wave equation One of the early investigations by D Alembert in 1747 resulted in an equation to describe the motion of the infinite taut string, a 2 φ (b φ) =0,a,b>0. t2 Here φ denotes the displacement of the string from its initial state and a, b are material parameters. This equation has since then become known as the wave equation. Although most wave propagation problems are not governed by the wave equation, it occurs in two fields: acoustics and elasticity. 2.1 Acoustics In the general acoustics problem we seek a description of the pressure disturbance u at a point x =(x 1,x 2,x 3 ) T in a medium Ω with density ρ and wave speed c. To allow for a medium with discontinuous density and wave speed we decompose the medium into a collection of sub-media in which parameters are smooth. Let Ω i be sub - media number i we denote by u i, ρ i and c i the pressure disturbance, density and wave speed in Ω i, respectively. The wave equation then takes the form 1 t 2 u 1 1 (ρ 1 u 1 )=0, x Ω 1, ρ 1 c t 2 u n 1 (ρ n u n )=0, x Ω n. ρ n c 2 n 2. (2.1) 9

14 Inevitably, an outgoing wave from a disturbance in one sub-media will at some time interact with interfaces to other sub - media. At such interfaces we require continuity of pressure disturbance and flux, u i u j =0, ρ i u i n i + ρ j u j n j =0, x Ω i Ω j. (2.2) Here the inward facing unit normals of sub - media i and j are denoted n i and n j, respectively. Hence, the equations of (2.1) are coupled via the interface conditions (2.2). On outer boundaries a variety of boundary conditions may be imposed. One such is the homogeneous Neumann boundary condition which models the impinging of a pressure wave on a hard surface. u n =0, x Ω. (2.3) A discretization of the general acoustics problem (2.1) - (2.3) is discussed in Paper I. As the following example illustrates the presence of boundaries and interfaces can give rise to interesting phenomena. Example1-Lovewaves Consider a two dimensional layer Ω 1 of thickness T overlying a half - plane Ω 2. Let the density in both media be constant and locate the interface between Ω 1 and Ω 2 at the line y = 0. The setup is displayed as an inset to Figure 2.1. At the upper boundary of the layer a homogeneous Neumann condition is imposed and at the interface the conditions (2.2) apply. A useful technique to analyze boundary and interface phenomena is to assume that the solution is periodic in time and all but one of the spatial dimensions i.e., let then from (2.1) u 1 = U 1 (y)e iξ(x ct), T y 0, u 2 = U 2 (y)e iξ(x ct),y 0, d 2 U 1 dy 2 + α 1U 1 =0, d2 U 2 dy 2 α 2U 2 =0. Here α 1 = ξ 2 ( c 2 /c ),α 2 = ξ 2 ( 1 c 2 /c 2 2). 10

15 The resulting solutions are u 1 = A 1 e i(ξx α1y ξct) + A 2 e i(ξx+α1y ξct), T y 0, (2.4) u 2 = B 1 e α2y e iξ(x ct),y 0. (2.5) The solution (2.4) represents plane waves propagating back and forth within the layer. The solution (2.5) gives a wave that travels harmonically along the interface and decays exponentially into the half - plane. Substituting the solutions (2.4) and (2.5) into the boundary and interface conditions yields u 1 n =0:A 1 e iα1t A 2 e iα1t =0, u 1 u 2 =0:A 1 + A 2 B 1 =0, u 1 n 1 u 2 n 2 =0: iα 1 ρa 1 + iρα 1 A 2 ρ 2 α 2 B 1 =0. (2.6) The condition for the existence of solutions to the system (2.6) is the vanishing of its determinant, D(c) =ρ 2 ( 1 c 2 /c 2 2) 1/2 ρ1 ( c 2 /c ) 1/2 tan ξt ( c 2 /c ) 1/2 =0. (2.7) Solutions (2.4) and (2.5) corresponding to a root c of (2.7) was used by Love [27] to explain the presence of large transverse components of displacement in the main tremor of an earthquake as a result of the layered structure of the earth. For this reason the solutions are called Love waves. A notable feature of the Love waves is that successive values of ξ will result in c = c(ξ). That is, Love waves are dispersive although the acoustic wave equation in the absence of interfaces and boundaries does not have dispersive solutions. A contour plot of the solution for one possible choice of layer thickness T and parameters ρ 1,ρ 2,c 1,c 2 is pictured in Figure Elasticity We introduce the equations of elastic wave propagation first in a infinite isotropic homogeneous medium. Boundaries and interfaces between sub - media are later introduced by considering the case of a half - space and two half-spaces in contact, respectively. The equations governing the material displacement u =(u 1,u 2,u 3 )atapointx =(x 1,x 2,x 3 ) due to an initial 11

16 Figure 2.1: Contour plot of Love waves. The setup of the layer over halfplane structure is displayed in the inset. disturbance in an infinite isotropic elastic body may be written as ( ) ρ 2 2 t 2 u 1 (λ + μ) x 2 u u u 3 + μ 2 u 1 =0, 1 x 1 x 2 x 1 x 3 ( ) ρ 2 2 t 2 u 2 (λ + μ) u x 2 x 1 x 2 u u 3 + μ 2 u 2 =0, 2 x 2 x 3 ( ) ρ 2 2 t 2 u 3 (λ + μ) u u x 3 x 1 x 3 x 2 x 2 u 3 + μ 2 u 3 =0. 3 (2.8) Here density and elastic constants for the material are ρ and the first and second Lamé parameters λ>0andμ>0. The wave equation is obtained from (2.8) by using a theorem by Helmholtz saying that a vector field can be resolved into the gradient of a scalar and the curl of a zero - divergence vector [35]. That is, we write u 1 ψ 1 ψ 1 u 2 = φ + ψ 2, ψ 2 =0 (2.9) u 3 ψ 3 ψ 3 12

17 Inserting (2.9) into (2.8) reduces to ) ρ (ρ 2 ψ 2 t 2 φ tt (λ +2μ) 2 t 2 1 μ 2 ψ 1 φ + ρ 2 ψ t 2 2 μ 2 ψ 2 = 0 (2.10) ρ 2 ψ t 2 3 μ 2 ψ 3 Hence, if φ and (ψ 1,ψ 2,ψ 3 ) T satisfies the wave equations 1 2 c 2 1 t 2 φ 2 φ =0, 1 2 c 2 2 t 2 ψ 1 2 ψ 1 =0, 1 2 c 2 2 t 2 ψ 2 2 ψ 2 =0, 1 2 t 2 ψ 3 2 ψ 3 =0, c 2 2 (2.11) then necessarily (2.10) holds. Here c 2 1 = λ+2μ ρ and c 2 2 = μ ρ. It now appears that there could exist values of φ and (ψ 1,ψ 2,ψ 3 ) T not satisfying (2.11) but satisfying (2.10). This possibility was excluded in [44] where it was established that the complete solution to (2.8) is given by the solution to (2.11). The presence of two distinct wave speeds in the wave equations of (2.11) suggests two different types of waves propagating in the elastic media. Example 2 - pressure and shear waves The volumetric and rotational disturbance in the elastic media, θ and ω, respectively, are computed by θ = u, ω = u/2. Now, (2.8) can be written as ρü (λ + μ) u + μ 2 u =0. (2.12) If the operations of divergence and curl are performed on (2.12) the two wave equations 1 2 c 1 t 2 θ 2 θ =0, 1 2 c 2 t 2 ω 2 ω =0, 13

18 for volumetric and rotational disturbances are obtained. The conclusion is that a change in volume, or a pressure (P-wave) disturbance, will travel at the velocity c 1 and that a rotational, or shear (S-wave) disturbance, travel at the velocity c 2 The stresses in the elastic media are computed by τ 11 = λθ +2μ u ( 1 u1,τ 12 = μ + u ) 2, x 1 x 2 x 1 τ 22 = λθ +2μ u ( 2 u2,τ 23 = μ + u ) 3, x 2 x 3 x 2 τ 33 = λθ +2μ u ( 3 u3,τ 13 = μ + u ) 1. x 3 x 1 x 3 Using (2.9) the stresses may also be expressed in φ and (ψ 1,ψ 2,ψ 3 ) An elastic half - space Consider wave propagation in an elastic half - space. Let the surface of the half - space be the plane x 2 = 0. A stress boundary condition imposes conditions on the normal and tangential stresses at the surface of the half - space. τ 22 = f 1,τ 12 = f 2,τ 23 = f 3. (2.13) That is, the wave equations (2.11) are coupled via the boundary condition. This boundary condition is of particular importance in seismology as it correctly models forces acting on the surface of the earth. With zero forcing the boundary condition models a free surface and is referred to as a traction free boundary condition. In the study of seismology one of the first observations was that earthquake tremors consisted of two early minor disturbances corresponding to pressure and shear waves. These disturbances were then followed by a later significantly damage causing tremor. This third arrival was not consistent with the current understanding of elastic wave theory. The question was therefor, whether the equations (2.11) together with (2.9) and (2.13) supported a third wave type. Example 3 - Lambś problem and the Rayleigh surface wave Consider a force periodic in the x 1 coordinate and time applied perpendicular to the surface of the half - space. Assuming that the force is independent of the spatial coordinate x 3 reduces the dimensionality of the problem by 14

19 one. The pressure and shear wave motions are then described by 1 c 2 1 t 2 φ 2 φ =0, 1 2 (x 1,x 2 ) (, ) [0, ). (2.14) t 2 ψ 3 2 ψ 3 =0, c In this case the stresses can be computed as ( ) τ 12 = μ 2 2 φ + 2 ψ x 1 x 2 x 2 2 ψ, 2 x 1 x 2 ( 2 ) ( φ τ 22 =(λ +2μ) x φ 2 φ 1 x 2 2μ 2 x 2 1 The forcing may be expressed by the conditions Solutions of the form satisfies (2.14), provided that ) 2 ψ 3. x 1 x 2 τ 12 =0,τ 22 = Z(k)e i(ωt kx1),x 2 =0. (2.15) φ = A(k)e ν1x2 ikx1,ψ 3 = B(k)e ν2x2 ikx1 (2.16) ν 2 1 = k 2 ω2 c 2 1,ν2 2 = k 2 ω2 c 2. 2 Inserting (2.16) into (2.15) results in 2A(k)iν 1 k (2k 2 ω2 c 2 2 (2k 2 ω2 c 2 2 ) B(k) =0, ) A(k)+2ikν 2 B(k) = Z(k) μ. (2.17) These equations can be solved for A(k) andb(k) to yield the complete solution, 2k 2 ω2 c A(k) = 2 Z(k) 2 D(k) μ,b(k) = 2ikν Z(k) (2.18) D(k) μ where ) 2 D(k) = (2k 2 ω2 4k 2 ν 1 ν 2 c

20 is the determinant of the system (2.17). In [25] Lamb studied surface disturbances generated by a time harmonic loading applied along a line coincident with the x 3 - axis. This solution can be obtained using the solution (2.16) and (2.18) by superposing an infinite number of stress distributions of the form (2.15). That is, put Z(k) = Q dk 2π and integrate with respect to k from to, obtaining τ 22 = Q e ikx1 dke iωt = Q 2π 2π δ(x 1)e iωt,x 2 =0. (2.19) The displacement at any point in the surface x 2 = 0 can now be written using (2.16), (2.18) and (2.9) ( ) u 1 = iq k 2k 2 ω2 2ν c 2 1 ν 2 2 e ikx1 dke iωt, 2πμ D(k) u 2 = iq x 2 =0. (2.20) ω 2 ν 2πμ c 2 dke iωt, 2 D(k)e ikx1 The integrals of (2.20) can be evaluated by residue calculus. A complete discussion is given in [11], pp There the expression u 1 = QA μ ei(ωt κ Rx 1) + iq ( ) ω 3/2 ) (ωt 2πμ B x 1 e i ωc1 x 1 c 1 + iq ( ) ω 3/2 ) (2.21) (ωt 2πμ C x 1 e i ωc2 x 1 + c 2 was derived. Here A, B and C are constants and κ R is the real zero of D(k). A similar expression holds for u 2. The first term of (2.21) represents a non - decaying displacement traveling harmonically along the surface with phase speed c R = κ R ω. The second and third terms correspond to pressure and shear waves decaying with distance from the source due to the x 3/2 1 term. Hence, a third wave type corresponding to the later tremor of an earthquake is supported by elastic wave theory. In fact, this third wave type was first discovered by Rayleigh [39] and is refereed to as the Rayleigh surface wave. In [39] it was also seen that the Rayleigh wave has all its energy concentrated close to the surface which explains the significantly damage causing tremor caused by this wave. The above problem and variants involving transient point and line surface or internal loadings are now commonly refereed to as Lambś problem and is a classic problem of elastic wave theory Paper II is devoted to a study of the Rayleigh surface wave where also a numerical solution of Lambś problem is presented 16

21 2.2.2 Two elastic half - spaces in contact Due to the layered structure of the earth, the propagation of elastic waves in an isotropic homogeneous half - space cannot represent all problems of practical interest. In a more general situation waves propagate in a media having a layered structure. The simplest case being two elastic half - spaces in contact. At the plane interface x 2 = 0 between an upper half - space, Ω = (, ) [0, ) (, ) and a lower half - space Ω =(, ) (, 0] (, ), we require continuity of displacement, normal and tangential stresses. Priming all quantities defined in Ω this may be written u u =0, τ 22 τ 22 =0, τ 12 τ 12 =0, τ 23 τ 23 =0. The problem of finding the displacement at any point as a function of time in either Ω or Ω, especially the phenomena which arise from influence of the interface is termed the fundamental problem of seismology by Cagniard [4]. One such phenomenon is the presence earlier refraction arrival in a slower media overlying a faster media. Example 4 - Refraction arrival Consider waves originating in the half - space with slower wave speed. It has then been observed that an earlier than predicted arrival in the slower media is apparent. The theoretical setup can be viewed as a variation of Lambś problem and analyzed similarly as in Example 3, see [11], pp Laboratory experiments have also been performed. In e.g., [38] two layers measuring 4 50 inches of Plexiglas and aluminum were welded together. A spark was used as point source 2 inches in on the upper boundary of the layer of Plexiglas and responses was recorded with 17 recorders spaced 2 inches apart. The setup can be viewed in the inset to Figure 2.2. Experiments were also done with the point source located on the boundary of the layer of aluminum. We perform the analogous numerical simulation in two spatial coordinates assuming that the layers are infinitely thin. Here the point 17

22 source is modeled by τ 12 =0 { 10 τ 22 = 4 (sin(2π2500t) 1/2sin(2π5000t)) δ(x 1 2),t (0, 1/2500), 0, else x 2 =0. (2.22) The results are compared in Figure 2.2, where the left figures display results when the source is located on the boundary of the layer of Plexiglas. The right figures show the results when the source is located on the boundary of the layer of aluminum. The results from the experiments are similar, a refraction arrival is apparent in the case of a point source on boundary of the slower layer of Plexiglas. No refraction arrival appears when the source is located on the boundary of aluminum. Discretization of the elastic wave equation and the interface conditions (2.14) is described in Paper III. 18

23 Figure 2.2: Top: results taken from [38], Bottom: results of a numerical simulation analogous to the practical experiment. In both cases the refraction arrival appears when the source is located on the boundary of the layer of Plexiglas. 19

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25 Chapter 3 Discretization of the wave equation By introducing the wave equation the study of wave phenomena in acoustics and elasticity consists of finding the solution to a system of partial differential equations. As described in the previous section boundary conditions supplementing the system of partial differential equations highly influence the behavior of the solution. Although the techniques used in Example 1 and 3 above are applicable in a variety problems the main assumption is that the solution is time harmonic, that the domain has planar boundaries and that the media parameters are piecewise constant. This is a severe restriction as the general application may include a transient interior or boundary forcing, a non-planar geometry and variable media parameters. To arrive at the solution in the general setting we discretize the equations, boundary conditions and the domain. This results in a discrete system of equations. The main difficulty is now to ascertain that the solution of the discrete system converges to the true solution as the discretization becomes finer. It is well known that convergence is guaranteed if the discretization is stable and accurate, see e.g., [13], pp 170. Also, early results [20] show the superiority of higher order methods (higher than 2) applied to wave propagation problems. High order accuracy and stability now become two crucial properties of a numerical approximation. Although both acoustic and elastic waves may be described by the wave equation, the formulation (2.11) has not traditionaly been used in numerical simulations of elastic wave motion. Instead the equations (2.8) have been discretized either directly or by first being rewritten to a first order system. 21

26 3.1 Previous work Early work on discretizing the acoustic and elastic wave equation was based on finite difference approximations of the second order systems (2.1) and (2.8) [8, 40, 1, 43]. Stability could only be guaranteed either by using an implicit scheme or only in the case of rectangular domains and constant media parameters. The main difficulty with the discretization of the elastic wave equation was to approximate the stress boundary condition in a stable manner. For high ratios between P - wave and S - wave velocities instabilities were especially bad. In [42] it is argued that for short time computations these methods with the stress boundary condition are still reliable but instabilities are reported for long time simulations. For media with discontinuous parameters the order of accuracy was reduced to first order. This is true in general if no special treatment of interfaces between different media are made in the discretization [2]. Other early finite difference methods involve rewriting the equations to first order systems [26, 47, 46]. These schemes were more successful in approximating the stress boundary condition although stability in these cases were proven only for homogeneous media in rectangular domains. More contemporary work on finite difference discretizations of the 2nd order formulation of the elastic wave equation includes provably stable approximations of the stress boundary conditions. In [3] a second order accurate energy conserving finite difference discretization of (2.8) with the stress boundary condition and non - planar geometries was constructed. This method was later extended to enable discontinuous media [37] where sub - grids with an overlap at interfaces of discontinuity was used. This method also allowed for refinement over sub - grids. The second order scheme was extended to a fourth order method for materials with smooth media parameters in [41]. Finite difference discretization of the acoustic wave equation on second order form with different boundary conditions was considered in [23, 24, 22]. There non - planar boundaries were treated by embedding the domain into a Cartesian grid. Those works where extended to enable media discontinuities in [21]. The possibly non - planar interface was there also embedded in the Cartesian grid and ghost points on each side of the interface are used to impose correct interface conditions. The acoustic wave equation on a collection of rectangular domains, each having constant wave speed and density was discretized with high order finite differences in [32]. There interfaces of discontinuity were treated using the projection method [36]. With the projection method no reduction in accuracy was present due to media interfaces. Recent work on solving the elastic wave equation on the 22

27 first order form has been considered in [18, 19] these schemes are provably stable and includes interface conditions that model frictional forces between different media. The first order formulation for wave propagation in an acoustic medium coupled to a porous media is described in [6]. There the interface between the different media was implemented using an immersed interface method [6]. Apart from finite difference methods recent finite volume [48], finite element [9], spectral element [34] and discontinuous Galerkin discretizations [17] on unstructured meshes have been developed. Here the main advantage is the simplicity to follow realistic topography and media interfaces, in particular if hanging nodes are alowed. However, the construction of a high quality unstructured mesh in 3 spatial dimensions is a non - trivial problem itself and and also requires extra bookkeeping and additional memory to keep track of the connectivity of the grid. Although the different boundary and interface conditions are easier to treat numerically in the first order formulation rewriting the second order systems as first order systems doubles the number of unknowns and may introduce side conditions that must be satisfied by the solution to the larger first order system. A discussion is given in [23]. For these reasons this thesis focuses on high order finite difference discretizations of the acoustic and elastic wave equations in 2nd order form. 3.2 SBP - SAT discretization of the wave equation We use summation - by - parts (SBP) finite difference operators [45, 33, 28] to discretize in the spatial coordinates. Boundary conditions are enforced via the simultaneous approximation term (SAT) technique [5]. SBP operators in conjunction with the SAT technique is commonly referred to as the SBP - SAT methodology. Previously the SBP - SAT methodology was used for the acoustic wave equation in [30], where the equation was solved approximately with different boundary conditions in one spatial dimension and constant media parameters. The work was further developed in [31] to include the case of materials with piecewise constant media. Both works extended the technique to a second order finite volume method in 3 spatial dimensions. The elastic wave equation on 2nd order formulation with stress, Dirichlet and energy absorbing boundary conditions was discretized with the SBP - SAT technique in [10]. There stencils of order of accuracy 2p in the interior and p in the vicinity of the boundary were used. Here p =1, 2, 3, 4. Experiments indicated a p + 2 global order of accuracy. 23

28 To introduce the concept of the SBP - SAT methodology we discretize thehalf-line[0, ) by introducing an equidistant grid with grid size h x j = jh, j =0, 1,... Let u be a function defined on the half - line, the value u(x j ) is denoted u j. A grid function corresponding to u is then u =(u 0,u 1,...). Let u and v be two grid functions. The scalar product of u and v is defined as usual by < u, v >= u i v i. If P is an operator acting on grid - functions its adjoint is denoted P. We can now define a 2p-th order narrow diagonal second derivative SBP operator acting on grid functions with the properties D 2 = H 1 ( M + BS), M = M 0, H = H 0, ) ( 2 u x 2 i=0 (D 2 u) j = + j O(hp ),j =0...n p, ( ) (D 2 u) j = 2 u x + 2 j O(h2p ),j = n p +1..., (Su) 0 = ( ( ) x u) 0 + O(hp+1 ), ( Bu )0 = u, Bu =0,j 0, j (3.1) where the integer n p depends on p. As an example we construct an SBP - SAT discretization of the wave equation on the half - line. Example 5 - Discretizing the wave equation Discretizing 2 t 2 u 2 u =0,x [0, ),t>0, x u =0,x=0,t>0, u(0,x)=u 0 (x),x [0, ), t u(0,x)=u 1(x),x [0, ), 24

29 in the spatial coordinate with the use of the second derivative SBP operator defined by (3.1) results in a system of ordinary differential equations with time being the continuous variable. The system may be written d 2 dt 2 u D 2u = SAT. (3.2) The discrete solution is subject to initial data u j (0) = u 0j,j =0, 1,, d dt u j(0) = u 1j,j =0, 1,. (3.3) The homogeneous Neumann boundary condition is imposed weakly via the penalty term SAT. This term is proportional to the difference of an approximation of the normal derivative and zero and thus penalizes the solution for not assuming the prescribed value at the boundary. The imposition is called weak since the boundary condition is not satisfied absolutely. We write the penalty term as SAT = σh 1 ( BSu 0 ). (3.4) Here the scalar σ is called a penalty parameter whose value must be determined by the stability requirement. Using (3.1) the system (3.2) can now be written as d 2 dt 2 u = ( H 1 M + BS + σ BS ) u = H 1 Au. (3.5) We see immediately from (3.1) that if σ = 1, then A = A and A 0. We define the energy of the system to be E(u) = t u,h u + u,au. (3.6) t Note that this is a semi - norm of the solution to (3.5). Theorem (Conservation of energy) If the penalty parameter σ equals 1, then the operator A of (3.5) is self adjoint, positive semi - definite and the solution to (3.5) satisfies E(u) =C, t 0, where C is a constant depending only on initial data (3.3). 25

30 A proof of this statement can be found in e.g., Paper III. The system (3.5) is now stable in the sense that solutions are bounded by initial data in the semi - norm (3.6) Using the SBP - SAT technique to discretize (2.1) and (2.8) with boundary and interface conditions leads to systems of the type (3.2). The penalty term SAT is determined through the type of boundary conditions that are to be approximated. The penalty term always contains at least one penalty parameter. The SBP properties are then used to determine the values of the penalty parameters such that the corresponding operator A of (3.5) is self - adjoint and positive semi - definite. The system (3.5) can then be discretized in time. In this thesis the classical Runge - Kutta method and time stepping schemes particularly constructed for such problems [12] have been used. 26

31 Chapter 4 Summary of papers 4.1 Paper I In this work we construct finite difference discretizations of the acoustic wave equation in complicated geometries and heterogeneous media. Non - planar geometries are handled by transforming the equation to a curvilinear coordinate system where the coordinate axes are aligned with the topography of the physical domain. Emphasis is placed on the accurate treatment of interfaces at which the underlying media parameters have jump discontinuities. To treat the jump discontinuities the domain is subdivided into blocks each with continuous media parameters. The equation on each block is then discretized with SBP operators and patched together via the SAT technique. This procedure enables the construction of a suitable discrete semi - norm in which the discrete solution is bounded by initial data and interior forcing. Numerical experiments aimed at verifying accuracy and stability are presented in two and three spatial dimensions. An application to a problem in underwater acoustics is also presented. This example makes full use of the features of the method and includes variable sound speed with a jump discontinuity, a three dimensional non - planar domain and a set of different outer boundary conditions. 4.2 Paper II This study builds upon a recent result that shows that the classical theory concerning accuracy and points per wavelength is not valid for surface waves in almost incompressible elastic materials. The grid size must instead be proportional to ( μ λ )(1/p) to achieve a certain accuracy. Here p is the order of 27

32 accuracy the scheme and μ and λ are the Lame parameters. This accuracy requirement becomes very restrictive close to the incompressible limit where μ λ 1, especially for low order methods. We present results concerning how to choose the number of grid points for 4th, 6th and 8th order summationby-parts finite difference schemes. The result is applied to Lambś problem in an almost incompressible material. 4.3 Paper III The first goal of this work is to construct finite difference schemes for simulating elastic wave motion in bodies that are welded in contact. The construction is similar to that of Paper I. Two different types of SBP operators are used, fully compatible and compatible. A proof that the numerical solution obtained by the scheme constructed with fully compatible SBP operators can be bounded by initial data in a semi - norm is presented. The corresponding proof for compatible SBP operators is not found. However, numerical experiments indicate that both types of schemes are stable. The numerical experiments also suggest that fully compatible operators give identical or better convergence and accuracy properties. The second goal is to construct the numerical experiments to illustrate the usefulness of the proposed method to simulations involving typical interface phenomena in elastic materials. In particular, mode conversion, refraction arrival phenomena and the Stoneley interface wave are studied. 28

33 Chapter 5 Summary and future work Numerical methods for the simulation of acoustic and elastic wave propagation have been proposed. The focus of the numerical schemes has been on stable approximation of boundary and interface conditions. One suggested use for the methods is then to study wave boundary interaction phenomena. One such study of the Rayleigh surface wave was presented. For the methods proposed to be useful in practice further generalizations needs to be considered. While the numerical scheme for the simulation of acoustic waves is implemented to handle three spatial dimensions, complicated geometries and arbitrary media with jump discontinuities, the implementation needs to be generalized to a large scale parallel computer. The codes that handle elastic wave propagation were only implemented in two spatial dimensions and Cartesian coordinates. However, the generalization to three spatial dimensions and general geometries should not pose a theoretical problem but more programming work needs to be done. In both the acoustic and elasticity case better absorbing boundary conditions or layers need to be added. An open question in the SBP - SAT framework used for second order equations is how to handle grid refinement. This is a key ingredient for efficiency of wave propagation codes as material inhomogeneities can cause the local wavelength to vary and thus lower the appropriate grid resolution locally. A successful strategy for first order systems was presented in [29] but problems involving second order equations were reported in [14]. As the Rayleigh surface wave was seen to suffer increasingly much from truncation errors as the elastic material became more incompressible it would be interesting, due to the similarity of the wave types, to analyze the Stoneley interface wave in a similar fashion. In such an investigation the 29

34 codes developed in this work can be used directly. The theoretical analysis would involve a discussion of the accuracy of the schemes at the boundary. In this thesis finite difference stencils with an accuracy reduction at boundaries have been used. Numerical experiments, that study both max error and l 2 error then suggest that the practical order reduction does not suffer as severely. In fact a rule of thumb is that if the interior stencil is of order 2p and the boundary stencil is of order p a global order of accuracy should be p + 2. It is unknown to the author if this has been proved for the elastic and acoustic wave equation and a detailed study, if it does not exist, needs to be done. With an implementation for the elastic wave equation that can handle curved geometries and the addition of gravity terms it would be interesting to study the influence of curvature and gravity on surface waves. 30

35 Acknowledgements I would like to thank my supervisor professor Gunilla Kreiss. Thank you for pointing me in a direction to genuinely interesting problems and letting me pursue the solutions using my own intuition. I would also like to thank Ken Mattsson for his introduction to the SBP - SAT methodology and Siyang Wang for his kind proofreading of this manuscript. Then there are numerous more persons that deserves thanks. To avoid the bemusement of leaving someone out I will not mention anyone more but only say: thanks all my friends! 31

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37 Bibliography [1] Alterman.Z., Karal.F.C.: Propagation of elastic waves in layered media by finite difference methods. Bulletin of the Seismological Society of America, 58(1): , [2] Brown. D.L.: A note on the numerical solution of the wave equation with piecewise smooth coefficients, Mathematics of Computations, 42(166): , [3] Appelö.D., Petersson.N.A.: A Stable Finite Difference Method for the Elastic Wave Equation on Complex Geometries with Free Surfaces. Communications in computational physics, Vol 5, No 1, pp [4] Cagniard.L.: Reflection and Refraction of Progressive Seismic Waves. McGraw - Hill, (1962). [5] Carpenter, M., Gottlieb, D., Abarbanel, S.: Time - Stable Boundary Conditions for Finite - Difference Schemes Solving Hyperbolic Systems: Methodology and Application to High - Order Compact Schemes, J. Comp Phys 111, (1994). [6] Chiavassa.G., Lombard.B.: Wave propagation across acoustic / Biot s media: a finite-difference method, Com Comp Phys 13-4 (2013). [7] Chiavassa.G., Lombard.B.: Time-domain numerical modeling wave propagation in 2D heterogeneous porous media, J Com Phys, 230 (2011). [8] Ciment.M., Leventhal.S.H.: Higher order compact implicit schemes for the wave equation, Math. Comput., 29 (1975). [9] Cohen.G., Fauqueux.S.: Mixed spectral finite elements for the linear elasticity system in unbounded domains, SIAM J. Sci. Comput., 26(3),

38 [10] Duru.K., Kreiss.G., Mattsson.K., Accurate and stable boundary treatments for the elastic wave equations in second order formulation. In Phd Thesis Kenneth Duru, Uppsala University, [11] Ewing.W.M., Jardetzky.W.S, Press.F.: Elastic Waves in Layered Media, McGraw - Hill, [12] Gilbert.J.C., Joly.P.: Higher order time stepping for second order hyperbolic problems and optimal CFL conditions, Num. Analys and Sci Comp for PDEs and their Challenging Applicat, vol. 16, pp. 67â93, Springer, [13] Gustaffsson.B., Kreiss.H.O., Oliger.J.: Time Dependent Problems and Difference Methods, Wiley [14] Gustafsson.M., Nissen.A., Korrmann.K., Stable difference methods for block - structured adaptive grids, Technical report , Department of Information Technology, Uppsala University, [15] Jensen.F.B., Ferla.C.M.: Numerical solutions of range - dependent benchmark problems in ocean acoustics. J. Acoust. Soc. Am 87, (1990). [16] Jensen.F.B., Kuperman.W.A., Porter.M.B., Schmidt.H.: Computational Ocean Acoustics, 2000 Springer - Verlag New York, Inc. [17] Käser.M., Dumbser.M.: An arbitrary high order discontinuous Galerkin method for elastic waves on unstructured meshes - I. The two - dimensional isotropic case, Geophysics. J. Int., 167, (2006). [18] Kozdon.J.E., Dunham.E.M., Nordstrà m.j.: Simulation of dynamic earthquake ruptures in complex geometries using high-order finite difference methods, J. Sci Comput, [19] Kozdon.J.E., Dunham.E.M., Nordstrà m.j.: Interaction of waves with frictional interfaces using summation-by-parts difference operators, 2. Extension to full elastodynamics, Technical Report, Department of Information Technology , Uppsala University. [20] Kreiss.H.-O., Oliger.J.: Comparison of accurate methods for the integration of hyperbolic equations. Tellus 24, (1972). [21] Kreiss.H.-O., Petersson.N.A.: An embedded boundary method for the wave equation with discontinuous coefficients, SIAM J. Sci. Comput. Vol. 28, No. 6,

39 [22] Kreiss.H.-O., Petersson.N. A.: A second order accurate embedded boundary method for the wave equation with Dirichlet data, SIAM J. Sci. Comput., 27 (2006). [23] Kreiss.H.-O., Petersson.N. A., Ystrom.J.: Difference approximations for the second order wave equation, SIAM J. Numer. Anal., 40 (2002). [24] Kreiss.H.-O., Petersson.N. A., Ystrom.J.: Difference approximations of the Neumann problem for the second order wave equation, SIAM J. Numer. Anal., 42 (2004). [25] Lamb.H.: On the Propagation of Tremors Over the Surface of an Elastic Solid, Phil. Trans. Roy. Soc. (London) A, vol [26] Levander.A.R.: Fourth-order finite-difference P-SV seismograms, Geophysics, 53: , [27] Love.A.E.H.: Some problems of geodynamics. Cambridge University Press (1926). [28] Mattsson, K.: Summation by parts operators for finite difference approximations of second - derivatives with variable coefficients, J Sci Comput, 51, (2012). [29] Mattsson.K., Carpenter.M.: Stable and accurate interpolation operators for high - order multiblock finite difference methods, SIAM J. sci. Comput, Vol. 32, No. 4, [30] Mattsson, K., Ham, F., Iccarino, G.: Stable Boundary Treatment for the Wave Equation on Second Order Form. J. Sci. Comput. 3, (2009). [31] Mattsson, K., Ham, F., Iaccarino, G.: Stable and accurate wave - propagation in discontinuous media, J. Comp Phys 227 (2008). [32] Mattsson.K., Nordström.J.: High order finite difference methods for wave propagation in discontinuous media, J. Comput. Phys., 220 (2006), pp [33] Mattsson, K., Nordström, J.: Summation by parts operators for finite difference approximations of second derivatives, J. Comp Phys 199 (2004). 35

40 [34] Mercerat.E.D., Vilotte.J.P., Sanchez-Sesma.F.J.: Triangular Spectral Element simulation of two-dimensional elastic wave propagation using unstructured triangular grids, Geophys. J. Int. 166, (2006). [35] Morse.P., Feshbach. H.: Methods of theoretical physics. Vols I and II. McGraw - Hill, New York (1953). [36] P.Olsson, Summation by parts, projections and stability. I, Math. Comp., 64 (1995). [37] Petersson. N.A., Sjögreen. B.: Stable grid refinement and singular source discretization for seismic wave simulations, Commun. Comput. Phys. 8, (2010). [38] Press.F., Oliver.J., Ewing.W.M.: Seismic Model Study of Refraction From a Layer of Finite Thickness, Geophysics, V XIX, No. 3 (1954) [39] Rayleigh. Lord.: On Waves Propagated Along the Surface Along the Plane Surface of an Elastic Solid. Proc. London Math. Soc. vol 17. [40] Shubin.G.R., Bell.J.B.: A modified equation approach to constructing fourth order methods for acoustic wave propagation, SIAM J. Sci. Stat. Comput. 8 (2) (1987). [41] Sjögreen.B., N.A.Petersson.: A Fourth Order Accurate Finite Difference Scheme for the Elastic Wave Equation in Second Order Formulation, Journal of Scientific computing, [42] Stacey.R.: New Finite - Difference Methods for Free Surfaces with a Stability Analysis. Bulletin of the Seismological Society of America, Vol. 84(1): , [43] Stephen. R.A.: Solutions to range dependent benchmark problems by the finite - difference method. J. Acoust.Soc.Am. 87(4), [44] Sternberg.E.: On the integration of the equations of motion in the classical theory of elasticity. Arch ration Mech.Analysis 6, 34 (1960). [45] Strand, B.: Summation by parts for finite difference approximations for d/dx, J. Comp Phys 110 (1994). [46] Strand.B.: Simulations of acoustic wave phenomena using high-order finite difference approximations, SIAM J. Sci. Comput., 20 (1999). 36

41 [47] Virieux.J.: P-SV wave propagation in heterogeneous media: Velocity - stress finite - difference method. Geophysics, 51: , [48] Zhang.C., LeVeque.R.: The immersed interface method for acoustic wave equations with discontinuous coefficients, Wave Motion, 25 (1997). 37

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45 Noname manuscript No. (will be inserted by the editor) Acoustic Wave Propagation in Complicated Geometries and Heterogeneous Media Kristoffer Virta Ken Mattsson the date of receipt and acceptance should be inserted later Abstract We construct finite difference discretizations of the acoustic wave equation in complicated geometries and heterogeneous media. Particular emphasis is placed on the accurate treatment of interfaces at which the underlying media parameters have jump discontinuities. Discontinuous media is treated by subdividing the domain into blocks with continuous media. The equation on each block is then discretized with finite difference operators satisfying a summation - by - parts property and patched together via the simultaneous approximation term method. The energy method is used to estimate a semi - norm of the numerical solution in terms data, showing that the discretization is stable. Numerical experiments in two and three spatial dimensions verifies the accuracy and stability properties of the schemes. Keywords Acoustic wave equation curvilinear grids high order methods strong stability energy estimates SBP - SAT 1 Introduction Linear second order hyperbolic systems occur frequently in applications of wave propagation problems. Typical examples are found in acoustics, elasticity and electromagnetism (see e.g., [23], p ). When approximating the equations with a finite difference scheme it has been known for a long time that the number of grid points per wavelength required for a certain accuracy decreases as the order of the scheme increases [7]. This makes high order finite difference methods particularly attractive when the length scales considered consists of several wavelengths. To a well - posed second order hyperbolic system there typically exists a corresponding semi-norm. The semi - norm of the solution is often called the energy of the system. It is in general possible to prove that the energy can be estimated in terms of data. For long time simulations it is important that the numerical solution also satisfies a corresponding K. Virta Division of Scientific Computing, Department of Information Technology, Uppsala, Sweden kristoffer.virta@it.uu.se K. Mattsson Division of Scientific Computing, Department of Information Technology, Uppsala, Sweden

46 2 Kristoffer Virta, Ken Mattsson discrete energy estimate. Such schemes are stable and the method of proving stability of a discretization via an energy estimate is called the energy method [3]. A class of finite difference operators that has been successful in constructing stable high order discretizations of time dependent problems are operators satisfying a summation - by - parts (SBP) property [20,13]. The simultaneous - approximation - term method (SAT) [2] is a technique to impose boundary conditions weakly by adding to the numerical scheme a forcing term proportional to the exact value at the boundary. The combination of SBP operators and the SAT methodology (SBP - SAT) results in schemes where the energy method can be applied. In realistic applications boundaries often have complicated shapes and are rarely flat. Another characteristic is the possible heterogeneity of the underlying medium. In particular, media parameters may have jump discontinuities across interfaces in the domain of interest. The existence of curved boundaries and medium heterogeneities makes the construction of stable and accurate finite difference schemes for hyperbolic problems challenging. The acoustic wave equation with general boundary conditions was discretized in [14]. In [15] the scheme was extended to treat piecewise constant media parameters. Both works used high order SBP operators together with the SAT technique in one spatial dimension, but extended the technique to a second order finite volume method on an unstructured mesh in three space dimensions. To treat complicated geometries within the finite difference framework a coordinate transformation is typically made to align the new coordinate axes with the curved geometry. This transformation introduces variable coefficients in second derivative terms. Variable coefficients may also arise from inhomogeneous media parameters. SBP operators to approximate variable coefficient second derivatives were constructed in [12]. The first main result in the present paper is the construction of high - order SBP - SAT discretizations of the second order acoustic wave equation in complicated geometries and heterogeneous media. In the simulation of underwater sound the acoustic wave equation is applicable [4]. Here a simplification of the equation such as parabolic equation models or ray methods are often used [5]. These methods work well when considering the solution in the farfield. In the context of shallow water ocean acoustics examples can be found where reflection and refraction phenomena as well as the effects of a 3 dimensional bottom topography becomes significant [22,21], making a 3D finite difference approximation of the acoustic wave equation a very useful tool. Here the ocean floor may consist of a layer of penetrable sediment on top of impenetrable bedrock. At the interface between water and sediment reflection and refraction occurs depending on the topography of the bottom and the difference in density and wave speed of the different materials. In the present work we generalize the results of [14], [15] and [12] to three spatial dimensions, complicated geometries and heterogeneous media. Stability is proved by the energy method. High order accuracy and stability are illustrated with numerical experiments in two and three spatial dimensions. The second main result is to solve a non-trivial problem in shallow water ocean acoustics. The strengths of the proposed method are high order accuracy, geometric flexibility and proof of stability. There are of course many other methods for solving second order hyperbolic systems in complicated geometries. Finite difference discretizations of the acoustic wave equation with different boundary conditions was considered in [10,11,9]. Here non - planar boundaries were treated by embedding the domain into a Cartesian grid. These works were generalized to handle material discontinuities in [8], there 2nd order accuracy was obtained. A recent finite difference method using SBP operators can be found

47 Acoustic Wave Propagation in Complicated Geometries and Heterogeneous Media 3 in [17] where a second order method is used to solve the elastic wave equation. Here interface conditions at media discontinuities was implemented using ghost points. In [18] SBP operators was used to solve the elastic wave equation in smoothly varying media. Although piecewise smooth media can be allowed, without any special treatment at media interfaces the order of accuracy is reduced to only first order [1]. Even though the method proposed in this paper may find usefulness in acoustics problems it is expected that an analogous construction of a scheme for the elastic wave equation with highly accurate treatment of piecewise smooth media and curved geometries can be made. The rest of the paper is outlined as follows. Section 2 introduces the acoustic wave equation in curvilinear coordinates proving well - posedness using the energy method. In Section 3 the semi - discretization is explained and strict stability of the semi - discretization is proved. Numerical experiments are presented in Section 4, conclusions and future research directions are discussed in Section 5. 2 The continuous problem Consider the two dimensional acoustic wave equation on a 2 - block domain Ω Ω, depicted to the left in Figure 1. In a Cartesian coordinate system (the left of Figure 1) the equation takes the form, 1 c v 2 tt = ρ 1 (ρ v)+g, c,ρ > 0, (ξ,η) Ω, 1 c v 2 tt = ρ 1 (ρ v )+g, c,ρ > 0, (ξ,η) Ω t 0. (1), Here the parameters c, c and ρ, ρ denote the wave speeds and densities in the domains Ω and Ω, respectively. The solution ( v, v ) is subject to initial data v(ξ,η,0) = v 0, v t (ξ,η,0) = v 1, (ξ,η) Ω, v (ξ,η, 0) = v 0, v t(ξ,η,0) = v 1, (ξ,η) Ω. (2) At the interface between Ω and Ω continuity of the solution and solution fluxes are required, v v =0, ρ v n + ρ v n =0, (ξ,η) Ω Ω. (3) On the outer boundaries the conditions α 1 ρ v n + α 2 v + α 3 v t =0, (ξ,η) Ω \ Ω Ω, α 1ρ v n + α 2v + α 3v t =0, (ξ,η) Ω \ Ω Ω (4) are imposed. Here n denotes the outward facing unit normal and α 1,α 1 {0, 1}, α 2,α 3,α 2,α 3 0, α 1 + α 2 + α 3 > 0,α 1 + α 2 + α 3 > 0. Define the acoustic energies of v and v to be ( Ē(v) = 1 ρ 2 Ω c v 2 2 t + ρ v v ) d Ω + Ω\ Ω Ω α 2 v 2 ds, Ē (v ( )= 1 ρ 2 Ω t + ρ v v ) d Ω + Ω \ Ω Ω α 2v 2 ds. c 2 v 2

48 4 Kristoffer Virta, Ken Mattsson We define the total acoustic energy to be Ē T (v,v )=Ē(v)+Ē (v ), note that this is a semi - norm of the solution to (1) - (4). The following lemma states that in this semi - norm the solution is bounded in terms of initial data and interior forcing. Lemma 1 Let Ψ(t) = 1 t ρc 2 g 2 d Ωds, Ψ (t) = 1 t ρ c 2 g 2 d Ω ds, 2 0 Ω 2 0 Ω then the solution to (1) - (4) satisfies the energy estimate Ē T (v, v ) 2 (( Ψ(t)+ Ψ (t) ) e t + Ē T (v 0,v 0) ). Proof: Let ( w g,w g) be the solution to (1) - (4) with non-zero forcing g,g and homogeneous initial data, v 0 = v 0 = v 1 = v 1 =0.Let ( w 0,w 0) be the solution to (1) ( - (4) with homogeneous forcing, g = g =0and non-zero initial data. By linearity v, v ) = ( w g,w g ) ( + w0,w 0 ) is the solution to (1) - (4) with arbitrary forcing and initial data. Multiplying the first of the equations (1) for ( w g,w g) by wgt ρ and the second by w gtρ, using the divergence theorem and (4) we get d dt ĒT (w g,w g)= w gt ρgd Ω + w gtρ g d Ω Ω Ω α 3 wgtds 2 α 3w gtds 2 Ω\ Ω Ω Ω \ Ω Ω ( + wgt ρ w g n + w gtρ w Ω Ω g n ) ds. At the interface w gt = w gt follows from w g = w g and by (3) the last integral vanishes. Now, d (w g,w g) 1 ρ 1 dtēt 2 Ω c 2 w2 gtd Ω + ρc 2 g 2 d Ω 2 Ω + 1 ρ w 2 2 Ω c 2 gtd Ω + 1 ρ c 2 g 2 d Ω 2 Ω Ē T (w g,w g)+ 1 ρc 2 g 2 1 d Ω + ρ c 2 g 2 d Ω. 2 Ω 2 Ω Integrating this inequality leads to, t Ē T (w g,w g) Ē T (w g,w g)ds + Ψ(t)+Ψ (t). Using Grönwall s lemma we get Ē T (w g,w g) ( Ψ(t)+ Ψ (t) ) e t. 0 Similarly, Finally, Ē T (w 0,w 0) Ē T (w 0 (0),w 0(0)) = Ē T (v 0,v 0). Ē T (v,v )=Ē T (w g + w 0,w g + w 0) 2 ( Ē T (w g,w g)+ē T (w 0,w 0) ) ( ( 2 Ψ(t)+ Ψ (t) ) ) e t + Ē T (v 0,v 0)

49 Acoustic Wave Propagation in Complicated Geometries and Heterogeneous Media 5 Fig. 1 The mapping between Cartesian (left) and curvilinear coordinates (right). Any growth in the solution is only due to the forcing functions g,g.hence,(1)-(4) is a well - posed problem [7]. In particular, with homogeneous or transient forcing the total acoustic energy is bounded for any t> The acoustic wave equation in curvilinear coordinates Before discretization of the model problem the equations for v, v are transformed to curvilinear coordinate systems. The transformations are done such that in the new coordinate systems the axes are aligned with the boundaries of the domains, Figure 1. Assume that there is a differentiable one to one mapping from Ω =[ 1, 0] [0, 1] to Ω ξ = ξ(x, y), η= η(x, y), (x, y) Ω, By the chain rule, on Ω we have the relations ξ = x ξ x + y ξ y, η = x η x + y η y (5) and on Ω x = x ξ ξ + x η η, y = x η ξ + η y η, (6) where x ξ etc., are referred to as the metric derivatives. Using (5) and (6) we find the metric derivatives, [ ] xξ y ξ = 1 [ ] ηy η x, (7) x η y η J ξ y ξ x where J = ξ xη y ξ yη x is the Jacobian of the mapping from Ω to Ω. Since the mapping is one to one J>0. Define F = a 11 x + a 12 y, G= a 22 y + a 12 x a 11 = ξ2 y +η2 y J, a 12 = ξxηy+ηx+ηy J, a 22 = ξ2 x +η2 x J, u(x, y, t) =v(ξ(x, y),η(x, y),t),f(x, y, t) =g(ξ(x, y),η(x, y),t) u 0 (x, y) =v 0 (ξ(x, y),η(x, y)),u 1 (x, y) =v 1 (ξ(x, y),η(x, y)). (8)

50 6 Kristoffer Virta, Ken Mattsson With a one - to - one mapping from Ω = [0, 1] [0, 1] to Ω we can define the similar expressions except that f, g,u,v,ξ,η, a 11,a 12,a 22,F,G are replaced by primed quantities. Using (5) - (8) (for details see e.g, [6]) (1) is transformed into J c u 2 tt = ρ 1 (ρf u)x + ρ 1 (ρgu)y + Jf, (x, y) Ω, J c u 2 tt = ρ 1 (ρ F u ) x + ρ 1 (ρ G u ) y + J f, (x, y) Ω. (9) The interface conditions (3) becomes u u =0, ρf u + ρ F u x =0, 0 <y<1 (10) =0, and the outer boundary conditions (4) transforms into α 1 ρf u + α 2 u + α 3 u t =0,x= 1, 0 <y<1, α 1 ρgu + α 2 u + α 3 u t =0, 1 <x<0,y = {0, 1}, α 1ρ F u + α 2u + α 3u t =0,x=1, 0 <y<1, α 1ρ G u + α 2u + α 3u t =0, 0 <x<1,y = {0, 1}. (11) Let Since [ ] [ a11 a A = 12,A a = 11 a ] 12 a 12 a 22 a 12 a,u = 22 [ ] Fu,U = Gu a 11 > 0, a 11 a 22 a 2 12 =(ξ xη y ξ yη x) 2 = J 2 > 0, [ F u ] G u. (12) A>0 similarly A > 0. In curvilinear coordinates the acoustic energies of u and u are then E(u) = ( ) ρj 2 c 2 u2 t + ρu T AU dxdy With α 2 u 2 (x, 0,t)dx + 1 E (u )= ( ρ J Ψ(t) = α 2u 2 (x, 0,t)dx + t α 2 u 2 (x, 1,t)dx + ) u 2 c 2 t + ρ U T A U dxdy α 2u 2 (x, 1,t)dx + ρc 2 fdxdyds,ψ (t) = α 2 u 2 ( 1,y,t)dy, α 2u 2 (1,y,t)dy. t 1 1 ρ c 2 f dxdyds, the energy estimate (2) becomes ( (Ψ(t)+Ψ E T (u, u )=E(u)+E (u ) 2 (t) ) ) e t + E T (f 1,f 1). (14) In this paper a stable semi - discretization of the model problem in curvilinear coordinates is constructed. To prove the stability of the numerical scheme the semi - discrete solution is shown to satisfy a discrete energy estimate mimicking (14). (13)

51 Acoustic Wave Propagation in Complicated Geometries and Heterogeneous Media 7 3 The semi-discrete problem 3.1 Definitions In one spatial dimension the interval [x L,x R ] is discretized using N equidistant grid points, x j = x L +(j 1)Δ x, j=1,...,n, Δ x = x L x R N 1. The value of a function u at x j is denoted u j, a one dimensional grid function corresponding to u is then given by u =[u 1,...,u N ] T. The first derivative u x is approximated by D 1 u.hered 1 is a 2p-th order accurate narrow diagonal first derivative SBP operator, (D 1 u) j =(u x) j + O(h p ),j =1...m p,j = N m p +1...N (D 1 u) j =(u x) j + O(h 2p ),j = m p +1...N m p, D 1 = H 1 Q, Q + Q T =diag( 1, 0,...,0, 1), H =diag(h 1,...,h N ) > 0. (15) Here the integer m p depends on p. The variable coefficient second derivative (b(x)u x) x,b(x) > 0, is approximated by D (b) 2 u,whered(b) 2 is a 2p-th order accurate narrow diagonal variable coefficient second derivative SBP operator compatible with D 1, (D (b) 2 u) j =((bu x) x) j + O(h p ),j =1...n p,j = N n p +1...N, (D (b) 2 u) j =((bu x) x) j + O(h 2p ),j = n p +1...N n p, D (b) 2 = H 1 ( M (b) + B(b) S), M (b) = D1 T HB (b) D 1 + R (b) 0, B (b) = diag(b 1,...,b N ), R (b)t = R (b) 0, (Sv) j =(v x) j + O(h p+1 ),j =1,N, B (b) = diag( b 1, 0,...,0,b N ), (16) where the integer n p depends on p. ThematrixH is the same in both D 1 and D (b) 2. For details on SBP narrow diagonal operators see e.g., [20, 13, 12]. Remark: Although termed 2p order accurate by (15) and (16) the local order of accuracy of the SBP operators is halved in the vicinity of boundaries. It has been proved that for a discretization of the Schrödinger equation using 2p-th order accurate narrow diagonal variable coefficient second derivative SBP operators for the spatial spatial derivatives the global order of accuracy is p +2 for p =2, 3 and 2 for p =1 [16]. This behavior has also been observed in the numerical experiments in [15], [12] and [14]. For this reason the expected order of accuracy of the method described in this paper, which is verified in Section 5, is 2, 4 and 5 corresponding to p =1, 2 and 3. The following lemma is the variable coefficient analogue of Lemma 2.3 of [15] and is central in the proof of stability of the scheme considered in this paper. Lemma 2 Let u be a 1 - dimensional grid function, 2p the order of the operator D (b) 2, l p an integer independent of Δ x but dependent on p and b L =min{b 1,...,b lp } and b R =min{b N lp+1,...,b N },

52 8 Kristoffer Virta, Ken Mattsson Table 1 l p, β p and γ p in Lemma 2 for 2nd, 4th and 6th order accurate narrow - diagonal compatible second - derivative SBP operators. 2nd order 4th order 6th order l p β p γ p then the part M (b) of (16) in the expression for D (b) 2 has the property u T M (b) u = Δ xβ pb L (Sv) Δ xβ pb R (Su) 2 N + u T M (b) u, (17) where M (b) is symmetric positive semi - definite and β p a positive constant independent of Δ x,butdependentonp. Proof: See appendix 1 Table 1 presents the values of β p and l p for the 2nd, 4th and 6th order narrow - diagonal compatible second - derivative SBP operators constructed in [12]. Note that the numbers β p does not differ much from the corresponding values of [15] and that as Δ x turns to zero b L, b R turns to b(x L ), b(x R ) respectively. It is, however, possible to construct functions, depending on Δ x, such that Lemma 2 never holds with b L = b(x L ) and b R = b(x R ).Hence,choosingb L,b R as the values of b in the boundary, as Lemma 2.3 in [15] suggests may render M (b) of (17) indefinite for a fixed Δ x. In two spatial dimensions the domain [x L,x R ] [y L,y R ] is discretized using N x N y grid points, x j = x L + jδ x,j =1,...,N x,δ x = xr xl N, x 1 y k = y L + kδ y,k =1,...,N y,δ y = yr yl N. y 1 The value of a function u at a grid point (x j,y k ) is denoted u j,k. Fixing either the index j or k the restriction of u to the points (x j,y k ),k =1...N y, (x j,y k ),j =1...N x is denoted by u j,: and v :,k, respectively. A two dimensional grid function corresponding to u is then u =[u T 1,:,...,u T N x,: ]T as illustrated in Figure 2. The Kronecker product of two matrices A and B of sizes p q and r s is the pr qs matrix a 1,1 B... a 1,q B C = A B = a p,1 B...a p,qb The Kronecker product has the properties. (A B)(C D) =(AC) (BD), if AC and BD are defined, (A B) 1 =(A 1 B 1 ), if A 1 and B 1 exist, (A B) T = A T B T.

53 Acoustic Wave Propagation in Complicated Geometries and Heterogeneous Media 9 Fig. 2 Rectangular domain On a rectangular grid in the (x, y) coordinate system, to distinguish whether an operator acting on grid functions is acting in the x or y direction a subscript x or y is used. Let I N be the N N identity we define the 2 - D operators of size N xn y N xn y D 1x = D 1 I Nx, D 1y = I Ny D 1, S x = S I Nx, S y = I Ny S, H x = H I Nx, H y = I Ny H, e W = e 1 I Ny, e S = I Nx e 1, e E = e Nx I Ny, e N = I Nx e Ny. (18) Here e 1,e N are the vectors [ 10 0 ] T [ ] T respective 00 1 (j) of length N. LetO N be the N N matrix with zero entries everywhere except unity on the jth diagonal entry. The 2 dimensional approximations D (b) 2x, D(b) 2y of x (b(x, y) x ) and y (b(x, y) y ) respectively is constructed using the 1 dimensional narrow - diagonal second - derivative SBP operators, Similarly define Ny D (b) 2x = j=0 N y R x (b) = i=0 N x D (b:,j) 2 O (j) N y, D (b) 2y = R (b:,j) O (j) N y, R (b) j=0 N x y = j=0 O (j) N x D (bj,:) 2. O (j) N x R (bj,:) (19) and Note that D (b) 2x = H 1 x D (b) 2y = H 1 y N y B x (b) S x = j=0 N x B (b:,j) S O (j) N y, B(b) y S y = ( D1xB T (b) H xd 1x R x (b) ( D1yB T (b) H yd 1y R y (b) + B(b) x + B(b) y j=0 S x)=hx 1 S y)=h 1 y O (j) N x B(b j,:) S. ( M x (b) ( M y (b) + B(b) x S x), + B(b) y S y). (20)

54 10 Kristoffer Virta, Ken Mattsson In the following the diagonal matrix with the values of a 2 dimensional grid function u on the diagonal Λ u := diag(u) will frequently be used. The notation u E, u W etc. is used to denote the vector consisting of the values of v at the east boundary, the west boundary of the grid etc. The 2 - dimensional second derivative SBP operators satisfy an analogue of Lemma 2, Lemma 3 Let u be a 2 - dimensional grid function, b(x, y) > 0, b Lj =min{b j,1,...,b j,lp },b Rj =min{b i,n lp+1,...,b j,nx }, j =1...N y, b L = diag(b L1,...,b LNy ),b R = diag(b R1,...,b RNy ). The part M (b) x of (20) in the expression for D (b) 2x has the property u T H ym x (b) u = Δ xβ p(s xu) T W Hb L (S xu) W + Δ xβ p(s xu) T EHb R (S xu) E + u T H y M (b) x u, where M (b) x is symmetric, positive semi - definite. Here l p and β p are as in Lemma 2. Proof: N y u T H ym x (b) u = H j,j u T :,jm (b:,j) u :,j j=1 N y ) = (Δ xβ pb Lj H j,j (Su :,j ) Δ xβ pb Rj H j,j (Su :,j ) 2 N x j=1 N y + u T (b :,jh M :,j) u :,j j=1 = Δ xβ p(s xu) T W Hb L (S xu) W + Δ xβ p(s xu) T EHb R (S xu) E + u T H y M (b) x u. Here Lemma 2 was used to get the second equality 3.2 The SBP-SAT approximation In this section we discretize (9) - (11) in the spatial coordinates. This yields a semi - discrete system in which time is the continuous variable. The semi - discrete solution is denoted ( u, u ). In the following, when the corresponding primed versions of expressions are defined analogously by priming unprimed quantities these will not be explicitly written down. The right hand side of (9) contains spatial derivatives of four types. These are approximated with the 2 dimensional SBP operators defined in (18) and (20) as (ρa 11 u x) x D (ρa11) 2x u, (ρa 12 u y) x D 1x Λ ρλ a12 D 1y u, (ρa 12 u x) y D 1y Λ ρλ a12 D 1x u, (ρa 22 u y) y D (ρa22) 2y u.

55 Acoustic Wave Propagation in Complicated Geometries and Heterogeneous Media 11 The components of the normal derivatives at boundaries or interfaces are approximated by a 11 u x B(a 11) x S xu, a 12 u y Λ a12 D 1y u, a 12 u x Λ a12 D 1x u,a 22 u y B(a 22) x S yu. Define Q = D (ρa11) 2x ( + D 1x Λ ρλ a12 D 1y + D (ρa22) 2y + D 1y Λ ρλ a12 D 1x, B(a 11) F (±) = G (±) = ) x S x ± Λ a12 D 1y, ( ) B(a 22) x S y ± Λ a12 D 1x. With this notation the right hand side of (9) is discretized as 1 ρ (ρf u)x + 1 ρ (ρgu)y + Jf Λ 1 ρ Qu + Λ J f, J c u 2 tt = ρ 1 (ρ F u ) x + ρ 1 (ρ G u ) y + J f Λ 1 ρ Q u + Λ J f. The discretization of the terms in the interface conditions (10) becomes u u u E u W ( = I 1, ( ρf u + ρ F u Λ ρf (+) u )E + Λ ρ F ( ) u ) = I 2 W and the boundary terms of (11) at the east respective north boundaries have the discrete counterpart ) α 1 ρf u + α 2 u + α 3 u t α 1 (Λ ρf (+) u + α 2u E + α 3 u te = M E, ) E α 1 ρgu + α 2 u + α 3 u t α 1 (Λ ρg (+) u + α 2u N + α 3 u tn = M N. N The boundary terms corresponding to the west respective south boundary are defined analogously. The semi - discretization of (9) together with (10) - (11) is written as Λ J Λ 2 c u tt = Λ 1 ρ Qu + Λ J f + SAT, Λ J Λ 2 c u tt = Λ 1 ρ Q u + Λ J f + SAT. (21) Thediscretesolution ( u, u ) is subject to the initial data u(0) = u 0, u t (0) = u 1, u (0) = u 0, u t(0) = u 1. (22) In (21) boundary and interface conditions are imposed weakly using the SAT method [2]. The imposition is done via the penalty terms SAT and SAT. The penalty terms are proportional to the difference between the discrete values at the boundaries or interface and the boundary or interface forcings. Introduce SAT = SAT I1 + SAT I2 + SAT MW + SAT MN + SAT MS SAT = SAT I 1 + SAT I 2 + SAT M E + SAT M N + SAT M S. Here the first two terms enforces interface conditions while the last three terms enforces outer boundary conditions. The approximation of the boundary conditions (10) differ slightly between Dirichlet (α 1 =0, α 3 =0) and mixed boundary conditions (α 1 =1).

56 12 Kristoffer Virta, Ken Mattsson In this section mixed boundary conditions are considered, whereas Dirichlet conditions are described in Appendix 2. The penalty terms are vritten as SAT I1 SAT I 1 SAT I2 SAT I 2 SAT ME SAT MW SAT MN SAT MS = Λ 1 ρ = Λ 1 ρ = Λ 1 ρ = Λ 1 ρ = Λ 1 ρ = Λ 1 ρ = Λ 1 ρ = Λ 1 ρ Hx 1 Hx 1 H 1 H 1 H 1 H 1 H 1 H 1 ( (τ I + σ I1 Hy 1 Λ ρf (+)) T Hy)e E I 1, ( Λ ρ F ( )) T Hy)e W I 1, ( τ I σ I1 Hy 1 x σ I2 e E I 2, x σ I2 e W I 2, x σ M (M E 0), x σ M (M W 0), y σ M (M N 0), y σ M (M S 0). (23) The penalty parameters τ I,σ I1,σ I2 stable semi-discretization. and σ M are determined in Theorem 1 to yield a 3.3 Stability of the discretization To prove the stability of the semi-discrete system (21) - (22) the energy method is used. The energy method consists of constructing a suitable semi-discrete energy, a semi-norm of the solution to (21) - (22). The semi - discrete energy is constructed such that an estimate mimicking (14) can be derived. Define [ ] ΛρH H = xh y 0, A = 0 Λ ρh xh y [ Λa11 Λ a12 Λ a12 Λ a22 ], U = [ ] D1x u. D 1y u The positive definiteness of A follows from that of A of (12) and H is a positive definite diagonal matrix. Hence, HA > 0. Discrete semi - norms of the solution to (21) - (22) analogous to (13) are then E(u) =u T t Λ J Λ ρλ 2 c H xh yu t + U T HAU + u T W Hu W + u T N Hu N + u T S Hu S, E (u )=u T t Λ J Λ ρ Λ 2 c HxHyu t + U T HA U + u T E Hu E + u T N Hu N + u T S Hu S. (24) In what follows (24) will be modified into a semi - norm in which a discrete energy estimate can be proved. To achieve this some lemmata are needed. Lemma 4 If λ j,k = 1 2 ((a 11j,k + a 22j,k ) ( ) (a 11j,k a 22j,k ) 2 +4a 2 1/2 ) 12j,k, j =1...N [ x,k ] =1...N y, Λλ 0 Λ =, 0 Λ λ then A Λ 0.

57 Acoustic Wave Propagation in Complicated Geometries and Heterogeneous Media 13 Proof: For any vector x of length 2N xn y,with [ x j,k = x (j 1)Nx+k x NxN y+(j 1)N x+k ], A j,k = [ ] a11j,k a 12j,k a 12j,k a 22j,k we get N x N y x T Ax = x T j,ka j,k x j,k. j=1 k=1 The matrix A j,k is positive definite and its smallest eigenvalue is λ j,k = 1 ( ) ) 1/2 (a 11j,k + a 22j,k ) ((a 11j,k a 22j,k ) 2 +4a j,k. Define Λ j,k =diag ( λ j,k,λ j,k ),weget Introduce N x N y 0 x T ( ) j,k Aj,k Λ j,k xj,k = x T (A Λ) x j=1 k=1 λ Lj =min{λ j,1,...,λ j,lp }, λ Rj =min{λ j,n lp+1,...,λ j,n }, j =1...Ny, λ L1 b 1L = β pδ xdiag(,..., ρ 1,1a ,1 λ R1 λ LNy ), ρ Ny,1a 2 11 Ny,1 λ RNy b 1R = β pδ xdiag(,..., ), ρ 1,Nx a 2 11 ρ Ny,Nx a 2 1,Nx 11 Ny,Nx λ 1,1 b 2L = γ pδ xdiag(,..., ρ 1,1a ,1 λ 1,Nx λ 1,Ny ), ρ Ny,1a 2 22 Ny,1 λ RNy,1 b 2R = γ pδ xdiag(,..., ). ρ 1,Nx a 2 22 ρ Ny,Nx a 2 1,Nx 22 Ny,Nx (25) Here the values of l p,β p,γ p are displayed in Table 1. Lemma 5 With E(u) and E (u ) defined by (24) the quantities Ẽ(u) =E(u)+u T H yr x (ρa11) u + u T H xr y (ρa22) u ( ) T ( ) Λ ρ B(a 11) x S xu Hb 1R Λ ρ B(a 11) x S xu E E (Λ ρλ a22 D 1y u) T E Hb 2R (Λ ρλ a22 D 1y u) E, Ẽ (u )=E (u )+u T H yr (ρ a 11 x ) u + u T H xr (ρ a 22 y ) u ( Λ ρ B(a 11 x ) S ) T ( xu W Hb 1L Λ ρ B(a 11 x ) S ) xu W ( Λ ρ Λ a 22 D 1y u ) T ( W Hb 2L Λρ Λ a D 1yu ) 22 W, (26) are semi- norms of u and u, respectively.

58 14 Kristoffer Virta, Ken Mattsson Proof: We can write the term U T HAU + u T H yr x (ρa11) u in Ẽ(u) as U T HAU + u T HyR x (ρa11) u = U T H (A Λ) U + u T HyR (ρ(a11 λ)) x u + U T HΛU + u T HyR x (ρλ) u = U T H (A Λ) U + u T HyR x (ρ(a11 λ)) u + u T H ym (ρλ) u + ( D 1y u ) T ΛρH xh yλ λ D 1y u Here the expressions (19) and (20) for R x (ρa11) By Lemma 4 and (20) the matrices (A Λ),R (ρ(a11 λ)) x and M (ρa11), respectively, has been used. are symmetric positive semi - definite. We use Lemma 3 and get ( ) T ( ) u T H ym (ρλ) u = u T H y M (ρλ) u + Λ ρ B(a 11) x S xu Hb 1R Λ ρ B(a 11) x S xu, E E where H y M (ρλ) is symmetric positive semi-definite. The matrix e Nx e T N x I Ny has either zeros or ones on the diagonal with ones only at positions corresponding to u E. Hence, ( D1y u ) T ΛρHxH yλ λ D 1y u = (Λ ρλ a22d 1y u) T E Hb 2R (Λ ρλ a22d 1y u) E + ( D 1y u ) ( ) T ΛρHxH y I Nx I Ny e Nx e T N x I Ny Λ λ D 1y u ( ) with H xh y I Nx I Ny e Nx e T N x I Ny symmetric positive semi - definite. This proves that Ẽ(u) is a semi - norm. That Ẽ (u ) is a semi-norm is proved analogously Lemma 6 below is used in order to determine the values of τ I of (23) for which (21) - (22) is stable. Lemma 6 For any integer n>0 let The matrix τ =diag(τ 1,...,τ n), c j = diag(c j1,...,c jn ),c jk > 0,j =1,...,4,k =1,...,n 1 2 = diag( 1 2,..., 1 2 ). τ τ τ τ T (τ,c 1,c 2,c 3,c 4 )= c c c c 4 is positive semi - definite if τ i 4c 1 1i 4c 1 2i 4c 1 3i 4c 1 4i. Proof: For n =1 T (τ,c 1,c 2,c 3,c 4 )=T(τ a,c 1, 0, 0, 0) + T (τ b, 0,c 2, 0, 0) +T (τ c, 0, 0,c 3, 0) + T (τ d, 0, 0, 0,c 4 )

59 Acoustic Wave Propagation in Complicated Geometries and Heterogeneous Media 15 where τ = τ a + τ b + τ c + τ d. By computing the principal minors of the matrices on the right hand side we get the conditions τ a 4c 1 1,...,τ d 4c 1 4 for the positive definiteness of the corresponding matrix. Hence, T (τ,c 1,c 2,c 3,c 4 ) 0 if τ 4c 1 1 4c 1 2 4c 1 3 4c 1 4. For n > 1 let x be a vector of length 6n, with x j = [ ] T x j x j+n x j+5n we get x T T (τ,c 1,c 2,c 3,c 4 )x = n x T j T (τ j,c 1j,c 2j,c 3j,c 4j )x j. j=1 By the case n =1the sum is positive or zero if τ l 4c 1 1l 4c 1 2l 4c 1 3l 4c 1 4l Define u E u H ( W ) Λ ρb (a11) S 0 H xu U I = E ( (Λ ρλ a12 D 1y u),h I = 0 0 H E Λ ρ B (a 11 ) S xu ) H H 0 ( W Λρ Λ D a12 1yu ) H W and let τ Ij 1 4b 1Rj,j 1 4b 2Rj,j Lemma4-6thenshowsthat 1 4b 1L j,j 1 4b 2L j,j,τ I =diag(τ I1,...,τ INy ). (27) E T ( u, u ) = Ẽ(u)+Ẽ (u )+U T I H I T (τ I,b 1R,b 2R,b 1 L,b 2 L )U I (28) is a semi - norm of the solution to (21) - (22). In analogy with the continuous setting (28) is denoted the total semi - discrete acoustic energy. Define Ψ(t) = 1 2 t 0 f T H xh yλ ρλ J Λ 2 cfdt, Ψ (t) = 1 2 t f T H xh yλ ρ Λ J Λ 2 c f dt. 0 The first main result of this paper proving the stability of (21) - (22) can now be stated. Theorem 1 Take τ I as in (27), σ I1 = 1 2, σ I 2 = 1 2 and σ M = 1. then the solution to (21) - (22) is bounded in the semi - norm (28) by initial data and interior forcing via the semi - discrete energy estimate E T ( u, u ) 2 ( (Ψ(t)+Ψ (t) ) e t + E T ( u0, u 0) ). (29) Proof: The proof is analogous to the proof of Lemma 1. Let ( w f, w f ) be a solution to (21) - (22) with non-zero forcing f, f and homogeneous initial data u 0 = u 0 = u 1 = u 1 =0.Let ( w 0, w 0) be a solution with homogeneous forcing, f1 = f 2 =0and non - zero initial data. By linearity ( u, u ) = ( w f, w f ) ( + w0, w 0) is the solution to (21) - (22) with arbitrary forcing and initial data. Multiplying the equations of (21) ( w f, w f ) by w T ft H xh yλ ρ and w f T t H xh yλρ,respectively, adding the transpose

60 16 Kristoffer Virta, Ken Mattsson of the result, using the properties of the SBP operators and the choice of parameters τ I,σ I1,σ I2,σ M we get after some algebra ( d E ( ) w dt f + E ( w f ) ) + U T I T (τ I, 0, 0, 0, 0)U I = ( ) α 3 wf T tn Hw ftn + wf T ts Hw fts + wf T tw Hw ftw ( ) α 3 w f T tn Hw f tn + w f T ts Hw f ts + w f T te Hw f te + w T f t H xh yλ ρλ J f + w T f t H xh yλ ρ ΛJ f 1 2 wt f t Λ J Λ ρλ 2 c H xh yw ft w T f t Λ J Λ ρ Λ 2 HxHyw f t f T H xh yλ ρλ J Λ 2 cf f T H xh yλρ Λ J Λ 2 c f. c UsingLemma5weget d dt E ( T wf, w f ) ( ET wf, w f ) f T H xh yλ ρλ J Λ 2 cf f T H xh yλ ρ Λ J Λ 2 c f. Integrating this inequality and using Grönwall s lemma leads to E T ( wf, w f ) ( Ψ(t)+Ψ (t) ) e t. Similarly, Finally, E T ( w0, w 0) ET ( u0, u 0). ( E T u, u ) ( = E T wf + w 0, w f + w 0) ( ( 2 ET wf, w f ) ( + ET w0, w 0 )) 2 ( (Ψ(t)+Ψ (t) ) e t + E T ( u0, u 0) ) In analogy with the continuous setting the total discrete acoustic energy is bounded in terms of interior and boundary forcing. In particular, with homogeneous or transient forcing the semi - discretization conserves the total discrete acoustic energy. Remark It is straightforward to extend the scheme to three dimensions. ( The acoustic ) wave equation in curvilinear coordinates now contains terms of the type ρ (i) a (i) 33 ( uz z and ρ (i) a (i) 13 )z ux. These terms are discretized by constructing the corresponding three dimensional operators analogously to the two dimensional case. The corresponding SAT terms and the three dimensional semi - discrete system can then be constructed. It is straightforward to prove lemmata corresponding to Lemma 3-5 and define a three dimensional discrete semi-norm in which stability can be proved.

61 Acoustic Wave Propagation in Complicated Geometries and Heterogeneous Media 17 4 Numerical experiments We present three numerical experiments. The first two evaluate the accuracy and rate convergence in two and three spatial dimensions. The last is an application on a problem in underwater acoustics. Errors are measured in l 2 norm, e l2 = (u u ) T (u u ) h. Here u is the approximate solution, u is the exact solution and h = Δ xδ y, h = Δ xδ yδ z for 2 and 3 spatial dimensions, respectively. For the time integration the classical fourth order Runge - Kutta method is used. 4.1 Verification of accuracy In the first experiment the error and rate of convergence of the numerical solution when both the density, wave speed and solution is smooth are evaluated. As a computational domain we take the unit cube (x, y, z) [0, 1] [0, 1] [0, 1]. The domain decomposition technique is applied by cutting the domain in two pieces along the surface z = sin(2πx)sin(2πy) and then patch it together with the SAT method,as illustrated in Figure 4.1. The subdivision of the unit cube in such a way introduces two blocks with a curved interface, therefore a transformation to a curvilinear coordinate system is first made according to Section 2. Homogeneous Neumann boundary conditions are enforced on the outer boundary and the material density and wave speed are ρ(x, y, z) =2+sin(2πx)sin(3πy)sin(4πz), c 2 (x, y, z) =2+sin(πx)sin(2πy)sin(3πz). Internal forcing and initial conditions are then chosen such that the exact, manufactured, solution becomes u(x, y, z, t) =cos(4πx)cos(3πy)cos(5πz)cos(4πt). The time step for the numerical integration is taken to be Δ t = in the computations. Table 4.1 gives the errors and rates of convergence of the numerical solution, computed using SPB operators of order 2, 4 and 6, evaluated in l 2 norm with different grid resolutions at time t =1.0. An equal number of points, N, isusedin each space dimension. As was remarked in Section 3 we here see the expected order of convergence of 2,4,5, p l2,inl 2 norm corresponding to SBP operators of order 2,4,6. In the second experiment errors and rate of convergence when the wave speed as well as the metric derivatives has a discontinuity across block interfaces are measured in an experiment on a 4 - block domain in two spatial dimensions, Figure 4. The density is constant in each block and wave speed is c =1in blocks Ω 1,Ω 2,Ω 3, and c =1/2 in Ω 4. Convergence is measured by constructing a reference solution. To be valid as a solution the reference solution need to have converged sufficiently and the solutions to

62 18 Kristoffer Virta, Ken Mattsson Fig. 3 The computational domain used in experiment 1, cut in two pieces along the surface z = sin(2πx)sin(2πy). Table 2 Accuracy and convergence properties measured in experiment 5.1 2nd order 4th order 6th order N log 10 e l2 p l2 log 10 e l2 p l2 log 10 e l2 p l which it is compared need to be sufficiently far from asymptotic convergence. For this purpose a reference solution is constructed with (2 1601) 2 grid points which is then compared to numerical solutions using up to (2 201) 2 grid points. The initial data is a Gaussian pulse u(x, 0) = 1.5e 40((x 1.5)2 +(y 1.4) 2),u t (x, 0) = 0, presented in Figure 4. The time step for the numerical integration is taken to be Δ t = in the computations. The convergence and errors are measured in l 2 norm at times T =1, 2, 3. Attimet =1the wavefront has reached the sharp corner at which the wave speed and all metric derivatives are discontinuous, Figure 5. Regardless of this jump in material properties and discontinuity of normal derivatives across the corner the expected order of convergence, p l2, can be observed, Table Application: Underwater acoustics As an application a variation of the ocean acoustics problem described in [22] is considered. A transient acoustic point source is placed in an three - dimensional oceanic environment, (x, y, z) [0, 2000] [0, 1000] [0, 425] consisting of a column of water overlaying a penetrable layer of sediment. The column of water is bounded above by the sea surface and the layer of sediment is bounded below by impenetrable bedrock.

63 Acoustic Wave Propagation in Complicated Geometries and Heterogeneous Media 19 Fig. 4 The 4-block physical domain and the initial data, a Gaussian pulse Table 3 Accuracy and convergence properties computed at times t = 1, 2, 3 in experiment 5.2 2nd order Time = 1 Time = 2 Time = 3 N log 10 e l2 p l2 log 10 e l2 p l2 log 10 e l2 p l th order Time = 1 Time = 2 Time = 3 N log 10 e l2 p l2 log 10 e l2 p l2 log 10 e l2 p l th order Time = 1 Time = 2 Time = 3 N log 10 e l2 p l2 log 10 e l2 p l2 log 10 e l2 p l (a) t =1 (b) t =2 (c) t =3 Fig. 5 The numerical solution of experiment 2 at time t =1, 2, 3.

64 20 Kristoffer Virta, Ken Mattsson The water depth is variable and the speed of sound in water is depth dependent. Figure 6 displays the structure of the oceanic environment and the sound speed depth dependence. The density in water is 1g/cm 3, the speed of sound and density in the layer of sediment is 1700m/s and 1.2g/cm 3, respectively. To correctly model the boundary at the sea surface a homogeneous Dirichlet condition (u =0) is applied at z =0the boundary between the layer of sediment and the impenetrable bedrock is modeled by a homogeneous Neumann condition ( u n =0)atz = 425. At vertical boundaries a radiation condition ( u n 1 c u t =0) is applied. The depth variability and the discontinuity in both sound speed and density at the interface between water and sediment makes this problem an ideal candidate for the methodology developed in this paper. The point source is f = δ(x x 0,y y 0,z z 0 )e 16π2 (t t 0) 2 cos(2πf ct), where δ is the Dirac delta function. Here the central frequency, f c,is35 Hz. The source is placed at (506.2, 500, 144.8). The shortest wave lengths occurring is estimated by the quotient of the smallest present wave speed and the central frequency as L = Solutions using 2nd and 6th order SBP operators are computed using two different grids. The grids are using 5 and 10 points per shortest wavelength respectively. The computations where done on a desktop machine using 4 Intel Xenon W GHz processors and 24 GB of available memory. We call the solution obtained with the 2nd order scheme on the grid with at least 5 points per wavelength the 2nd order 5 point grid solution. The other solutions are termed similarly. To compare the performance between the different schemes a time series of snap-shots of the solution in different regions of the vertical cross-section (y = 500) through the source is constructed. The time series displays the propagation of the acoustic wavefront through out the domain. In all snap-shots the solution has been scaled with the distance form the source to compensate for spherical spreading. The time step for the numerical integration is taken to be Δ t = 1 min{δ x,δ y,δ z} in the computations. From the accuracy and convergence results of the previous numerical experiments we conclude that the 6th order 10 point grid solution is closest to the true solution. The time series obtained using the 2nd order scheme is shown in Figure 7. To evaluate the quality of the solution we also plot a contour of the 6th order 10 point grid solution at the corresponding times. The 2nd order 5 point grid solution shows significant differences, higher modes are clearly under resolved and the wave front has not traveled the same distance. The 2nd order 10 point grid solution corresponds seemingly well at the first times but in later times the solution can be seen by comparing the location of corresponding features in the solutions to be approximately 180 degrees out of phase. The results of the computations using the 6th order SBP operators are displayed in Figure 8. Here the solutions are similar at all times. The solution on the 5 point grid does not suffer from under resolved modes or visible phase errors. To compute the 6th order 5 point grid solution takes 1.5 hours and requires 0.5 GB of memory. In order to obtain a reasonable solution with the second order scheme further refinement of the grid would be necessary. An approach that is unfeasible due to limitations in memory and computational performance. For this reason such a computation is not done here. The above experiment illustrates the significant advantage of using higher order methods for hyperbolic wave propagation problems in three spatial dimensions. We also conclude that with the usage of high order methods realistic wave propagation phenomena can be simulated on smaller scale computers whereas 2nd order methods would need a larger scale computer to give a reliable solution.

65 Acoustic Wave Propagation in Complicated Geometries and Heterogeneous Media 21 Fig. 6 Left: the sound speed profile in water. Right: the bottom topography and the vertical cross section y = 500 of the domain. 5 Conclusions We have developed finite difference discretizations of the acoustic wave equation in complicated three dimensional geometries and heterogeneous materials. The treatment of complicated geometries was done by transforming the underlying equation to a curvilinear coordinate system where the coordinate axes are aligned with the topography of the physical domain. To treat variable coefficients in second derivatives arising from such coordinate transformation as well as a heterogeneous media the discretizations were constructed using variable coefficient second derivative SBP operators of orders 2,4,and 6. Jump discontinuities in media parameters were made possible by subdividing the domain into blocks where each parameter is smoothly varying and then patch them together using the SAT methodology. Our analysis shows that the discrete solution can bounded in a semi - norm by initial conditions and interior forcing. The proof of this property was done via the energy method. Hence, the discretization is stable. The convergence and accuracy of the method have been confirmed by numerical experiments in two and three spatial dimensions. As an application an acoustics problem in the simulation of underwater sound was considered. The properties of this problem made the generality and accuracy of our method a suitable choice of solution strategy. In particular, the underwater acoustics problem was stated in a three dimensional oceanic environment with a curved ocean floor and a jump discontinuity in material parameters at the interface between water and bottom sediments. Future work includes generalizing the approach of the present paper to treat problems of wave motion in elastic solids. Here a numerical solver with the generality of our method naturally finds usefulness in for example seismology due to the curved and layered structure of the earth. We expect that the SBP - SAT technique in conjunction with the energy method should be successful in constructing a stable and accurate discretization of the equations of elastic waves. A Proof of Lemma 2 The proof of Lemma 2 uses the structure of the 2p-th order accurate narrow diagonal variable coefficient second derivative SBP operator constructed in [12]. For reference the operators D 1

66 22 Kristoffer Virta, Ken Mattsson (a) 2nd order, 5 points per shortest (b) 2nd order, 10 points per shortest wavelength wavelength Fig. 7 Time series using 2nd order SBP operators, t = 0.3, 0.5, 0.8, 1.1, 1.4

67 Acoustic Wave Propagation in Complicated Geometries and Heterogeneous Media (a) 6th order, 5 points per shortest (b) 6th order, 10 points per shortest wavelength wavelength Fig. 8 Time series using 6th order SBP operators, t = 0.3, 0.5, 0.8, 1.1,

68 24 Kristoffer Virta, Ken Mattsson and D (b) 2 for the case p =2are formulated explicitly. The norm H is given by H = Δx diag( 17 48, 59 48, 43 48, 49 49, 1,...,1, 48 48, 43 48, 59 48, ). The 4-th order accurate narrow diagonal first derivative SBP operator D 1 is: D 1 = Δx With the lower right 4 6 block obtained by rotating the upper left 4 6 block 180 degrees and changing the signs of the elements. The part M (b) of the 4-th order accurate narrow diagonal variable coefficient second derivative SBP operator compatible with D 1 is M (b) = D1 T HB(b) D 1 + Δx5 18 DT 3 C3B(b) Δx7 3 D DT 4 C4B(b) 4 D4. Where B (b) =diag(b 1,...,b N )=B (b) 4, B (b) 3 =diag( b 1+b 2, b i+b i+1,..., b N 1+b N ), C 3 =diag(0, 0, , , 1,...,1, , , 0, 0) C 4 =diag(0, 0, , , 1,...,1, , , 0, 0), d D 3 = 1 d 2 d 3 d 4 d 5 d with d 1 = d 4 = ,d2 =,d3 = , ,d5 =,d6 = the lower right 3 6 block of D 3 is obtained by rotating the upper left 3 6 block 180 degrees and changing the signs of the elements, d D 4 = 1 d 2 d 3 d 4 d 5 d the lower right 3 6 block of D 4 is obtained by rotating the upper left 3 6 block 180 degrees. The matrix S is given by S =

69 Acoustic Wave Propagation in Complicated Geometries and Heterogeneous Media 25 A.1 Proof of Lemma 2 Let l p be an positive integer, b L as in the lemma and B L the matrix with the only non-zero elements B Li,i = b L,i =1...l p.forp =2define B (b) = B B L, B (b) 3 = B 3 B L, B (b) 4 = B 4 B L, M (b) = D1 T H BD 1 + Δx5 18 DT 3 C3 B 3D 3 + Δx7 144 DT 4 C4 B 4D 4, M L = D1 T HB LD 1 + Δx5 18 DT 3 C3B LD 3 + Δx7 144 DT 4 C4B LD 4. By the choice of B L these matrices are symmetric positive semi - definite. Now, S T M L S 1 = Δxb L ML.Where M L is independent of Δx and b L. By the construction of B L the only non - zero elements of M L are located in a upper left square block A of size independent of Δx but dependent on l p,byconstructiona = A T 0. Forl p =1,... compute A and construct the matrix B with B ij = A ij,b 11 = A 11 β. β is then chosen by computing the eigenvalues of B as the largest number such that B 0. Notethatalsoβ is independent of Δx. Achoice of l p =4gives a value of β as in Table 1. Let M L be the matrix resulting from subtracting Δxb L β from the first diagonal element of Δxb L ML.Weget u T M (b) u = u T ( M (b) + M L )u = u T ( M (b) + S T ML S)u + βδx ( B (bl) Su) 2 1 b. L Similarly we can derive the second term of (17) by constructing the corresponding M R e.t.c., Then taking M (b) = M (b) + M L + M R proves the lemma for p =2. The proof for p =1and p =3follows the same arguments as for p =2. (30) B Dirichlet boundary conditions Assume a homogeneous Dirichlet boundary condition at the west boundary. The corresponding SAT term is then ( ) SAT DW = Λ 1 ρ H 1 x τ DW + σ D Hy 1 F( )T H y e W (u W 0). (31) Similar to the discretization of the interface conditions homogeneous Dirichlet boundary conditions adds a term to the total semi - discrete acoustic energy. The following lemma, proved similarly to lemma 6, determines the parameter τ D for this term do be positive semi - definite. Lemma 7 With the same notation as in Lemma 6 and 1 =diag(1,...,1) the matrix τ 1 1 T D = 1 c 1 0 (32) 1 0 c 2 is positive semi - definite if τ j 1 c 1j 1 c 2j. Define and let u W U D = (B (ρa11) S xu) W,H D = H H 0 (33) (Λ ρλ a12 D 1yu) W 0 0 H τ Dj 1 b 1Rj,j 1 b 2Rj,j,τ D =diag(τ D1,...,τ DNy ) (34) where b 1Rj,j and b 2Rj,j are defined by (25), then the addition of the term U T D H DT D (τ D,b 1R,b 2R )U D to the total semi - discrete energy also defines a semi norm of the solution to (21) - (22). Now with a Dirichlet boundary condition. An energy estimate in this semi - norm follows as in Theorem 1.

70 26 Kristoffer Virta, Ken Mattsson References 1. Brown, D.L.: A note on the numerical solution of the wave equation with piecewise smooth coefficients, Mathematics of Computations, 42(166): , Carpenter, M., Gottlieb, D., Abarbanel, S.: Time - Stable Boundary Conditions for Finite - Difference Schemes Solving Hyperbolic Systems: Methodology and Application to High - Order Compact Schemes, J. Comp Phys 111, (1994) 3. Gustafsson, B., Kreiss, H-O., Oliger, J.: Time Dependent Problems and Difference Methods, Wiley, Jensen, F.B., Ferla, C.M.: Numerical solutions of range - dependent benchmark problems in ocean acoustics, J. Acoust. Soc. Am 87, (1990) 5. Jensen, F.B., Kuperman, W.A., Porter, M.B., Schmidt, H.: Computational Ocean Acoustics, 2000 Springer-Verlag New York, Inc 6. Knupp, P., Steinberg, S.: Fundamentals of grid generation, CRC Press, Kreiss, H - O., Oliger, J.: Comparison of accurate methods for the integration of hyperbolic equations, Tellus, 24 (1972) 8. Kreiss.H.-O., Petersson.N.A.: An embedded boundary method for the wave equation with discontinuous coefficients, SIAM J. Sci. Comput. Vol. 28, No. 6, Kreiss.H.-O., Petersson.N. A.: A second order accurate embedded boundary method for the wave equation with Dirichlet data, SIAM J. Sci. Comput., 27 (2006). 10. Kreiss.H.-O., Petersson.N. A., Ystrom.J.: Difference approximations for the second order wave equation, SIAM J. Numer. Anal., 40 (2002). 11. Kreiss.H.-O., Petersson.N. A., Ystrom.J.: Difference approximations of the Neumann problem for the second order wave equation, SIAM J. Numer. Anal., 42 (2004). 12. Mattsson, K.: Summation by parts operators for finite difference approximations of second - derivatives with variable coefficients, J Sci Comput, 51, (2012) 13. Mattsson, K., Nordström, J.: Summation by parts operators for finite difference approximations of second derivatives, J. Comp Phys 199 (2004) 14. Mattsson, K., Ham, F., Iccarino, G.: Stable Boundary Treatment for the Wave Equation on Second Order Form. J. Sci. Comput. 3, (2009) 15. Mattsson, K., Ham, F., Iaccarino, G.: Stable and accurate wave - propagation in discontinuous media, J. Comp Phys 227 (2008) 16. Nissen, A., Kreiss, G., Gerritsen, M.: High Order Stable Finite Difference Methods for the Schrödinger Equation. J. Sci. Comput. 55, (2013) 17. Petersson, N. A., Sjögreen, N.: Stable grid refinement and singular source discretization for seismic wave simulations, Communications in Computational Physics Sjögreen, B., Petersson, N. A.: A Fourth Order Accurate Finite Difference Scheme for the Elastic Wave Equation in Second Order Formulation, Journal of Scientific computing Sjögren, B., Petersson, N.A.: A Fourth Order Accurate Finite Difference Scheme for the Elastic Wave Equation in Second Order Formulation, 20. Strand, B.: Summation by parts for finite difference approximations for d/dx, J. Comp Phys 110 (1994) 21. Sturm, F.: Investigation of 3-D benchmark problems in underwater acoustics: a uniform approach 22. Sturm, F., Ivansson, S., Jiang, Y., Chapman, N.R.: Case Study Concerning 3-D Out-of- Plane Sound Propagation in Shallow Water, Theoretical and Computational Acoustics Whitham, G. B.: Linear and nonlinear waves, Wiley, 1974

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73 Surface Waves in Almost Incompressible Elastic Materials Kristoffer. Virta 1,, Gunilla. Kreiss 1 1 Division of Scientific Computing, Department of Information Technology, Uppsala. kristoffer.virta@it.uu.se Abstract A recent study shows that the classical theory concerning accuracy and points per wavelength is not valid for surface waves in almost incompressible elastic materials. The grid size must instead be proportional to ( μ λ )(1/p) to achieve a certain accuracy. Here p is the order of accuracy the scheme and μ and λ are the Lame parameters. This accuracy requirement becomes very restrictive close to the incompressible limit where μ 1, λ especially for low order methods. We present results concerning how to choose the number of grid points for 4th, 6th and 8th order summationby-parts finite difference schemes. The result is applied to Lambs problem in an almost incompressible material. 1 Introduction Consider the half - plane problem for the two - dimensional elastic wave equation in a homogeneous isotropic material. With time scaled to give unit density the displacement field (u, v) is governed by u tt = μδu +(λ + μ)(u x + v y ) x, v tt = μδv +(λ + μ)(u x + v y ) y, (x, y) (, ) [0, ),t 0, (1) where λ > 0 and μ > 0 are the first and second Lame parameters of the material. We assume that both Lame parameters are constant. Initial data for (u, v) and (u t,v t ) is given at t =0. On the boundary y =0we consider conditions on the normal and tangential stresses v y + λ λ+2μ u x = g 1 (x, t), y =0,t>0. (2) u y + v x = g 2 (x, t), With g 1 = g 2 =0(2) is called a traction free boundary condition. The elastic energy, a semi - norm of the solution to (1), is given by E(t) = 1 ( u 2 2 t + v 2 ) ( t + λ (ux + v y ) 2 + μ 2u 2 x +2v2 y +(u y + v x ) 2) dxdy. 0 (3) 1

74 The elastic energy satisfies (see e.g., [7], pp ) d E(t) = (v t (λ +2μ) g 1 + u t μg 2 ) dt y=0 dx. (4) In particular, with a traction free boundary condition the elastic energy is constant, E(t) =E(0),t 0,g 1 = g 2 =0. (5) It is well known that (1) admits compressional and shear waves. This becomes transparent when considering the simpler set of equations equivalent to (1), φ tt =(λ +2μ)Δφ, H tt = μδh, (x, y) (, ) [0, ),t 0. (6) Here the equations for φ and H, governs the propagation of compressional and shear waves with phase velocities λ +2μ and μ, respectively. The displacement field (u, v) is obtained via u = φ x + H y, v = φ y H x. (7) The boundary condition (2) in terms of φ and H becomes, φ xx + φ yy 2μ λ+2μ (φ xx + H xy )=g 1 (x, t), y =0,t>0. (8) 2φ xy + H yy H xx = g 2 (x, t), For a discussion on how to arrive at (6) - (8) from (1) - (2) see [7], pp The elastic wave equation with a traction free boundary condition also admits Rayleigh surface waves. These waves travel harmonically along the surface of the half - plane, whereas the amplitude decay exponentially into the domain. The phase velocity c R of the waves satisfies c R / μ<1. The exact value of the quotient depends on μ/λ, but an approximation is given in [13] by c R / μ ( ν)/(1 + ν) < 1. Hereν = λ/2(λ + μ). Hence, the Rayleigh surface waves always travel slower than both the compressional and shear waves. In many applications the period of the solution is given through boundary and internal forcing and can be considered as known. Then, as the phase velocity of a wave is defined by the ratio of its length and period, the shortest present wavelengths becomes proportional to μ. According to the classical theory in [8] an accurate numerical solution is obtained if the shortest wave length is not smaller than a constant number of grid sizes, where the constant depends on the order of accuracy of the numerical method. This predicts that the grid size should be proportional to μ. In a recent paper by H - O. Kreiss and N.A. Petersson [9] materials with μ λ are studied. There it is shown that the classical theory is inadequate when simulating surface waves. Instead it is proved that the grid size must be proportional to (μ/λ) 1/p in order to achieve an accurate solution. Here p is the order of accuracy of the numerical method. This requirement becomes very restrictive close to the incompressible limit μ/λ 1, especially for low order methods. The theory in [9] was supported by numerical experiments using 2nd and 4th order discretizations of (1) - (2) with μ/λ as small as Another discretization of (1) - (2) was constructed in [3]. The discretization uses summation - by - parts (SBP) finite difference operators of 2

75 orders 2, 4, 6, 8 [12, 11] to discretize the right hand side of (1). The method uses the simultaneous - approximation - term (SAT) method [1] to approximate the boundary conditions (2). By using the properties of the SBP operators stability of the resulting scheme was proven by constructing a discrete semi - norm of the discrete solution with the property of mimicking (4). In particular, the discretization with a traction free boundary condition mimics (5) to machine precision. Accuracy and convergence of the discretization was verified by using a standing wave solution. In this paper we continue in the lines of [9] and use the code developed in [3] to further study simulation of surface waves in almost incompressible materials. In particular we study materials in which μ/λ < In the concluding section of [9] remarks are made on the use of methods of higher order than 4. It is there concluded that numerical experiments must be performed to evaluate how small μ/λ has to be to compensate for the higher complexity of higher order methods. As an introductory example we therefor let a Rayleigh surface wave propagate in the half - plane y 0 with a traction free boundary condition at y =0. The wave clings to the surface and decays exponentially in y, see Figure 1. We take λ =1and μ =10 4. The solution is scaled such that the surface wave has unit wavelength. The resulting period of the solution is then T = In the numerical experiment the x - direction is made 1 - periodic. The performance of methods using 4th and 8th order SBP operators are then compared. We use N x points per surface wave length and compute until time T/2. In Figure 2 the relative max error as a function of time is displayed for the different methods on a series of finer grids. Note that to achieve a relative max error of at most 5% the method using 4th order SBP operators require 101 grid points per surface wavelength. This is approximately 10 times the number of points predicted by the classical theory. The figure showing the results for the method using 8th order SBP operators shows that only 21 grid points per surface wavelength is needed to make the relative max error less than 5%. The originality of this work follows in Section 3 and 4. Section 3 presents numerical tests on the performance of higher order methods with μ/λ as small as These results are used in section 4 to estimate the number of points per smallest wavelength needed to accurately approximate a version of Lambs problem ([10]) in a almost incompressible material. In an appendix we derive an analytic expression for the Rayleigh surface wave and discuss its sensitivity to a boundary truncation error in a numerical approximation. This presentation is analogous to the one given in [9] but differs in that the theory is obtained via the equations (6) - (8) rather than (1) - (2). Concluding remarks are given in section 5. 2 The numerical method The elastic wave equation on the second order form (1) was discretized in [3]. To approximate spatial operators high order SBP operators were used. In [3] it was shown how to impose a traction free boundary condition weakly with the SAT technique. A Dirichlet condition was imposed strongly by injecting data at the boundary. Stability of the numerical scheme was proved with the energy method by showing that the discrete system satisfies a discrete energy estimate mimicking (5). The discretization and proof of the energy estimate was done 3

76 Figure 1: Plot of the (u, v) components of the Rayleigh surface wave at t =52.34 for a material with λ =1and μ =10 4. The u component is shown to the right and the v component to the left th order SBP operators th order SBP operators Nx = 51 Nx = 101 Nx= Nx = 21 Nx = 41 Nx= Figure 2: Relative max error as a function of time for the Rayleigh surface wave in a material with λ =1and μ =10 4. Results from a scheme using 4th and 8th order SBP operators are shown on the left and right, respectively. The number of points per wavelength is increased from top to bottom. Note that the grids are finer for the computations using 4th order SBP operators. 4

77 for general SBP operators without any restrictions on the order of accuracy. In this paper we consider numerical schemes constructed with 2p -thordersbp operators [12, 11] for p =2, 3, 4. Although termed 2p - th order accurate the local order of accuracy is only p at a constant number of points in the vicinity of the boundary of the domain. It has been shown in [6] for a discretization of the Schrödinger equation using 2p - th order SBP operators the global order of accuracy is p +2. This was also observed in the numerical experiments of [3]. In the numerical experiments we impose periodic boundary conditions at x = ±L x in the x - direction. At a distance L y below the traction free surface at y =0we either impose a Dirichlet condition, when the exact solution is known, or use the perfectly matched layer (PML) constructed in [2] to absorb outgoing waves. The discretization of spatial derivatives with SBP operators, enforcing of a traction free boundary condition with the SAT technique and a Dirichlet condition at y = L y results in a semi - discrete system of the type [ u = v] 1 [ ] u h 2 Q. (9) v tt Here u and v are vectors with approximative values of u and v at the grid points of the computational domain and h is the grid size. Q is a matrix with elements independent of h. To discretize (9) in time we use the 4th order time stepping scheme of [5]. This scheme was designed for a system of the type (9) in that it is not rewritten to a system first order in time. In [5] it was shown that if a time step k is chosen as C k = h (10) ( Q Q 1 ) 1/4 stability is guaranteed provided that a discrete energy estimate exists. Here C is a constant depending on the order of accuracy of the spatial discretization. 5

78 Using the PML results in a system of the type u tt v tt = P u v. (11) w w t Here w contains auxiliary variables arising from the addition of the PML. In the presence of a PML the system (11) is rewritten as a first order system in time and the classical Runge - Kutta 4 scheme is used to integrate in time. 3 A numerical study for different values of μ/λ To perform reliable numerical computations it is of importance to know the number of grid points per surface wavelength needed to obtain a certain accuracy in an approximate solution. In this section we study surface waves in materials with varying μ/λ. We are interested in the performance of schemes using SBP operators of different orders to discretize (1) - (2). The numerical study is performed as follows. For a given value of μ/λ we determine the number of grid points, P R, per surface wavelength needed to achieve a relative max error of at most 5% after having propagated for 10 periods in time. As the number of grid points per surface wavelength needed for accuracy is proportional to (λ/μ) 1/p where p is the order of accuracy of the method, high order of accuracy is expected to become more influential as μ/λ decreases. The higher the order of the scheme the more computational effort is required. For this reason execution times are recorded. It is then possible to conclude how small μ/λ must be to compensate for the higher complexity of higher order schemes. We use (22) and (7) to derive an analytic expression of a Rayleigh surface wave. We are computing in real arithmetic, therefor we use the real part of the displacement field (u, v), ) ( (u ) ) T ( ξe αy + β2 +ξ 2 Re v = A 2 2ξ e βy sin(ξ(x c R t)) ),A ( αe αy + β2 +ξ 2 2 R. 2β e βy cos(ξ(x c R t)) (12) Here the phase velocity c R is given by (23). We keep the wavelength fixed at L R =1by choosing ξ =2π. Thevaluesofα and β then follows from (14). The constant A 2 is arbitrary but we take A 2 = 1 2π. For simplicity we keep λ =1 fixed. The period, T, of the solution is then proportional to 1/ μ.weconsider a domain which is periodic in the x - direction. The computational domain is chosen to contain exactly one wavelength of the solution. At the boundary y =0 a traction free boundary condition is imposed. The time step is chosen according to (10). The computational domain is truncated at y =10by imposing exact boundary data given by the exact solution (12). The numerical computations are made on a single Intel Xenon W Ghz processor. Results for different values of μ/λ are reported in Table 2. We see that for 10 3 μ/λ 10 2 the schemes using 6th and 8th order SBP operators perform similarly whereas the scheme using 4th order operators need significantly more computational time to achieve a 5% relative error, in particular for the case μ =10 3. The errors obtained with the 4th order method are of the same magnitude as those obtained in [9] for the same values of μ/λ. For10 5 μ/λ 10 4 the 6th order scheme 6

79 is clearly disadvantageous compared to the 8th order scheme, it uses more than 4 times the amount of time to get a relative error of maximum 5% for μ/λ in this interval. For μ/λ 10 6 the required number of points per surface wavelength used by the 8th order scheme has increased very much above the value predicted by the classical theory and for such materials even higher order methods would be needed for an efficient numerical method. These computations verifies the theory of [9] for schemes of higher accuracy than 4 and predicts how small μ/λ must be for the different higher order methods to be more efficient when surface waves are present in simulations. Case P R e 4 T 4 e 6 T 6 e 8 T 8 μ = T = μ = T = μ = T = μ = T = μ = T = Table 1: The leftmost column displays the value of μ/λ and the period, T,ofthe solution. The rest of the columns report the number of grid points per surface wavelength and the corresponding relative max errors, e p and execution times, T p for schemes using p-th order SBP operators. 4 Application: Lambs problem in almost incompressible material We solve a version of Lambs problem [10] in which the surface of a half - space is subjected to a periodic array of line sources with loading normal to the surface. Lamb proved in [10] that under these conditions compressional, shear and 7

80 Rayleigh waves are generated. The stress forcing of (2) is g 1 (x, t) =f(t)δ(x km),m >0,k =0, ±1,..., g 2 (x, t) =0, where M is the distance between the sources and δ the Dirac delta function. We let f be the wavelet given by { sin(2πωt) 1 f(t) = 2 sin(4πωt), 0 t 1 ω 0, else. f is shown as an inset in Fig 3(a) with ω =1. With λ =1,μ =10 3 the (a) Numerical solution (b) Relative error Figure 3: The numerical solution (a) and the relative error (b) at time t =3.2 Rayleigh phase velocity becomes c R μ = The highest significant frequency with ω =1in the time function f is 2. The corresponding shortest wavelength of the Rayleigh wave is L min = cr μ 2 = We choose the domain [ 2L min, 2L min ] [0, 6L min ], M =4L min and solve numerically until time t =3.2. Figure 3(a) shows the magnitude of the displacement field. Periodic boundary conditions are applied at the vertical boundaries and the domain is truncated above with a perfectly matched layer [2]. To estimate the required number of points per wavelength to achieve a relative max error of at most 5% with a 6th order method we consult Table 2 to conclude that 25 points per wavelength should suffice. To ascertain this claim a reference solution with 200 points per shortest wavelength is constructed. As a comparison a solution using 10 points per shortest wavelength, a quantity predicted by the classical theory to yield a relative error lower than 5%, is also computed. The results presented in Table 2 verifies the claim for this application. Figure (3(b)) shows the relative 8

81 error in the magnitude of the displacement field at time t =3.2. It is interesting to see that the main bulk of the error is seen to be located in the vicinity of the surface. This is in accordance with the theory presented in [9], which predicts that the Rayleigh waves are much more sensitive to discretization errors than the shear and pressure waves. P e Table 2: Points per shortest wavelength and corresponding relative max error. 5 Conclusions We have studied numerical difficulties in the simulation of surface waves in almost incompressible elastic materials. The work as been greatly influenced by the theory of H-O. Kreiss and N.A. Petersson in [9]. Here they showed that the number of grid points per wavelength of the surface wave needed for accuracy is proportional to (μ/λ) 1/p,wherep is the order of accuracy of the method. This is opposing the classical theory which suggest a proportionality to μ. This requirement becomes more restrictive as the elastic material becomes more incompressible, μ/λ 0. In this work we have used a SBP + SAT discretization of the elastic wave equation in a half - plane to study surface waves in materials in which μ/λ is as small as The main goal was to investigate how small the quotient μ/λ must be to compensate for the higher complexity of higher order methods. In particular we have used methods of orders higher than 4. The results of this study was then used in an application where we numerically solved a version of Lambs problem in an almost incompressible material. A synopsis of the existence of a Rayleigh surface wave and its sensitivity to boundary truncation was given in an appendix. This presentation is analogous to the one given in [9] with the difference that the results where developed from the formulation (6) - (8) rather than the formulation (1) - (2). In the case of elastic wave propagation in two half - planes in welded contact the existence of Stoneley interface waves with much similarity to the Rayleigh surface wave can be proved [4]. In a ongoing study the authors aim to investigate numerical difficulties in the simulation of the Stoneley interface wave. It is then believed that an approach similar to the one given in the appendix may be fruitful. A The Rayleigh surface wave and sensitivity to boundary truncation errors A.1 The Rayleigh surface wave Consider the half - plane problem (6) - (8) with a traction free boundary condition, g 1 = g 2 =0.Weexaminetheexistenceofsolutionsofthetype φ = f(y)e st+iξx, H = h(y)e st+iξx, ξ R \{0}. (13) f <, h <, 9

82 Inserting (13) into (6) we get, ( f α 2 f =0,α 2 = ξ 2 + h β 2 h =0,β 2 = ( ξ 2 + s2 μ s2 λ+2μ ). ), (14) The solution to (14) is, f = A 1 e αy + A 2 e αy, h = B 1 e βy + B 2 e βy. (15) φ and H of (13) then becomes, φ = A 1 e iξx+αy+st + A 2 e iξx αy+st, H = B 1 e iξx+βy+st + B 2 e iξx βy+st. (16) Letting A 1 = B 1 =0and inserting the expressions (16) for φ and H into the boundary conditions (8) we get, ( β 2 + ξ 2) A 2 +2iβξB 2 =0 2iαξA 2 +(β 2 + ξ 2 (17) )B 2 =0 The linear system (17) has a solution if and only if its determinant is zero, D = ( β 2 + ξ 2) 2 4αβξ 2 =0. (18) Using the expressions (14) for α and β we can write (18) in the form ( ( ) ) 2 D = 4ξ 4 s 2 s (λ+2μ)ξ +1 2 s 2 μξ ξ 2 μ +1 = 4ξ 4 ϕ( s) =0, (19) where ϕ( s) = ) 2 1+ s 2 1+ (1+ μ s2 λ +2μ s2, s = s 2 ξ μ. (20) Since we exclude ξ =0, the zeros of the determinant (19) are the solutions of ϕ( s) =0. The function ϕ( s) was investigated in [9] its properties can be summarized in the following lemma, Lemma 1 The function ϕ( s) has exactly three roots s =0and s = s 0 = ±iω 0. ω 0 depends on μ/λ and 0 <ω 0 < 1. Furthermore, ϕ( s 0 ) is bounded away from zero for all μ/λ. Values of s 0 and ϕ( s 0 ) for some values of μ/λ are calculated in Table 1. Using the fact 0 <ω 0 < 1 in the expression (20) for s we get ( ) ( ) α 2 = ξ 2 ξ2 μω 2 0 λ+2μ >ξ 2 1 μ λ+2μ > 0, ( ) ( ) (21) β 2 = ξ 2 ξ2 μω0 2 μ >ξ 2 1 μ μ =0. Hence, φ = A 2 e αy e μt), iξ(x±ω0 H = B 2 e βy e μt), α, β > 0,ξ R \{0},ω R (22) iξ(x±ω0 10

83 μ/λ s 0 ϕ( s 0 ) i i i i Table 3: The roots s 0 of ϕ( s) and magnitude of ϕ( s 0 ) for some values of μ/λ. represents Rayleigh surface waves with amplitude that decays exponentially in the y - direction. The waves travel harmonically along the x - axis with phase velocity c R = ω 0 μ. (23) Note that A 1 and B 1 necessarily vanishes, otherwise the solutions would have an unbounded amplitude for increasing y. In case of the root s =0(17) gives A 2 /B 2 =1/i so that φ = A 2 e iξ(x y), H = ia 2 e iξ(x y). The relation (7) then gives u = v =0.Thatis,theroot s =0corresponds to a displacement field that vanishes everywhere. A.2 Sensitivity to boundary truncation errors The truncation errors arising from a discretization of the traction free boundary condition can be thought of as a perturbation of the homogeneous boundary condition by introducing non-zero boundary forcing functions g 1,g 2 in (2) and (7). Typically g 1 and g 2 depends on derivatives of the continuous solution and the grid size. We again consider a solution of the form (16) with A 1 = B 1 =0, φ = A 2 e iξx αy+st, H = B 2 e iξx βy+st (24). Inserting this solution into the now inhomogeneous boundary conditions (8) we get after some algebra, (1+ s2 2 i ) A 2 + i 1+ s 2 B 2 = (λ +2μ) g 1 1+ μ s2 λ +2μ A 2 2μξ 2 (25) ) (1+ s2 B 2 = g 2 2 2ξ 2, (26) where s = s ξ μ. The determinant of this system is ϕ( s) defined by (20). Hence, (25) - (26) becomes singular exactly at the roots of ϕ( s). Eliminating B 2 from (26) and inserting it into (25) gives, ϕ( s)a 2 = λ +2μ ( ) g1 (1+ s2 2ξ 2 i ) 1+ sg 2. (27) μ 2 Let the grid size be h. Discretizing (2) with a second order accurate method the principal part of the truncation errors becomes g 1 = τ 11 h 2 u xxx + τ 12 h 2 v yyy, g 2 = τ 21 h 2 v xxx + τ 22 h 2 y =0. (28) u yyy, 11

84 By (7), g 1 = τ 11 h 2 (φ xxxx + H yxxx )+τ 12 h 2 (φ yyyy H xyyy ), g 2 = τ 21 h 2 (φ yxxx H xxxx )+τ 22 h 2 y =0. (29) (φ xyyy + H yyyy ), Using (24) the boundary forcing functions becomes, g 1 = τ 11 h 2 ( ξ 4 A 2 iξ 3 βb 2 ) + τ12 h 2 ( α 4 A 2 iξβ 3 B 2 ), g 2 = τ 21 h 2 ( iξ 3 αa 2 ξ 4 B 2 ) + τ22 h 2 ( iξα 3 + β 4 B 2 ). (30) Since the right-hand side of (25) is proportional to g 1 /μ while the right - hand side of (26) is independent of μ, forμ λ the main effect comes form g 1. To simplify we therefor assume that g 2 =0. Using (25) with g 2 =0we eliminate B 2 from g 1, ( )) ( )) (1+ g 1 = τ 11 h 2 ξ 4 iξ 3 β i μ s2 λ+2μ (1+ A 2 + τ 12 h 2 α 4 iξβ 3 i μ s2 λ+2μ 1+ s2 2 g 2 =0. (31) The solution formula (27) can then be written in the form of an eigenvalue problem ϕ( s)a 2 = θ( s)a 2, (32) where θ( s) = (λ +2μ) h2 ξ 2 (( β τ 11 2μ ξ + τ β 3 ) 12 )(1+ μ s2 α ξ 3 + (τ 4 )) 11 + τ 12 )(1+ s2 λ +2μ ξ 4. 2 (33) This eigenvalue problem arises as a consequence of introducing truncation errors in a discretization of the traction free boundary condition. The eigenvalues s of this problem determines the phase velocities of surface waves in a numerical solution of (1) - (2). The phase velocity of the Rayleigh surface wave was determined by the roots s 0 of the function ϕ( s). We now investigate how sensitive the difference s s 0 is to truncation errors. We have for μ/λ 1, s ,ϕ( s 0 ) ±2.12i, α ξ = 1+ μ s2 λ +2μ 1, β ξ = 1+ s (34) Therefor θ( s 0 ) λh2 ξ 2 2μ (0.3τ τ (τ 11 + τ 12 )). (35) Taylor expanding (32) about s 0 gives We get, s s 0 θ( s 0) ϕ ( s 0 ) i λh2 ξ 2 2μ 1+ s2 2 A 2 ( s s 0 ) ϕ ( s 0 ) θ( s 0 ). (36) (0.3τ τ (τ 11 + τ 12 )). (37)

85 To achieve a relative error in the phase velocity of size ɛ, with0 <ɛ 1 we must choose the grid size h such that λh 2 ξ 2 τ μ = ɛ, τ = i (0.3τ τ (τ 11 + τ 12 )). (38) 4.24 If the computational grid has P R points per surface wavelength L R =2π/ ξ we get h = L R,h ξ = 2π τ λ,p R =2π P R P R ɛ μ. (39) That is, as μ/λ 0 the number of points per surface wavelength must be proportional to λ/μ to maintain an relative error in the phase velocity of ɛ. For a p-th order method the leading order truncation errors terms are g 1 = τ 11 hp p+1 u x p+1 + τ 12 hp p+1 v y p+1 (40) and (38) is replaced by λh p ξ 2 τ = ɛ. (41) μ The number of grid points required to maintain an error in the phase velocity of ɛ now becomes P R =2π ( τ ɛ λ μ ) 1/p. (42) Hence, as μ/λ 0 the number of grid points per surface wave length grows much slower for larger p. References [1] M.Carpenter, D.Gottlieb and S.Abarbanel, Time - Stable Boundary Conditions for Finite - Difference Schemes Solving Hyperbolic Systems: Methodology and Application to High - Order Compact Schemes, J.CompPhys 111, (1994) [2] K.Duru, G. Kreiss, A Well - Posed and Discretely Stable Perfectly Matched Layer for Elastic Wave Equations in Second Order Formulation, Commun. Comput. Phys Vol 11, pp Editions, London, [3] K.Duru, G.Kreiss, K.Mattsson, Accurate and Stable Boundary Treatments for Elastic Wave Equations in Second Order Formulation, submitted manuscript. [4] W.M.Ewing, W.S.Jardetzky, F. Press, Elastic Waves in Layered Media, McGraw - Hill [5] J. C.Gilbert, P. Joly, Higher order time stepping for second order hy- perbolic problems and optimal CFL conditions, Num. Analys and Sci Comp for PDEs and their Challenging Applicat, vol. 16, pp. 67âĂŞ93, Springer,

86 [6] A. Nissen, G. Kreiss, M. Gerritsen, High Order Stable Finite Difference Methods for the Schrödinger Equation, J.Sci.Comput.55, (2013) [7] K.F.Graff, Wave Motion In Elastic Solids, Dover Publications. [8] H-O.Kreiss and J.Oliger, Comparison of Accurate Methods for the Integration of Hyperbolic Equations, Tellus, 24, [9] H - O.Kreiss and N.A.Petersson, Boundary Estimates for the Elastic Wave Equation in Almost Incompressible Materials, SIAM J. Numer Anal, V 50, No. 3. [10] H.Lamb, On the Propagation of Tremors Over the Surface of an Elastic Solid, Phil. Trans. R. Soc, A203, [11] K.Mattsson, J.Nordstr ö m, Summation by parts operators for finite difference approximations of second derivatives, J. Comp Phys 199 (2004). [12] B Strand, Summation by parts for finite difference approximations for d/dx, J. Comp Phys 110 (1994). [13] I.A.Viktorov, Rayleigh and Lamb Waves: Physical Theory and Applications, Plenum Press, New York

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89 High Order Finite Difference Schemes for the Elastic Wave Equation in Discontinuous Media Kristoffer Virta 1,, Kenneth Duru 2 1 Division of Scientific Computing, Department of Information Technology, Uppsala. 2 Department of Geophysics Stanford University, Stanford, CA. kristoffer.virta@it.uu.se Abstract Finite difference schemes for the simulation of elastic waves in materials with jump discontinuities are presented. The key feature is the highly accurate treatment of interfaces where media discontinuities arise. The schemes are constructed using finite difference operators satisfying a summation - by - parts property together with a penalty technique to impose interface conditions at the material discontinuity. Two types of operators are used, termed fully compatible or compatible. Stability is proved for the first case by bounding the numerical solution by initial data in a suitably constructed semi - norm. Numerical experiments indicate that the schemes using compatible operators are also stable. However, the numerical studies suggests that fully compatible operators give identical or better convergence and accuracy properties. The numerical experiments are also constructed to illustrate the usefulness of the proposed method to simulations involving typical interface phenomena in elastic materials. 1 Introduction The elastic wave equation governs the propagation of seismic waves resulting from earthquakes and other seismic events. Other applications include waves in plates, beams and solid material structures. In the general setting the media can be described by piecewise smooth functions with jump discontinuities. This is especially true in seismological problems due to the layered structure of the earth. Large contrasts in media parameters may also be found in solid mechanics devices composed of different materials in welded contact. The presence of material discontinuities gives rise to reflection and refraction phenomena. For example earlier than expected arrivals of seismic waves, so called refraction arrivals, have been shown to exist in the presence of a material discontinuity [20, 6]. Similar to the Rayleigh surface wave [21] an analogous Stoneley interface wave may under certain circumstances travel along media interfaces [23]. Also, as in the case of the traction free boundary a distinguishing characteristic of 1

90 wave - interface interaction is that mode conversion occurs. That is, an incident wave type, either pressure or shear, is converted into two wave types, pressure and shear on reflection and refraction [8], pp In this paper, we consider the following problem: two elastic 2 dimensional half - planes are in contact at a line interface. We seek an approximation of the displacement field due to an initial disturbance as a function of time. We are particularly interested in the numerical simulation of the phenomena which arise from the influence of the line interface. To arrive at a solution to the problem we construct finite difference approximations of the elastic wave equation. The finite difference method has proven to be an efficient and easy to implement technique to approximate the elastic wave equation. However, the presence of jump discontinuities in media parameters makes the design of stable and accurate numerical methods challenging. It has been shown that smoothing or ignoring the material discontinuities in a numerical approximation reduces the formal order of accuracy to one [1]. That is, in order to preserve high order accuracy interfaces of jump discontinuities must be given special treatment. When using finite differences for simulation of wave propagation problems it has been known for a long time that high order methods (higher than 2) are superior to lower order methods [12]. It is also well known that stability together with accuracy guaranties convergence of the numerical scheme [9], pp 170. Therefor a minimal requirement of the numerical scheme is high order accuracy and stability. To arrive at the approximate solution we use summation - by - parts (SBP) finite difference operators [24, 17] to approximate spatial derivatives. To maintain high accuracy in the presence of material discontinuities the required conditions at media interfaces are enforced with the simultaneous - approximation - term (SAT) methodology [3]. The elastic wave equation contains both second derivative terms such as / x(b u x) and mixed derivative terms like / x(b u y), where b>0. To discretize these terms approximations of both first and second derivatives are required. Then certain compatibility conditions of the two approximations needs to be fullfilled. We use SBP approximations of first and second derivatives that are either compatible or fully compatible, the definitions will become clear in Section 3. This results in the case of fully compatible operators in a scheme which is proved to be stable in the sense that the corresponding approximative solution is bounded in a semi - norm by initial data. Stability in the case of compatible SBP operators is investigated by numerical experiments. The experiments indicate stability also in this case. Discretizing the elastic wave equation has been done in several previous works. Benchmark problems which includes liquid - solid interfaces was considered in [22]. There material interfaces were not specified by any numerical boundary conditions. In [25] the equations were rewritten to a first order system (1st order velocity - stress formulation) prior to discretizing. However, stability could only be shown for homogeneous materials. A recent and promising strategy includes the use of ghost points to enforce the correct conditions at the interface is described in [19]. In that paper focus was put on the stable 2

91 treatment of hanging nodes in a grid refinement and correct discretization of singular sources. Typical features of a discontinuous media was only considered in one experiment where a finite layer was put on top of a infinite half - space. In that experiment the results showed good agreement with a semi - analytical solution. The elastic wave equation with the traction free boundary condition was discretized using the SBP - SAT approach in [5]. The present study should be seen as a direct continuation of the work in [5]. The general problem of seismic wave propagation includes non - planar boundaries. In the finite difference framework a coordinate transformation to curvilinear coordinates is therefor made before discretizing the equations. The general media might also be anisotropic. The theory in this paper is therefor presented in the general case of anisotropic media and in curvilinear coordinates. In order to focus on interface phenomena the numerical experiments are all carried out in isotropic media with piecewise constant media parameters and includes examples of mode conversion, refraction arrivals and the Stoneley interface wave. However, with the theory developed the generalization to curved geometries and anisotropic media is straightforward. Using an unstructured mesh greatly simplifies the handling of complicated geometries. In particular if hanging nodes are allowed. Methods with this approach includes discontinuous Galerkin [11] and finite element [4] discretizations of the elastic wave equation. Generating a high quality unstructured mesh can however be a non - trivial problem in itself and also requires extra bookkeeping and additional memory to keep track of the connectivity of the grid. It would therefor be interesting to study the performance of the method proposed in this paper on problems that includes curved geometries. The rest of the paper is outlined as follows. In Section 2 the equations and interface conditions are introduced in the general setting. Section 3 introduces the necessary definitions used to describe the discretization. Stability of the numerical scheme using fully compatible SBP operators is also proved here. In Section 4 numerical experiments are presented. The purpose of the experiments are twofold. One, to verify stability and accuracy, two, to illustrate the usefulness of the proposed method on simulation of phenomena typical to wave motion in discontinuous elastic media. In particular, numerical examples of mode conversion, refraction arrivals and the Stoneley interface wave will be given. Section 5 concludes and mentions future work. 2 Two elastic half - planes in contact Consider waves propagating in two elastic half - planes. The half - planes are in contact at y = 0. We denote the displacement field U =(u 1,u 2 ) T in the upper half-plane Ω = (, ) [0, ) andu =(u 1,u 2 )T in the lower half-plane Ω =(, ) (, 0]. Let the positive quantities ρ, c 11,c 12,c 22,c 33 and ρ, c 11,c 12,c 22,c 33 be the density and elastic coefficients in Ω and Ω, respectively. The elastic coefficients may also include metric terms coming from a 3

92 transformation to curvilinear coordinates. The elastic coefficients satisfy c 11 c 22 c 2 12 > 0,c 11 c 22 c 2 12 > 0. The wave motion is governed by ρu tt =(AU x ) x +(BU y ) y +(CU y ) x + ( C T U x, (x, y) Ω, )y ρ U tt =(A U x ) x + ( ) B U y + ( ) C U x y + ( C T U x x )y, (x, y) t>0, (1) Ω, where [ ] [ ] [ ] c11 0 c c12 A =,B =,C =. 0 c 33 0 c 22 c 33 0 The matrices A,B and C are defined analogously by the corresponding primed entries. In the following, we let unprimed quantities be defined in Ω and primed primed quantities be defined in Ω. When expressions are identical except for primed quantities we only write down the unprimed version. The solution U is subject to initial data U(x, y, 0) = U 0,U t (x, y, 0) = U 1, (x, y) Ω. (2) Here U 0 L 2 (Ω),U 1 H0 1 (Ω). From the solution of (1) we can compute normal and tangential stresses (τ 22,τ 12 ) T (τ 22,τ 12 ) T = BU y + C T U x. At the interface y = 0 continuity of normal stresses, tangential stresses and displacements are required, τ 22 τ 22 =0, τ 12 τ 12 =0, <x<,y =0. (3) U U =0, Define the potential energy matrix [ ] A C P = C T. B Note that P is symmetric positive semi - definite, see [5]. We define the elastic energies by E (t) := 0 ρut T U t dxdy + [ ] T [ ] Ux Ux 0 P dxdy, U y U y E (t) := 0 ρ U t T U t dxdy + [ ] 0 U T [ ] x U U y P x U y dxdy. The total elastic energy is given by the sum of the elastic energies E T (t) =E (t)+e (t). It is straightforward to show that the elastic wave equation (1) with the interface conditions (3) satisfy E T (t) =E T (0),t 0. Hence, the total elastic energy is conserved. Thus, (1) together with (2) - (3) is a well posed problem. 4

93 3 Spatial discretization In one spatial dimension the half - line [0, ) is discretized by introducing an equidistant grid with grid size h x j = jh, j =0, 1,... Let u be a function defined on the half - line, the value u(x j ) is denoted u j.a one dimensional grid function corresponding to u is then u =(u 0,u 1,...). Let u and v be two grid functions. The scalar product of u and v is defined as usual by < u, v >= u i v i. If L is an operator acting on grid - functions its adjoint is denoted L. The first derivative u x is approximated by D 1 u. Here D 1 is a 2p-th order diagonal first derivative SBP operator, i=0 (D 1 u) j =(u x ) j + O(h p ),j =0...m p, (D 1 u) j =(u x ) j + O(h 2p ),j = m p +1..., D 1 = H 1 Q, < u, (Q + Q ) u >= nu 2 0, < u,hu >= j=0 u2 j h jh, h j > 0, (4) where the integer m p depends on p and n is the inward facing unit normal of the half - line. The SBP operator is termed diagonal due to the effect of multiplying each element of a grid function by a scalar when the operator H is applied c.f., multiplying a diagonal matrix with a vector. The variable coefficient second derivative (b(x)u x ) x,b(x) > 0, is approximated by D (b) 2 u,whered(b) 2 is a 2p-th order diagonal variable coefficient second derivative SBP operator compatible with D 1, (D (b) 2 u) j =((bu x ) x ) j + O(h p ),j =0...n p, (D (b) 2 u) j =((bu x ) x ) j + O(h 2p ),j = n p +1..., D (b) 2 = H 1 ( M (b) + B (b) S), M ( (b) = D1 HB(b) D 1 + R (b) 0, B (b) u ) j = b ju j, (5) R (b) = R (b) 0, (Su) 0 =(v x ) 0 + O(h p+1 ), ( { nb0 u u) B(b) 0,j =0, 0,j 0, j = where the integer n p depends on p. The operator H is the same in both definitions of D 1 and D (b) 2. We also define a diagonal second derivative SBP operator 5

94 fully compatible with D 1 by requiring S = D 1 in (5). Then one order of accuracy is lost at the point exactly on the boundary i.e., (D (b) 2 u) 0 =((bu x ) x ) 0 + O(h p 1 ), (Su) 0 =(v x ) 0 + O(h p ). For details on SBP operators see e.g., [24, 17, 15]. The grid, grid functions and operators corresponding to the half - line (, 0] are defined analogously. We will also consider corresponding grid functions and operators on the line (, ) discretized by x j = jh,j =0, ±1,... Now boundaries are absent and the corresponding SBP operators are defined by the interior schemes of (4) - (5). To emphasize that it is the interior stencil that is used we use the notation L for operators acting on grid functions defined on the discretized line. Here L is any operator acting on grid functions defined on the discretized half - line. Figure 1: The function, grid function restricted to x = h j and y = y k. In two spatial dimensions the upper half - plane is discretized by a two - dimensional equidistant grid with grid size h, x j = jh,j =0, ±1,..., y k = kh, k =0, 1,... The value u(x j,y k ) of a function defined on the half-plane is denoted u j,k atwo - dimensional grid function corresponding to u is then u whose entry u j,k is u j,k. If u and v are two dimensional grid functions defined on the discretized half - plane the scalar product of u and v is computed as u, v = j= k=0 6 u j,k v j,k.

95 Fixing either the index j or k the one dimensional restrictions of u or u to the points (x j,y k ),k =0, 1,..., (x j,y k ),j =0, ±1,... are denoted u j,:, u :,k and u j,:, u :,k, respectively, see Figure 1. Let L be an operator acting on one dimensional grid functions. We extend L to two dimensional operators acting in the x direction by repeated use of L on u :,k and the y direction by repeated use of L in on u j,:. To distinguish whether an operator acting on two - dimensional grid functions is acting in the x or y direction a subscript x or y is used. That is, (L x u) j,k := ( Lu:,k )j, (L yu) j,k := (Lu j,: ) k. The grid, grid functions and operators defined on a discretization of the lower half - plane are defined analogously. We use the following operators to extract the boundary values of grid functions defined on either the upper or lower half plane and map it to a grid function on either the upper or lower half plane such that the only non - zeros values are located at the boundary, (E ±2± u) 0,k = u 0,k, (E ±2± u) j,k =0,j 0. For example, the operator E +2 takes a grid function u defined on the upper discrete half - plane and maps it to a grid function defined on the lower half - plane which is zero everywhere except at the boundary, where it takes the corresponding values of u. Notethat E +2+ = E +2+,E 2 = E 2,E +2 = E 2+. (6) Multiplication of grid functions is defined by element - wise multiplication. Multiplication from the left of a grid function and an operator is defined as (vlu) i,j = v i,j (Lu) i,j. 3.1 Spatial discretization of the equations We now discretize (1) together with (2) and (3) in the spatial coordinates. The result is a semi-discrete system in which time is the continuous variable. The discrete solutions are denoted U =(u 1, u 2 ) T and U =(u 1, u 2 )T. The right hand side of (1) contains spatial derivatives of four types. These are discretized with the two - dimensional extensions of the SBP operators defined in (4) - (5) 7

96 as [ ] D (c11) (AU x ) x 2x 0 0 D (c33) U =: P xx U, 2x [ ] 0 D (CU y ) x 1x c 12 D 1y U =: P D 1x c 33 D 1y 0 yx U, [ ] D (c33) 2y 0 (BU y ) y 0 D (c22) U =: P yy U, 2y [ ] ( C T ) U x y 0 D 1y c 33 D 1x U =: P D 1y c 12 D 1x 0 xy U. Define [ c33 S T = 1y c 12 D 1x ] c 33 D 1x. c 22 S 1y (7) Normal and tangential stresses are then approximated by [ ] [ ] τ22 t22 TU =. τ 12 t 12 (8) Introduce Q = P xx + P yy + P yx + P xy. (9) The right hand side of (1) can then be discretized as (AU x ) x +(BU y ) y +(CU y ) x + ( C T U x )y QU, (A U x) x + ( ) B U y ) C U y y ) C T U x x y Q U. (10) With this notation the semi-discrete system may be written ρu tt = QU + SAT, ρ U tt = Q U + SAT. (11) The semi-discrete solution is subject to initial data U(0) = U 0, U t (0) = U 1, U (0) = U 0, U t(0) = U 1. In (11) the interface conditions (3) are imposed weakly via the penalty terms SAT and SAT. The penalty terms are proportional to the difference between discrete values of the displacements and stresses at the interfaces. Introduce SAT = SAT D + SAT S, SAT = SAT D + SAT S (12) and define I D1 = E +2+ u 1 E 2+ u 1, I D 1 = E 2 u 1 E +2 u 1, I D2 = E +2+ u 2 E 2+ u 2, I D 2 = E 2 u 2 E +2 u 2, I S22 = E +2+ t 22 E 2+ t 22, I S 22 = E 2 t 22 E +2 t 22, I S12 = E +2+ t 12 E 2+ t 12, I S 21 = E 2 t 12 E +2 t 12. 8

97 Thepenaltyterms(12)canthenbewrittenas [ ( H 1 y σ11 I SAT D = D1 + σ 2 (c 33 S 1y ) I D1 + σ 2 Hx 1 (c 12D 1x ) )] ( H x I D2 Hy 1 σ12 I D2 + σ 2 Hx 1 (c 33 D 1x ) H x I D1 + σ 2 (c 22 S 1y ) ) I D2 [ ( H SAT D 1 y σ11 I = D1 σ 2 (c 33 S 1y) I D 1 σ 2 Hx 1 (c 12 D 1x) )] ( H x I D 2 Hy 1 σ12 I D2 σ 2 Hx 1 (c 33D 1x ) H x I D 1 σ 2 (c 22S 1y ) ) I D [ 2 σ3 Hy SAT S = 1I ] S 22 σ 3 Hy 1, I S12 [ ] SAT S = σ3 Hy 1 I S 22 σ 3 H 1. y I S 12 (13) Here the terms SAT D and SAT S enforces the continuity of displacements and stresses, respectively. The penalty parameters σ 11,σ 12,σ 2 and σ 3 are determined below to yield a stable discretization of (1) together with (2) and (3). 3.2 Stability of the discretization Introduce the operators H = [ ] Hx H y 0, Ẽ 0 H x H ±2± = y [ ] Hx E ±2± 0. 0 H x E ±2± Using the properties (4) - (5) we can write (9) as Q = H 1 Q H 1Ẽ +2+T, Q = H 1 Q H ) ( Ẽ 2 1 where Q = P xx + P yy + P yx + P xy, [ D P xx = 1x H x H y c 11 D 1x 0 0 D 1x H xh y c 33 D 1x ] + [ D P yy = 1y H x H y c 33 D 1y 0 [ 0 D1x P yx = H ] xh y c 12 D 1y D1xH, x H y c 33 D 1y 0 [ ] 0 D P xy = 1yH x H y c 33 D 1x D1y H. xh y c 12 D 1x 0 Note that Define B 1 = B 3 = 0 D 1y H xh y c 22 D 1y ] + T [ H y R x (c11) 0 0 H y R (c33) [ H x R (c33) y 0 0 H x R (c22) y x ], ], P xx = P xx, P yy = P yy, P yx = P xy. (14) ] [, B 2 = 0 Ẽ 2 ], B 4 = [Ẽ+2+ 0 [Ẽ Ẽ 2 0 Ẽ 2+ Ẽ+2 0 [ 0 Ẽ 2+ Ẽ +2 0 ], ]. 9

98 the terms (13) can then be written as [ ] [ ] 1 ( SATD H 0 SAT D = 0 H [ ] 1 ( [ ] H 0 T 0 σ 2 0 H 0 T (B 3 + B 4 ) [ ] [ ] 1 SATS H 0 = ( σ 3 )(B 3 + B 4 ) 0 H With SAT S [ ] Q 0 M = 0 Q [ ] [ ]) σ11 0 U (B 0 σ 1 + B 2 ) 12 U [ U U ]), [ T 0 0 T ][ U U ]. [ ] [ ] T 0 σ B 3 0 T (B 0 σ 1 + B 2 ) 12 σ 2 [ T 0 0 T ] (B 3 + B 4 ) σ 3 (B 3 + B 4 ) the system (11) can be written [ ρu ρ U ] tt = [ T 0 0 T ] (15) [ ] 1 [ ] H 0 U M 0 H U. (16) We now prove that the penalty parameters σ 2 and σ 3 can be chosen such that the operator M is self - adjoint. Lemma 1 (Self adjointness of the spatial operator) If σ 2 = 1 2, σ 3 = 1 2, then M given by (16) is self - adjoint. Proof: By (14) and (6) Q = Q, B 1 = B 1, B 2 = B 2, B 3 = B 3, B 4 = B 4. Write M = M 1 + M 2 + M 3 + M 4 where [ ] [ ] Q 0 Q 0 M 1 = = 0 Q 0 Q = M 1, [ ] [ ] σ11 0 M 2 = (B 0 σ 1 + B 2 )= (B 1 + B 2 ) σ11 0 = M 12 0 σ 2, 12 [ ] [ ] T 0 T 0 M 3 =(1 σ 3 ) B 3 0 T σ 2 0 T B 3 ( [ ] T 0 =1/2 B 3 0 T + [ T 0 0 T ] B 3 ) = M 3, [ ] [ ] T 0 T 0 M 4 = σ 2 0 T ( B 4 ) σ 3 B 4 0 T ([ ] [ ]) T 0 T 0 = 1/2 0 T B 4 + B 4 0 T = M 4. Hence M = M As a consequence of Lemma 1 we have the following corollary. 10

99 Corollary 1 (Conservation of energy) All real-valued solutions (U, U ) to (16) satisfy E(t) =C, (17) where [ ] [ ][ ] [ ] [ ] U ρ H 0 U U U E(t) = U, t 0 ρ H U + U, M U (18) t and C is a constant depending only on the initial data. Proof: Lemma 2 gives 1 d 2 dt ( [ ] U U ( = 1 2 = 1 [ ] d U 2 dt U [ ][ ] ρ H 0 U, t 0 ρ H U [ ] U U, M t t ) [ U U ] + M [ U U ], [ ] U, M U. [ U U ] By integrating this expression in time we get (17) For the quantity (18) to be an energy we need M 0. We will use the following calculations. [ ] [ U U U, M 1 ] U = U, QU + U, Q U. (19) t ) Here [ ] [ ][ ] U, QU D1x u = 1, D 1y u H c11 c 12 D1x u 1 2 c 12 c 22 D 1y u 2 [ ] [ ][ ] D1y u + 1, D 1x u H c33 c 33 D1y u 1 2 c 33 c 33 D 1x u 2 + u 1,H y R x (c11) u 1 + u 2,H y R x (c33) u 2 + u 2,H x R y (c22) u 2 + u 1,H x R y (c33) u 1 0. (20) The inequality holds since c 11 c 22 c 2 12 > 0, c 33 > 0and R x (c11),r x (c33) R x (c22) R x (c33) 0. The first term of (20) can be computed as [ ] [ ][ ] D1x u 1, D 1y u H c11 c 12 D1x u 1 2 c 12 c 22 D 1y u 2 [ ] T [ ][ ] (D1x u 1 ) = i,j c11i,j c 12i,j (D1x u 1 ) i,j h (D 1y u 2 ) i,j c 12i,j c 22i,j (D 1y u 2 ) i h j h 2 0. i,j i= j=0 Define ) 1/2 ) c i = 1 2 ( (c11i,0 + c 22i,0 ) ( (c11i,0 c 22i,0 ) 2 +4c 2 12i,0 11

100 this is the smallest eigenvalue of the matrix [ ] c11i,0 c 12i,0. c 12i,0 c 22i,0 Since c 11 c 22 c 2 12 > 0, c i > 0and [ ] c11i,0 c i c 12i,0 0. c 12i,0 c 22i,0 c i We get [ ] [ ][ ] D1x u 1, D 1y u H c11 c 12 D1x u 1 2 c 12 c 22 D 1y u 2 [ ] T [ ][ ] (D1x u 1 ) = i,j c11i,j c 12i,j (D1x u 1 ) i,j h (D 1y u 2 ) i= j=1 i,j c 12i,j c 22i,j (D 1y u 2 ) i h j h 2 i,j [ ] T [ ][ ] (D1x u 1 ) + i,0 c11i,0 c i c 12i,0 (D1x u 1 ) i,0 h (D 1y u 2 ) i= i,0 c 12i,0 c 22i,0 c i (D 1y u 2 ) i h 0 h 2 i,0 + c i (D 1x u 1 ) 2 i,0 h ih 0 h 2 + c i (D 1y u 2 ) 2 i,0 h ih 0 h 2 0. i= Similarly the second term of (20) can be computed as [ ] [ ][ ] D1y u 1, D 1x u H c33 c 33 D1y u 1 2 c 33 c 33 D 1x u 2 [ ] T [ ][ ] (D1y u 1 ) = i,j c33i,j c 33i,j (D1y u 1 ) i,j h (D 1x u 2 ) i= j=1 i,j c 33i,j c 33i,j (D 1x u 2 ) i h j h 2 i,j ) 2 + c 33i,0 ((D 1y u 1 ) i,0 +(D 1x u 2 ) i,0 hi h 0 h 2 0. i= (21) (22) 12

101 The second term of (19) is computed analogously. We also compute [ ] [ U U U, M 2 ] U = + [ ] [ U U U, M 3 ] U = + [ ] [ U U U, M 4 ] U = + i= i= i= i= i= i= [ u1i,0 u ] [ ][ ] σ 11i,0 σ 11i,0 u1i,0 1 i,0 σ 11i,0 σ 11i,0 u h i h 0 h 1 i,0 [ u2i,0 u ] [ ][ ] σ 12i,0 σ 12i,0 u2i,0 2 i,0 σ 12i,0 σ 12i,0 u h i h 0 h, 2 i,0 [ u1i,0 [ ][ ] u 1 0 t22i,0 1 i,0] 0 1 t h i h 0 h 22 i,0 [ u2i,0 [ ][ ] u 1 0 t12i,0 2 i,0] 0 1 t h i h 0 h 12 i,0 [ u1i,0 [ ][ ] u 0 1 t22i,0 1 i,0] 1 0 t h i h 0 h 22 i,0 [ u2i,0 [ ][ ] u 0 1 t12i,0 2 i,0] 1 0 t h i h 0 h. 12 i,0 (23) So far we have made no assumptions on the SBP operators used. We now prove that M 0 in the case of fully compatible SBP operators, that is with S = D 1 in the definition (5) of the diagonal second derivative SBP operators. Lemma 2 (Ellipticity, case of fully compatible SBP operators) Using fully compatible SBP operators, for all real - valued grid functions U and U if [ ] [ ] U U U, M U 0, σ 22i,0 < c 33 i,0 4h 0 h c 33 i,0 4h 0 h, σ 12i,0 < c2 12 i,0 4 c i h 0 h c2 22 i,0 4 c i h 0 h c 2 12i,0 4 c i h 0 h c 2 22i,0 4 c i h 0h. Proof: We have [ ] [ ] [ ] [ [ ] [ U U U U U U U, M U = U, M 1 ] U + + U, M 4 ]. U 13

102 Here M = M 1 + M 2 + M 3 + M 4 as in the proof of Lemma 2. Adding the terms of (19), (23) and using (21) -(22) we get after some algebra. [ ] [ ] U U U, M U = i= j=1 i= j= i= j=1 [ (D1x u 1 ) i,j [ ] T [ (D1y u 1 ) i,j c33i,j c 33i,j 1 i= j= i= i= (D 1x u 2 ) i,j ] T [ ][ ] c11i,j c 12i,j (D1x u 1 ) i,j h (D 1y u 2 ) i,j c 33i,j c 33i,j (D 1y u 2 ) i h j h 2 i,j c 33i,j [ (D1x u 1 ) ] T [ i,j c 11i,j c 12 i,j (D 1y u 2 ) i,j c 33 i,j 1 [ (D1y u 1 ) ] T [ i,j c 33i,j c 33 i,j c 33i,j ][ (D1y u 1 ) i,j (D 1x u 2 ) i,j c 33 i,j ][ (D1x u 1 ) i,j (D 1y u 2 ) i,j ] h i h j h 2 ] h i h j h 2 ] h i h j h 2 ][ (D1y u (D 1x u 2 ) i,j c 33 i,j c 1 ) i,j 33 i,j (D 1x u 2 ) i,j [ ] T [ ][ ] (D1x u 1 ) i,0 c11i,0 c i c 12i,0 (D1x u 1 ) i,0 h (D 1y u 2 ) i,0 c 12i,0 c 22i,0 c i (D 1y u 2 ) i h 0 h 2 i,0 u 1i,0 u 1 i,0 t 22i,0 t 22 i,0 T σ 11i,0 σ 11i,0 1/2 1/2 σ 11i,0 σ 11i,0 1/2 1/2 h 1/2 1/2 0h c 33i,0 0 1/2 1/2 0 h 0h c 33 i,0 u 1i,0 u 1 i,0 t 22i,0 t 22 i,0 h ih (24) T u 2i,0 u 2 i,0 (D + 1x u 1 ) i,0 i= (D 1y u 2 ) i,0 (D 1x u 1 ) i,0 (D 1y u 2 ) i,0 σ 12i,0 σ 12i,0 1/2 1/2 1/2 1/2 σ 12i,0 σ 12i,0 1/2 1/2 1/2 1/2 u c 1/2 1/2 1h 0h 2i, u c 2 12 i,0 2 i,0 c 1/2 1/2 0 1h 0h (D 0 0 1x u 1 ) i,0 c 2 22 i,0 c 1/2 1/ h0h (D 1y u 2 ) h i h (25) i,0 0 c 2 12 i,0 (D 1x u 1 ) i,0 (D 1y u 2 1/2 1/ ) i,0 c 1 h0h c 2 22 i,0 + u 1,R x (c11) u 1 + u 2,R x (c33) u 2 + u 2,R y (c22) + u 1,R x (c11) u 1 + u 2,R x (c33) u 2 + u 2,R y (c22) u 2 + u 1,R (c33) y u 1 u 2 + u 1,R (c33) y u 1. 14

103 The first five and the last eight terms of this sum are non - negative. The matrices of (24) and (25) also appeared in [27]. There it was shown that if σ 22i,0 < c 33 i,0 4h 0 h c 33 i,0 4h 0 h, σ 12i,0 < c 12 i,0 4 c i h 0 h c 22 i,0 4 c i h 0 h c 12 i,0 4 c i h 0h c 22 i,0 4 c i h 0h then the matrices are positive semi - definite. Hence with this choice of parameters M 0 Lemma 1, Corollary 1 and Lemma 2 proves the main result of this paper, Theorem 1 The system (16) obtained with fully compatible SBP operators is stable in the sense that its solution is bounded by initial data in the semi - norm (18) if the penalty parameters σ 11,σ 12,σ 2 and σ 3 are chosen as in Lemma 1 and 2. The proof that M 0 depends on S = D 1 in the approximation of the stresses by (7) and (8). For the case of compatible SBP operators S D 1 and the proof does not hold. Intuitively it would seem to be preferable to use compatible SBP operators, if such a discretization is stable. This is due to the fact that fully compatible SBP operators loses one order of accuracy in exactly one point at the boundary. In the next section we investigate both discretizations where it seems, although we can not prove it, that a scheme using fully compatible SBP operators is also stable. However, the numerical experiments also show, rather counterintuitive, that the schemes using fully compatible operators have identical or better accuracy and convergence properties. 4 Numerical experiments In this section we present experiments with the numerical method described above. In particular, we will use schemes that have been constructed using 4th and 6th order diagonal fully compatible and compatible SBP operators. We choose the material parameters to represent two anisotropic elastic medium. In particular, if λ and μ are the first and second Lamé parameters, respectively. Then c 11 = c 22 = λ +2μ, c 12 = λ, c 33 = μ. To integrate numerically in time we use the 4th order scheme described in [7] which was designed for systems on the form (16). We will consider three experiments that serve as verification of stability and accuracy of the proposed method, as well as its applicability to interface phenomena in layered elastic materials. 4.1 Mode conversion at a line interface In a first experiment we consider a compressional plane wave of unit amplitude propagating with angle θ and temporal frequency ω 2π in the negative y-direction., 15

104 The displacement field is given by ( ) U (in) ξ P = e η i(γξx γηy ωt), θ (0, π ω ),ξ = sin(θ),η =cos(θ),γ = 2 x (, ),y [0, )., λ +2μ We assume an interface between two different materials at y =0. TheLame parameters and densities are λ, μ, ρ and λ,μ,ρ in the half-planes y>0and y<0, respectively. When this wave encounters the interface between the two different materials it will be split into reflected, compressional and shear waves, U = U (in) + U (refl) P + U (refl) S, ( ) U (refl) ξ P = A refl e η i(γξx+γηy ωt), ( ) U (refl) γ2 η x (, ),y (, 0] S = B 2 refl e γξ i(γξx+γ2η2y ωt), η 2 =cos(θ 2 ), sin(θ 2 )= γ ξ,γ 2 = ω, γ 2 μ and refracted compressional and shear waves, U = U (refr) P + U (refr) S, ( ) U (refr) γ1 ξ P = A 1 refr e γ 3 η i(γ1ξ1x γ3η3y ωt), 3 η 3 =cos(θ 3 ), sin(θ 3 )= γ 1 ω ξ 1,γ 3 = γ 3 λ +2μ, x (, ),y (, 0], ( ) U (refr) γ4 η S = B 4 refr e γ 1 ξ i(γ1ξ1x γ4η4y ωt), 1 η 4 =cos(θ 4 ), sin(θ 4 )= γ 1 γ 4 ξ 1,γ 4 = ω μ, see Figure 2. The constants A refl,b refl,a refr,b refr are obtained by inserting U and U into the conditions (3) and solving the resulting linear system. For a more detailed discussion see [8], pp We choose θ = π 4, ω =2π, λ = μ =1,λ =1/2,μ =2andρ = ρ = 1. We use this exact solution to verify stability and accuracy of the numerical schemes described in this work. The computational domain is taken to be (x, y) [ 2π γξ, 2π 2π 2π γξ ] [ γ 2η 2, γ 3η 3 ]. Initial data for the numerical scheme is taken as the real part of the analytic solution at time t = 0 and exact data is imposed at the outer boundaries. The solution is computed for 100 periods until t = 100 and the discrete max - error is measured. We use 2N N grid points. Figures 3(a) - 3(b) display the max error in the numerical solution as a function of time for t 100 obtained with schemes using compatible SBP operators. Figures 3(c) - 3(d) display the corresponding measurements obtained with fully compatible SBP operators. These figures illustrate the stability and accuracy of the schemes. Numerical 16

105 values of the rate of convergence measured at time t = 100 are shown in Table 1. Stability was only proved for the schemes using fully compatible SBP operators. The computations done with compatible SBP operators indicates that these schemes are also stable. Note that the computations for this example suggest that the 6th order fully compatible SBP operators yield schemes that have slightly better accuracy and convergence properties than the corresponding 6th order compatible versions even though the formal order of accuracy is one order higher at exactly one point for the compatible SBP operators. Compatible SBP operators Fully compatible SBP operators N p 4 p 6 p 4 p Table 1: Rate of convergence of the solution in max norm. p 4 and p 6 gives the measured convergence using 4th and 6th order operators, respectively. Figure 2: Reflection and refraction of a plane pressure wave impinging on the interface between two elastic half-spaces. 4.2 Refraction arrivals Elastic wave motions can be significantly affected by a layered structure of the underlying media. We illustrate this by solving a version of Lambś problem [14] on a domain consisting of two materials in welded contact. The domain is composed of a plate of Plexiglas welded together with a plate of aluminum, 17

106 TH ORDER COMPATIBLE SBP OPERATORS 6TH ORDER COMPATIBLE SBP OPERATORS N = 21 N = 41 N = 81 N = 161 MAX ERROR MAX ERROR N = 21 N = 41 N = 81 N = 161 TIME (a) 4th order, compatible 4TH ORDER FULLY COMPATIBLE SBP OPERATORS TIME (b) 6th order, compatible 6TH ORDER FULLY COMPATIBLE SBP OPERATORS N = 21 N = 41 N = 81 N = 161 MAX ERROR MAX ERROR N = 21 N = 41 N = 81 N = TIME (c) 4th order, fully compatible TIME (d) 6th order, fully compatible Figure 3: Max error as function of time for a plane wave impinging on an interface where discontinuities in the Lamé parameters occur. Results from methods using fourth and sixth order SBP operators are shown from left to right. The number of grid points increases from top to bottom. 18

107 both plates measure 50 by 4 inches. The plate of Plexiglas has pressure - wave velocity Cp Pl = 2630m/s, shear - wave velocity Cs Pl = 1195m/s and density ρ Pl = 1190kg/m 3, the plate of aluminum has pressure -wave velocity Cp Al = 6460m/s, shear - wave velocity Cs Al = 3100m/s and density ρ Al = 2700kg/m 3. Initially, the displacements and velocities are zero and the problem is forced by adding a source term { 10 f = 4 (sin(2π250t) 1/2sin(2π500t)) δ(x 4),t (0, 1/250) 0, else to the normal stresses at the upper horizontal traction free boundary of either the plate of Plexiglas or the plate of aluminum. The source term is located 4 inches from the left corner of the upper plate, see Figure 4. On the vertical boundaries a traction free boundary condition is imposed. The response is then recorded as a function of time by receivers spaced 2 inches apart, see Figures 4 and 5. A generalization of this problem in which two semi - infinite half - planes was considered is discussed in [6]. There it is shown that refraction arrivals are generated if a source is placed in the half - space of lower pressure and shear - wave speeds as the wave front impinges on the interface to the faster media. Refraction arrivals was also measured experimentally in [20] where a spark served as a point source on the boundary of a plate of Plexiglas welded together with a plate of aluminum or vice versa, as described above. Here refraction arrivals was observed only in the case of a source on the boundary of the slower plate of Plexiglas, in accordance with the theory in [6]. The same behavior is observed in the present numerical experiment. In the case of the source located on the boundary of the plate of Plexiglas an earlier refraction arrival is measured at the receivers located 12 inches and further away from the source. In the other case the direct wave is always the first arrival, see Figures 4 and 5. For comparison the results of the corresponding laboratory experiment presented in [20] are displayed in Figure 6. Features of the recordings are similar, in both experiments the refraction arrival first becomes apparent at the recorder spaced 12 inches away from the source. 4.3 The Stoneley interface wave A Stoneley interface wave satisfies the elastic wave equation in two - homogeneous half - planes welded in contact. Define λ +2μ μ α =,β = ρ ρ. To simplify we assume that the Stoneley wave is 2π - periodic in the x - direction. The component of the Stoneley wave in the half - plane y 0 can then be written U = Ae y ( ) 1 c 2 cos(x c S /α2 S t) 1 c 2 S /α 2 sin(x c S t) + Be y ( ) y 0. (26) 1 c 2 S 1 c 2 /β2 S /β 2 cos(x c S t), sin(x c S t) 19

108 Figure 4: The source is placed at the upper boundary of the plate of Plexiglas. A refraction arrival becomes apparent 12 inches from the source. The arrows from the source displays the path of the direct wave and the refracted wave, respectively. Figure 5: The source is placed at the upper boundary of the plate of aluminum. The direct wave is now the first arrival. The arrows from the source displays the path of the direct wave and the refracted wave, respectively. 20

109 Figure 6: Results from the laboratory experiment presented in [20]. Top figure: the source is placed at the upper boundary of the plate of aluminum. Bottom figure: the source is placed at the boundary of the plate of Plexiglas. 21

110 The component of the Stoneley wave in the half - plane y 0 is written U = Ce y ( ) 1 c cos(x c 2 S /α 2 S t) 1 c 2 S /α 2 sin(x c S t) + De y ( ) y 0. (27) 1 c 2 S 1 c 2 /β 2 S /β 2 cos(x c S t), sin(x c S t) As in Experiment 4.1 the constants A, B, C and D are determined through the solution to the linear system arising form inserting (26) and (27) into the interface conditions (3). This system now depends on the Stoneley phase velocity c S and a solution exists iff c S satisfies the dispersion relation 1 1 c2 S β c2 S β 2 det 1 c2 S α c2 S α 1 2 ρc 2 S 2ρβ2 2ρβ 1 2 c2 S β 2 ρ c 2 S +2ρβ 2 2ρ β 2 =0. 1 c2 S β 2 2ρβ 1 2 c2 S α 2 ρβ 2 (2 c2 S β 2 ) 2ρ β 2 1 c2 S α ρ β 2 (2 c2 2 S β ) 2 (28) The existence of a root c S to this equation was first considered by Stoneley [23], where existence was shown for some special values of the choice of Lamé parameters and densities of the half - planes. Further investigations are discussed in [6], pp and the references therein. It has been shown that a root corresponding to the correct Stoneley phase speed can exist when β β 1and that c S < min{β,β } < min{α, α } always. Hence, the solutions (26) and (27), if they exist, represent waves traveling harmonically along the interface between the half - planes and decays exponentially into the domain. Numerical simulations involving the Rayleigh surface wave was discussed in [13] and later in the SBP-SAT context in [26]. There difficulties of the simulation of surface waves was encountered, in particular as the material becomes almost λ incompressible i.e., μ. The purpose of this experiment is to evaluate the performance of the current method when simulating Stoneley waves and to further study the difference in accuracy between schemes using 6th order compatible and fully compatible operators observed in Experiment 4.1. In this study both half - planes are far from incompressible. We take λ = μ = ρ =1 and λ =3,μ =2,ρ =1.98. For this case a root c S = to (28) is found. Figure 7 displays vertical and horizontal components of the Stoneley interface wave for this choice of parameters. The computational domain is taken to contain exactly one wavelength, 2π, in the x - direction. Periodic boundary boundary conditions are imposed at x =0andx =2π and exact Dirichlet data is imposed at y =4π and y = 4π. Initial data for the computations is taken from (26) and (27) at time t = 0. We use a grid size 22

111 Figure 7: The Stoneley surface wave as a function of (x, y) att =0. Theu 1 component is show to the left and the u 2 component to the right. of h =2π/(N 1) for N =21, 41, 81, 161 and the max error in the approximate solutions are computed as a function of time for 5 temporal periods. The temporal period is given by 2π/c S The computations are done for each N and each of the two schemes. Figure 8 compares the errors obtained with each scheme. A dashed and full line corresponds to the scheme using fully compatible SBP operators and compatible SBP operators, respectively. It becomes evident that the scheme using fully compatible operators converges faster and has a smaller error. For N = 161 the error is more than one magnitude smaller. 5 Conclusions High order accurate finite difference schemes for the two - dimensional elastic wave equation in a domain consisting of two elastic half - planes in contact have been presented. The key ingredient in the discretization was fully compatible or compatible finite - difference operators approximating first and second derivatives. Stability of the numerical scheme was proven for the discretization using fully compatible operators. Stability for the schemes using compatible operators was only observed in numerical experiments. However, the convergence and accuracy properties was observed to be identical or in some cases better for the case of fully compatible operators. The numerical experiments served to 23

112 Figure 8: Max errors as functions of time. A dashed line corresponds to the errors obtained with a scheme using fully compatible 6th order operators. A full line corresponds to the errors obtained with a scheme using compatible 6th order operators. verify convergence, stability and accuracy as well as study the performance of the proposed method on key features arising from a line interface such as mode conversion, refraction arrivals and Stoneley interface waves. The current method needs several generalizations to be useful in practice. Even though the theory allows for general curvilinear coordinates and isotropic media, the computations were all done i Cartesian coordinates, piecewise constant anisotropic media and two spatial dimensions. A generalization to three spatial dimensions, complicated domains and general media should not pose a theoretical problem, but would require more programming work. An open question in the SBP - SAT context is how to treat grid refinement for equations containing second derivatives in space. A study involving grid refinement for first order systems show good results using SBP preserving interpolation operators [16] but problems has been encountered when applied to higher order equations [10]. For the method to be significantly more efficient grid refinement is a key feature. Also, an analytical study of the behavior of simulations involving Stoneley interface waves in almost incompressible media would be interesting, as it was seen in [13] and [26] that the similar Rayleigh suffers increasingly from truncation errors as the media becomes more incompressible. 24

113 References [1] Brown. D.L.: A note on the numerical solution of the wave equation with piecewise smooth coefficients, Mathematics of Computations, 42(166): , [2] Cagniard.L., Flinn.E.A., Dix.C. H.: Reflection and refraction of progressive seismic waves, McGraw Hill Book Company Inc., (1962). [3] Carpenter, M., Gottlieb, D., Abarbanel, S.: Time - Stable Boundary Conditions for Finite - Difference Schemes Solving Hyperbolic Systems: Methodology and Application to High - Order Compact Schemes, J. Comp Phys 111, (1994). [4] Cohen.G., Fauqueux.S.: Mixed spectral finite elements for the linear elasticity system in unbounded domains, SIAM J. Sci. Comput., 26(3), [5] Duru.K, Kreiss.G, Mattsson.K.: Accurate and Stable Boundary Treatments for Elastic Wave Equations in Second Order Formulation, submitted manuscript, (2012). [6] Ewing.W.M., Jardetzky.W.S., Press.F.: Elastic Waves In Layered Media, McGraw Hill Book Company Inc., (1957). [7] Gilbert.J.C., Joly.P.: Higher order time stepping for second order hyperbolic problems and optimal CFL conditions, Num. Analys and Sci Comp for PDEs and their Challenging Applicat, vol. 16, Springer, [8] Graff.K.F.: Wave Motion In Elastic Solids, Dover Publications. [9] Gustaffsson.B., Kreiss.H.O., Oliger.J.: Time Dependent Problems and Difference Methods, Wiley [10] Gustafsson.M., Nissen.A., Korrmann.K., Stable difference methods for block - structured adaptive grids, Technical report , Department of Information Technology, Uppsala University, [11] Käser.M., Dumbser.M.: An arbitrary high order discontinuous Galerkin method for elastic waves on unstructured meshes - I. The two - dimensional isotropic case, Geophysics. J. Int., 167, (2006). [12] Kreiss, H. -O., Oliger, J.: Comparison of accurate methods for the integration of hyperbolic equations, Tellus, 24 (1972) [13] Kreiss. H.-O., Petersson.N.A.: Boundary Estimates for the Elastic Wave Equation in Almost Incompressible Materials, SIAM J. Numer Anal, V 50, No. 3. [14] Lamb.H.: On the Propagation of Tremors Over the Surface of an Elastic Solid, Phil. Trans. Roy. Soc. (London) A, vol

114 [15] Mattsson, K.: Summation by parts operators for finite difference approximations of second - derivatives with variable coefficients, J Sci Comput, 51, (2012) [16] Mattsson.K., Carpenter.M.: Stable and accurate interpolation operators for high - order multiblock finite difference methods, SIAM J. sci. Comput, Vol. 32, No. 4, [17] Mattsson, K., Nordström, J.: Summation by parts operators for finite difference approximations of second derivatives, J. Comp Phys 199 (2004) [18] Mattsson.K., Ham.F., Iaccarino.G.: Stable and accurate wave propagation in discontinuous media, J. Comp. Phys, 218 (2006) pp [19] Petersson. N.A., Sjögreen. B.: Stable grid refinement and singular source discretization for seismic wave simulations, Commun. Comput. Phys. 8, (2010). [20] Press.F., Oliver.J., Ewing.M.: Seismic Model Study of Refraction From a Layer of Finite Thickness, Geophysics, V XIX, No. 3 (1954) [21] Rayleigh. Lord.: On Waves Propagated Along the Surface Along the Plane Surface of an Elastic Solid. Proc. London Math. Soc. vol 17. [22] Stephen.R.A.: Solutions to range - dependent benchmark problems by the finite-difference method, J. Acoust. Soc. Am. 87, [23] Stoneley.R.: Elastic Waves at the Surface of Separation of Two Solids, Proc. Roy. Soc. (London),A,vol.6, [24] Strand, B.: Summation by parts for finite difference approximations for d/dx, J. Comp Phys 110 (1994) [25] Virieux.J.: P-SV wave propagation in heterogeneous media: Velocity - stress finite - difference method. Geophysics, 51: , [26] Virta.K., Kreiss.G.: Surface Waves in Almost Incompressible Elastic Materials, Technical report, arxiv: [physics.geo-ph]. [27] Virta.K., Mattsson.K.: Acoustic Wave Propagation in Complicated Geometries and Heterogeneous Media. Manuscript subitted to J Sci Comput. 26

115 Recent licentiate theses from the Department of Information Technology Emil Kieri: Numerical Quantum Dynamics Johannes Åman Pohjola: Bells and Whistles: Advanced Language Features in Psi-Calculi Daniel Elfverson: On Discontinuous Galerkin Multiscale Methods Marcus Holm: Scientific Computing on Hybrid Architectures Olov Rosén: Parallelization of Stochastic Estimation Algorithms on Multicore Computational Platforms Andreas Sembrant: Efficient Techniques for Detecting and Exploiting Runtime Phases Palle Raabjerg: Extending Psi-calculi and their Formal Proofs Margarida Martins da Silva: System Identification and Control for General Anesthesia based on Parsimonious Wiener Models Martin Tillenius: Leveraging Multicore Processors for Scientific Computing Egi Hidayat: On Identification of Endocrine Systems Soma Tayamon: Nonlinear System Identification with Applications to Selective Catalytic Reduction Systems Magnus Gustafsson: Towards an Adaptive Solver for High-Dimensional PDE Problems on Clusters of Multicore Processors Department of Information Technology, Uppsala University, Sweden