How To Write Aneastic Wave Propagation On Unstructured Meshes In D And 3D

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1 Geophys. J. Int. ( 4, Arbitrary High Order Finite Voume Schemes for Seismic Wave Propagation on Unstructured Meshes in D and 3D Michae Dumbser,, Martin Käser, Josep de a Puente 3 Department of Civi and Environmenta Engineering, University of Trento, Itay Institut für Aerodynamik und Gasdynamik, Universität Stuttgart, Germany 3 Department of Earth and Environmenta Sciences, Geophysics Section, Ludwig-Maximiians-Universität, München, Germany Accepted 999 November. Received 999 October ; in origina form 999 August 3 SUMMARY We present a new numerica method to sove the heterogeneous aneastic seismic wave equations with arbitrary high order of accuracy in space and time on unstructured trianguar and tetrahedra meshes in two and three space dimensions, respectivey. Using the veocity-stress formuation provides a inear hyperboic system of equations with source terms that is competed by additiona equations for the aneastic functions incuding the strain history of the materia. These additiona equations resut from the rheoogica mode of the generaized Maxwe body and permit the incorporation of reaistic attenuation properties of viscoeastic materia accounting for the behaviour of eastic soids and viscous ßuids. The proposed method reies on the Finite Voume (FV approach where ce-averaged quantities are evoved in time by computing numerica ßuxes at the eement interfaces. The basic ingredient of the numerica ßux function is the soution of Generaized Riemann Probems at the eement interfaces according to the ADER approach of Toro et a., where the initia data is piecewise poynomia instead of piecewise constant as it was in the origina Þrst order FV scheme deveoped by Godunov. The ADER approach automaticay produces a scheme of uniform high order of accuracy in space and time. The high order poynomias in space, needed as input for the numerica ßux function, are obtained using a reconstruction operator acting on the ce averages. This reconstruction operator uses some techniques originay deveoped in the Discontinuous Gaerkin (DG Finite Eement framework, namey hierarchica orthogona basis functions in a reference eement. In particuar, in this artice we pay specia attention to underine the differences as we as the points in common with the ADER-DG schemes previousy deveoped by the authors. The numerica convergence anaysis demonstrates that the proposed Finite Voume schemes provide very high order of accuracy even on unstructured tetrahedra meshes whie computationa cost for a desired accuracy can be reduced when appying higher order reconstructions. Appications to a series of we-acknowedged eastic and aneastic test cases and comparisons with anaytic and numerica reference soutions, obtained by different we-estabished numerica methods, conþrm the performance of the proposed method. Therefore, the deveopment of the highy accurate ADER-FV approach for tetrahedra meshes incuding viscoeastic materia provides a nove, ßexibe and efþcient numerica technique to approach three-dimensiona wave propagation probems incuding reaistic attenuation and compex geometry. Key words: viscoeasticity, attenuation, Finite Voume schemes, high order accuracy, unstructured meshes, ADER approach INTRODUCTION Today numerica seismoogy can provide computer simuations of the propagation of seismic waves within the earth interior, that represent an invauabe too for the understanding of the wave phenomena, their generation and their consequences. However, the simuation of a compete, highy accurate three-dimensiona wave Þed in reaistic media with compex geometry is sti a great chaenge. After the semina work of Madariaga (97 and Virieux (984; 98 a number of different methods have been deveoped and a vast amount of pubications on the simuation of seismic wave propagation can be found in the iterature. However, the improved knowedge of the subsurface

2 Michae Dumbser, Martin Käser, Josep de a Puente structure and the necessity to hande geometricay compicated geoogica features has driven the deveopment of numerica methods that use non-reguar, unstructured meshes that provide the required geometrica ßexibiity. First approaches, e.g. in (Braun & Sambridge 995; Käser & Ige ; Käser, Ige, Sambridge & Braun ; Zhang 997, provided numerica schemes with accuracies too ow to be appied to reaistic arge scae probems. However, after the Spectra Eement Methods (SEM was introduced in the Þed of numerica seismoogy in (Prioo, Carcione & Seriani 994; Seriani 998, this spatiay high order accurate scheme was further deveoped in (Komatitsch & Viotte 998; Komatitsch & Tromp 999; Komatitsch & Tromp. Later, a new numerica method based on a Discontinuous Gaerkin (DG approach in combination with a nove time integration scheme using Arbitrary high order DERivatives (ADER was introduced in (Käser & Dumbser a; Dumbser & Käser a to simuate eastic wave propagation of unstructured trianguar and tetrahedra meshes with arbitrary high order of accuracy. Due to the increased accuracy, it became important to incorporate second-order effects such as attenuation and dispersion to correcty mode the wave ampitudes and phases of a fuy three-dimensiona seismic wave Þed. A successfu mode for reaistic attenuation is the approximation of the materia as a viscoeastic medium. Hereby, it is important that the composition of the earth s poycrystaine materia and the superposition of the microscopic physica attenuation processes eads to a ßat attenuation band, see (Liu, Anderson & Kanamori 97; Stein & Wysession 3. The correct numerica treatment of a viscoeastic medium is outined in (Moczo, Kristek & Haada 4. Day & Minster (984 transformed the stress-strain reation in the time domain into a differentia form and obtained n differentia equations for n additiona interna variabes, which repace the convoution integra. Emmerich & Korn (987; 99 improved this approach and showed that their method is superior in accuracy and computationa efþciency and appied the viscoeastic modes for the P-SV case. Independenty, a different approach in (Carcione, Kosoff & Kosoff 988; Carcione & Cavaini 994 introduced additiona Þrst order differentia equations for memory variabes. Recent work by Moczo & Kristek (5 reviewed both modes and showed that indeed both approaches are equivaent. Moczo et a. (997 presented a hybrid two-step method for simuating P-SV seismic motion in inhomogeneous viscoeastic structures with free surface topography combining discrete-wavenumber (DW (Bouchon 98, Þnite eement (FE (Marfurt 984 and Þnite-difference (FD (Moczo & Bard 993 methods. Finay, in (Käser, Dumbser, de a Puente & Ige viscoeastic attenuation was incorporated into the ADER-DG schemes. In this paper, we introduce an arbitrary high order ADER Finite Voume (ADER-FV scheme on unstructured trianguar and tetrahedra meshes incuding eastic and viscoeastic media. In contrast to previous FV approaches, e.g. (Dormy & Tarantoa 995; Wang ; Wang & Liu ; Tadi 4; Wang & Liu 4; Wang, Zhang & Liu 4 the proposed ADER-FV method is based on a new and efþcient reconstruction operator for unstructured meshes in D and 3D deveoped by Dumbser and Käser in (Dumbser & Käser b and the soution of Generaized Riemann Probems (GRP (Toro & Titarev at the eement interfaces for ßux computation. To our knowedge, the method presented in (Dumbser & Käser b is the Þrst Finite Voume scheme on three-dimensiona unstructured meshes of order higher than two. Former work on high order Finite Voume schemes on unstructured meshes was restricted to two space dimensions, see for exampe (Abgra 994; Friedrich 998; Oivier-Gooch & Van Atena ; Hu & Shu 999; Käser & Iske 5. The Þna formuation of the ADER-FV scheme differs from the ADER-DG scheme (Käser & Dumbser a; Dumbser & Käser a; Käser, Dumbser, de a Puente & Ige ony in the use of the reconstruction operator to obtain high order spatia accuracy. Once the reconstruction is done, the impementation of the ADER-FV scheme is essentiay the same as for ADER-DG methods. The advantage of Finite Voume methods, however, is that they aow consideraby arger time steps than ADER-DG schemes and that one singe time step is cheaper than a corresponding time step for ADER-DG schemes. The inconvenience is, that ADER-FV schemes are ess accurate. However, if a Þne mesh is needed for resoving sma features in compex geometries, sometimes the accuracy provided by ADER-DG is not usefu due to the Þne mesh. In this case, a ess expensive Finite Voume scheme as described in this artice may be the better choice. The paper is structured as foows. In Section we introduce the system of the three-dimensiona aneastic wave equations in veocitystress formuation incuding attenuation due to viscoeasticity. The reconstruction operator needed for the Finite Voume scheme is brießy expained in Section 3 and the resuting ADER Finite Voume scheme based on this reconstruction is derived in Section 4. Two important boundary conditions are discussed in Section 5. In Section we show numericay the convergence properties of the proposed scheme and in Section 7 we compare the ADER-FV scheme deveoped in this artice with the ADER-DG method previousy presented by the authors in (Käser & Dumbser a; Dumbser & Käser a; Käser, Dumbser, de a Puente & Ige. Finay, in Section 8, we present the numerica resuts obtained with the ADER-FV method for the two-dimensiona Lamb s probem and the three-dimensiona test cases LOH. and LOH.3 proposed by the PaciÞc Earthquake Engineering Research Center (Day, Bieak, Dreger, Graves, Larsen, Osen & Pitarka 3 providing anaytic and numerica reference soutions obtained by we-estabished codes of other research institutions. In particuar, we compare the corresponding resuts of the ADER-FV proposed in this artice with numerica resuts obtained with the ADER-DG method previousy pubished by the authors in (Dumbser & Käser a; Käser, Dumbser, de a Puente & Ige.

3 ANELASTIC WAVE EQUATIONS Arbitrary High Order Finite Voume Schemes for Seismic Wave Propagation 3 The aneastic wave propagation can be described by modifying the constitutive reation, i.e. Hooke s Law, as shown in (Moczo, Kristek & Haada 4 and transforming it into the frequency domain. The reation between stresses σ =(σ xx,σ yy,σ zz,σ xy,σ yz,σ xz T and strains ε =(ε xx,ε yy,ε zz,ε xy,ε yz,ε xz T in the case of inear viscoeasticity can then be written as σ i(ω =M ij(ω ε j(ω ( where M ij is a matrix incuding compex, frequency-dependent viscoeastic modui. In genera M ij has independent entries, however, for the isotropic case they reduce to the two Lamé parameters λ = λ(ω and µ = µ(ω. The rheoogica mode that deþnes the parameters of M ij has to have a physicay feasibe expression that, in addition, reproduces the expected resuts of stress and strain damping as we as experimenta observation of strain response to stress oads. In (Liu, Anderson & Kanamori 97 a superposition of different reaxation mechanisms is proposed as a way to fuþ both conditions. As introduced in (Emmerich & Korn 987 and ceary outined in (Moczo & Kristek 5 we can express the viscoeastic modui as a combination of n mechanisms (so-caed Maxwe bodies as ( n λ(ω =λ U Y λ ω, ( ω + iω ( n µ(ω =µ U Y µ ω, (3 ω + iω = where λ U = im ω λ(ω and µ U = im ω µ(ω are the unreaxed Lamé parameters as used in purey eastic media. The Y λ and Y µ are the aneastic coefþcients to be determined and ω are the reaxation frequencies of the different mechanisms. In genera, given a viscoeastic moduus, e.g. the shear moduus µ(ω, the quaity factor Q(ω is deþned as Q µ(ω = Re(µ(ω Im(µ(ω. (4 Inserting the shear moduus µ(ω from (3 into (4 eads to Q µ (ω = n = = ω ω + ω Q µ (ω ω + ω Y µ. (5 The equation (5 can be used to Þt anyq(ω-aw as shown in (Emmerich & Korn 987; Moczo, Kristek & Haada 4. Observations show, that the quaity factor Q is approximatey constant over a arge frequency range of interest for most geophysica appications. They propose, that good approximations can be obtained by choosing n reaxation frequencies ω, =,..., n, that equidistanty cover the ogarithmic frequency range of interest. They suggest to use n known vaues Q( ω k at frequencies ω k, k =,..., n, with ω = ω and ω n = ω n and sove the overdetermined system in (5 for the aneastic coefþcients Y µ by the east square method. A more detaied discussion of the choice of frequency ranges and the corresponding samping frequencies can be found in (Graves & Day 3. In practice and anaogous to the seismic P- and S-wave veocities, we have quaity factors Q P and Q S that describe the different degree of attenuation for the different wave types. Therefore, from (5 we can aso derive aneastic coefþcients Y P S-wave propagation by soving the systems and Y S for viscoeastic P- and n Q ω ω k + ω Q ν (ω k ν (ω k = ω = + Y ω ν, with ν = P, S, and k =,..., n. ( k In the foowing, however, it is more convenient to express the aneastic coefþcients in terms of the Lamé parameters λ and µ, which are obtained by the transformation ( Y λ = + µ Y P µ λ λ Y S, Y µ = Y S, (7 foowing directy from ( and (3 as the reation of physica parameters, e.g. eastic parameters or veocities, corresponds to the purey eastic case due to the inearity of the expressions in ( and (3. As shown in (Kristek & Moczo 3; Moczo & Kristek 5 we deþne a new set of variabes, which are independent of the materia properties, caed the aneastic functions ϑ =( ϑ xx, ϑ yy, ϑ zz, ϑ xy, ϑ yz, ϑ xz T, which contain the time history of the strain in the form t ϑ j(t =ω ε j(τe ω(t τ dτ. (8 Using (8 and appying the inverse Fourier transform to the viscoeastic moduus M ij as outined in detai in (Moczo & Kristek 5 the stress-strain reation ( can be written in the time domain in the form n σ ij = λε kk δ ij +µε ij (λy λ ϑ kkδ ij +µy µ ϑ ij, with i, j, k [x, y, z] (9 = where δ ij is the Kronecker Deta and the equa-index summation convention appies to the index kk. The viscoeastic constitutive reation in (9 represents the eastic part minus the aneastic part depending on the aneastic coefþcients Y λ and Y µ and the aneastic functions

4 4 Michae Dumbser, Martin Käser, Josep de a Puente ϑ ij. The remaining probem is the evoution of the aneastic functions ϑ ij in (8 in time. In fact, equation (8 is the soution of the partia differentia equation ϑ j t (t+ω ϑ j (t =ω ε j, ( which competes the inear, hyperboic system of the aneastic wave equations. However, to express the equation system in the veocity-stress formuation it is convenient to redeþne the aneastic functions in the form ϑ j = t ϑ j. ( Finay, using the equations of motion, the deþnition of strain ε j and equations (9, ( and ( we can formuate the system of the aneastic wave equations as σxx (λ +µ u λ v λ w = n (λy t x y z λ +µy µ ϑ xx λy λ ϑ yy λy λ ϑ zz, = t σyy λ u (λ +µ v λ w = n λy x y z λ ϑ xx (λy λ +µy µ ϑ yy λy λ ϑ zz, = t σzz λ u λ v (λ +µ w = n x y z = σxy µ( v + u = n µy µ t x y ϑ xy, = t σyz µ( v + w = n z y = t σxz µ( u + w = n z x = ρ u t x σxx y σxy σxz =, z ρ v t x σxy y σyy σyz =, z ρ w t x σxz y σyz σzz =, z t ϑ xx ω x u = ωϑ xx, λy λ ϑ xx λy λ ϑ yy (λy λ +µy µ ϑ zz, µy µ ϑ yz, µy µ ϑ xz, t ϑ yy ω v = y ωϑ yy, t ϑ zz ω z w = ωϑ zz, ( t ϑ xy ω( v + u = x y ωϑ xy, t ϑ yz ω( v + w = z y ωϑ yz, t ϑ xz ω( u + w = z x ωϑ xz,.. t ϑn xx ω n x u = ωnϑn xx, t ϑn yy ω n y v = ωnϑn yy, t ϑn zz ω n w = z ωnϑn zz, t ϑn xy ωn( v + u = x y ωnϑn xy, t ϑn yz ωn( v + w = z y ωnϑn yz, t ϑn xz ωn( u + w = z x ωnϑn xz where n is the number of mechanisms used to approximate a frequency-independent Q-aw and ρ is the density. Note, that each mechanism adds further equations, i.e. one for each stress component. Therefore, the system of the purey eastic three-dimensiona wave equations consisting of 9 equations increases by n equations in the aneastic case, when n mechanisms are used. Furthermore, the aneasticity adds reactive source terms on the right hand side of (. In the foowing, we assume that the viscoeastic materia is described with the same number n of mechanisms throughout the computationa

5 Arbitrary High Order Finite Voume Schemes for Seismic Wave Propagation 5 domain. Therefore, the notation wi be identica to previous work (Dumbser & Käser a treating the purey eastic case. The above system ( of n v =9+n variabes and equations can be written in the more compact form U p t U q + Ǎ pq x + U q ˇB pq y + Č U q pq z z = Ě pq U q. (3 To obtain the two-dimensiona case, we simpy set =and remove the corresponding equations for the stresses σzz, σxz and σyz, the memory variabes ϑ zz, ϑ xz and ϑ yz as we as for the veocity w from the state vector U p and the Jacobians. Note, that the dimensions of the variabe vector U p, the Jacobian matrices Ǎ, ˇB, Č and the source matrix Ě depend on the number n of attenuation mechanisms. To keep the notation as simpe as possibe and without oss of generaity, in the foowing we assume that the order of the equations in (3 is such, that p, q [,..., 9] denote the eastic part and p, q [,..., n v], denote the aneastic part of the system as presented in (. As the Jacobian matrices Ǎ, ˇB and Č as we as the source matrix Ě are sparse and show some particuar symmetry pattern and as their dimensions may become impractica for notation, we wi use the bock-matrix syntax. Therefore, we decompose the Jacobian matrices as foows: Ǎ pq = [ A A a ] [ R nv nv, ˇB pq = B B a ] [ R nv nv, Č pq = C C a ] R nv nv, (4 where A, B, C R 9 9 are the Jacobians of the purey eastic part as given in (Dumbser & Käser a. The matrices A a, B a, C a incude the aneastic part and exhibit themseves a bock structure of the form: A a = A. A n Rn 9, B a = B. B n Rn 9, C a = C. C n Rn 9, (5 where each sub-matrix A,B,C R 9, with =,..., n, contains the reaxation frequency ω of the -th mechanism in the form: A = ω, ( B = ω, (7 C = ω. (8 The matrix Ě in (3 representing a reaction source that coupes the aneastic functions to the origina eastic system can be decomposed as with E of the bock structure Ě pq = [ E E ] R nv nv, (9 E =[E,...,E n] R 9 n, (

6 Michae Dumbser, Martin Käser, Josep de a Puente where each matrix E R 9, with =,..., n, contains the aneastic coefþcients Y λ and Y µ E = of the -th mechanism in the form: λy λ λy λ λy λ µy µ λy λ λy λ λy λ µy µ µy µ µy µ. ( µy µ λy λ µy µ λy λ λy λ The matrix E in (9 is a diagona matrix and has the structure E = E... E n Rn n, ( where each matrix E R, with =,..., n, is itsef a diagona matrix containing ony the reaxation frequency ω of the -th mechanism on its diagona, i.e. E = ω I with I R denoting the identity matrix. Since for ßux computation we need to rotate the data into a coordinate system aigned with the face norma and since the numerica ßux furthermore requires the absoute vaue Ǎ pq of matrix Ǎpq,in the foowing we wi have a coser ook at the absoute vaue matrix Ǎpq and the rotation matrix Ť pq. Simiar to (4 we Þnd that [ Ǎpq = A A ] R nv nv, (3 where A R 9 9 is identica to the one of the purey eastic part as given in (Dumbser & Käser a and has the form c p λ/(c pρ λ/(c pρ c s A =, (4 c s c p c s c s λ+µ with c p = and c ρ s = µ the P- and S-wave veocities of the unreaxed purey eastic materia. ρ The matrix A incudes the aneastic part and exhibits itsef a simiar bock structure as in (5 of the form: A = A. A n Rn 9, (5 where each sub-matrix A R 9, with =,..., n, contains the oca unreaxed materia parameters and the reaxation frequency ω of the -th attenuation mechanism in the form: = ω A /(c pρ /(c sρ /(c sρ. ( To compute the rotation matrices we reca that the aneastic functions ϑ are tensors ike the stresses and thus the rotation matrix Ťpq for the fu aneastic system ( has the form Ť pq = T t T v T a R nv nv, (7

7 Arbitrary High Order Finite Voume Schemes for Seismic Wave Propagation 7 where T t R is the rotation matrix responsibe for the stress tensor rotation as in the purey eastic part and is given as n x s x t x n xs x s xt x n xt x n y s y t y n ys y s yt y n yt y T t n z s z t z n zs z s zt z n zt z =, (8 n yn x s ys x t yt x n ys x + n xs y s yt x + s xt y n yt x + n xt y n zn y s zs y t zt y n zs y + n ys z s zt y + s yt z n zt y + n yt z n zn x s zs x t zt x n zs x + n xs z s zt x + s xt z n zt x + n xt z with the components of the norma vector n =(n x,n y,n z T and the two tangentia vectors s =(s x,s y,s z T and t =(t x,t y,t z T, which ie in the pane determined by the boundary face of the tetrahedron and are orthogona to each other and the norma vector n as shown in (Dumbser & Käser a. The matrix T v R 3 3 is the rotation matrix responsibe for the veocity vector rotation as in the purey eastic part and is given as T v = n x s x t x n y s y t y n z s z t z The matrix T a in (7 is a bock diagona matrix and has the structure T t T a =... T t where each of the n sub-matrices T t is the tensor rotation matrix given in (8.. (9 Rn n, (3 3 RECONSTRUCTION ALGORITHM The main ingredient of the proposed arbitrary high order Finite Voume scheme is the new reconstruction agorithm proposed in (Dumbser & Käser b that makes use of techniques deveoped originay in the Discontinuous Gaerkin (DG framework. Whereas the method proposed in (Dumbser & Käser b even incudes a non-inear WENO reconstruction agorithm based on severa stencis to ensure monotonicity of discontinuous soutions, we wi restrict the reconstruction operator in this artice to a inear one, based ony on one centra stenci. The computationa domain Ω is discretised by conforming eements T (m, indexed by a unique mono-index m ranging from to the tota number of eements E. The eements are chosen to be trianges in D and tetrahedrons in 3D. The union of a eements is caed the trianguation or tetrahedrization of the domain, respectivey, E T Ω = T (m. (3 m= As usua for Finite Voume schemes, data is represented by the ce averages of the state vector U p inside an eement T (m, ū (m p = U pdv, (3 T (m T (m where T (m denotes the voume of the eement. In order to achieve high order of accuracy for a Finite Voume scheme, we need to reconstruct higher order poynomias W p from the given ce averages ū p. We write the reconstruction poynomia for eement T (m as W p (m (ξ,η,ζ =ŵ (m p Ψ (ξ,η, ζ, (33 where ξ, η and ζ are the coordinates in a reference coordinate system, see Figure, where aso the reference eements T E are deþned. Throughout the whoe paper we use cassica tensor notation, which impies summation over each index appearing twice. Whereas the reconstructed degrees of freedom ŵ (m p are not space-dependent, the reconstruction basis functions Ψ are poynomias of degree M and depend on space. The index ranges from to its maximum vaue L, where L = (M +(M +and L = (M +(M +(M +3 are the numbers of reconstructed degrees of freedom in D and 3D, respectivey, depending on the order of the reconstruction. We use the hierarchica orthogona reconstruction basis functions that are given e.g. in (Dubiner 99; Cockburn, Karniadakis & Shu or in Appendix A for trianges in D and tetrahedrons in 3D. The transformation from the physica coordinate system x y z into the reference coordinate system ξ η ζ is in three space dimensions deþned by ( ( ( x = X (m + X (m X (m ξ + X (m 3 X (m η + X (m 4 X (m ζ, ( ( ( y = Y (m + Y (m Y (m ξ + Y (m 3 Y (m η + Y (m 4 Y (m ζ, ( ( ( z = Z (m + Z (m Z (m ξ + Z (m 3 Z (m η + Z (m 4 Z (m ζ,

8 8 Michae Dumbser, Martin Käser, Josep de a Puente z 4 ζ y 3 T (m η 3 T (m 3 4 y T E 3 η T E x ξ x ξ Figure. Transformation from the physica triange and tetrahedron T (m to the canonica reference triange T E with nodes (,, (, and (, and the canonica reference tetrahedron T E with nodes (,,, (,,, (,,, (,,. where X (m i, Y (m i and Z (m i denote the physica vertex coordinates of the considered eement T (m. In two space dimensions, the same transformation appies for x and y, setting ζ =. As short notation for the mapping and its inverse mapping from ξ ξ, η, ζ to x =(x, y, z and vice versa with respect to the eement T (m, we simpy write ( x = x T (m, ξ, ξ = ξ (T (m, x. (34 Via the inverse mapping given in (34 for the vector ξ, the eement T (m is transformed to the unit eement T E, whose voume is T E = in two dimensions and T E = in three space dimensions, respectivey. Furthermore, J ij = xi (35 ξ j is the Jacobian matrix of the transformation and J = J ij its determinant, being equa to twice the triange surface in D and equa to six times the tetrahedron voume in 3D. For performing the reconstruction on eement T (m, we now choose a reconstruction stenci S (m = n e j= that contains a tota number of n e eements. Here j n e is a oca index, counting the eements in the stenci, and (k j is the mapping from the oca index j to the goba indexation of the eements in T Ω. We set by deþnition (k =(m and thus the Þrst eement in the stenci is aways the considered eement T (m for which reconstruction is to be done. For ease of notation, we write in the foowing ony (k, meaning (k j. We then appy the inverse mapping (34 with respect to eement T (m to a the eements T (k S (m, where the transformed eements are in the foowing denoted as T (k. We emphasize that for a eements T (k S (m the mapping with respect to the Þrst eement in the stenci is appied, so m is constant for each stenci and therefore the appied mapping formua is the same for a eements in S (m. We note in particuar that the transformed eement of the Þrst eement in the stenci is of course the canonica reference eement, hence T (k = T (m = T E. The stenci transformed in that way is denoted S (m = T (k, see a two- and three-dimensiona exampe in Figs. and 3. The reconstruction must be conservative, at east in the eement considered for reconstruction. Initiay, we even require integra conservation for W p (m in a eements T (k S (m. In the physica coordinate system we thus have ( ( (m Wp ξ T (m, x dv = T (k ū (k p, T (k S (m. (37 T (k After transforming a eements of the stenci using (34 and taking into account that the degrees of freedom ŵ (m p do not depend on space, we obtain the intermediate resut ( J Ψ ξ dξdηdζ ŵ (m = J T (k ū (k p, T (k S (m. (38 T (k p The Jacobian determinant appears on both sides of eqn. (38 and thus cances out. Pease note that in the genera case this is ony possibe for trianges and tetrahedrons with straight edges, to which we restrict ourseves in this paper. Genera poyhedra eements or even curved boundaries are not considered here. The canceing of the Jacobian determinants automaticay cances scaing effects of the probem and avoids i-conditioned reconstruction matrices as reported by Abgra in (Abgra 994. Abgra and Friedrich (Friedrich 998 used barycentric coordinates in order to avoid this probem, whereas we use a hierarchica orthogona basis as commony used in the Discontinuous Gaerkin Þnite eement framework. T (k j (3

9 Arbitrary High Order Finite Voume Schemes for Seismic Wave Propagation 9 a.5 b y. η x ξ Figure. Exampes of an origina stenci S (m (a and the corresponding transformed stenci S (m (b in D for the reconstruction of a poynomia of degree 3 with n e =5. a b z y 4 5 x X - - Z Y ζ η - ξ X - Z -3 Y Figure 3. Exampes of an origina stenci S (m (a and the corresponding transformed stenci S (m (b in 3D for the reconstruction of a poynomia of degree with n e =. During the reconstruction step, the basis poynomias are continuousy ( extended over the whoe stenci. In more detai, this extension means that during reconstruction the poynomia term given by Ψ ξ is not ony vaid inside the reference eement T E, but aso in a the other eements in the transformed stenci S (m. After the reconstructed poynomia for eement T (m has been obtained, the basis poynomias are again restricted to the considered eement T (m. We emphasize that the integration on the eft hand side has to be done over the transformed eements T (k. In order to do this integration, the trick now consists in doing another coordinate transformation to a second reference coordinate system using the vertices of the transformed eement T (k as parameter of another mapping from the Þrst ξ η ζ reference system to the second ξ η ζ reference coordinate system. For convenience, we denote Ξ=( ξ, η, ζ. The mapping and its inverse are then denoted as ξ = ξ ( T (k, Ξ (, Ξ= Ξ T (k, ξ, (39 and the Jacobian determinant of this mapping is caed J. Thus, eqn. (38 becomes after the second transformation J J ( ( Ψ ξ T (k, Ξ d ξd ηd ζ ŵ (m p = J J T E ū (k p, T (k S (m, (4 T E where again a Jacobian determinants cance out! The Þna set of reconstruction equations is ( ( Ψ ξ T (k, Ξ d ξd ηd ζ ŵ (m p = T E ū (k p, T (k S (m. (4 T E In order to compute the integra on the eft hand side of (4, we use cassica mutidimensiona Gaussian quadrature of appropriate order. For an exhaustive overview of such mutidimensiona quadrature formuae see (Stroud 97.

10 Michae Dumbser, Martin Käser, Josep de a Puente For convenience, we introduce the simpiþed tensor notation A j ŵ p =ū pj, (4 with A j = ( ( Ψ ξ T (k j, Ξ T d ξd ηd ζ and ū pj =ū (k j p. (43 E T E The number of reconstructed degrees of freedom is L and therefore we need at east n e = L eements in the stenci. Unfortunatey, if we choose n e = L so that the matrix A j becomes square, the resuting scheme may become unstabe on genera meshes. Therefore, we are forced to use more eements than the necessary minimum. The use of enarged reconstruction stencis for robustness purposes has aready been reported previousy in the iterature, see e.g. (Barth & Frederickson 99; Oivier-Gooch & Van Atena ; Käser & Iske 5. Furthermore, due to geometrica issues, the reconstruction matrix may be not invertibe. This may happen for exampe when a eements are aigned on a straight ine. Therefore, the stenci construction agorithm shoud avoid such cases. In our particuar impementation, we compute the singuar vaues of the matrix A j and check if some of them are zero. If so, we continue adding eements unti none of the singuar vaues is zero. In order to Þx parameters once and for a, since we are interested in a very genera agorithm, we usuay choose n e =.5L in D and n e =L in 3D. This means that we take between 5% and % more eements than the minimum necessary for reconstruction. The reconstruction stenci is generated for each eement T (m according to the foowing agorithm: We recursivey add successivey the Neumann neighbours (i.e. the direct side neighbours of the eement T (m and a Neumann neighbours of the eements added to the stenci so far, unti the desired number of stenci eements n e is reached. This procedure guarantees a rather centra reconstruction stenci which is needed for inear stabiity issues of the scheme. For an exampe of centra stencis see Figure (a and (b in two dimensions and Figure 3 (a and (b in three dimensions. As conþrmed by the numerica resuts in Section 8, this agorithm works equay we at the boundaries of the computationa domain, where the stencis are biased to one side. Since (4 becomes overdetermined with our choice n e > L we use a constrained east-squares technique in order to sove (4 respecting conservation in the Þrst eement T (m of the stenci. Due to the specia choice of the reconstruction basis functions, the equaity, which is written in tensor notation C ŵ p = R iū pi. (44 The vectors C and R i contain ony zeros except of the entries C =and R =. The east-squares soution of (4 with the constraint (44 couped via a Lagrangian mutipier λ p is obtained according to (Dumbser & Käser b as ( ( ( A j A jk C ŵ pk A j ū pj =. (45 C δ k λ p R iū pi constraint becomes simpy ŵ p =ū (k p =ū (m p Here, δ k is the Kronecker symbo. The matrix on the eft hand side of (45 wi be caed reconstruction matrix in the foowing and in order to increase the speed of the agorithm, it is inverted and stored for each eement of T Ω so that the unknown vector of the reconstructed degrees of freedom ŵ p can be easiy cacuated for each component p by a simpe matrix-vector mutipication of the inverse reconstruction matrix and the vector of known ce averages ū pj of the stenci S (m. We repeat that reconstruction is done component-wise for each variabe p of the governing equations (3. We note that most of the memory requirements of our proposed scheme are due to the storage of the inverse reconstruction matrices. 4 FINITE VOLUME DISCRETISATION OF THE ANELASTIC WAVE EQUATIONS 4. Semi-Discrete Finite Voume Scheme The genera semi-discrete form of the Finite Voume scheme is obtained by integration of (3 over an eement T (m, integration by ( parts and inserting a numerica ßux Fp h W + p,wp n in norma direction, U p t dv + h ( Fp W q,w + q nds= Ě pqu qdv. (4 T (m T (m T (m The numerica ßux is a function of the boundary extrapoated vaues W p + and Wp at the eement interfaces. In the case of a Þrst order Finite Voume scheme, these vaues correspond to the ce averages in the eement T (m and the neighbours, respectivey. In the case of a higher order Finite Voume scheme, the vaues W p + and Wp are obtained from a high order poynomia reconstruction as shown in the previous Section 3. The ßux can be written very easiy in a coordinate system which is aigned with the outward pointing unit norma vector n = (n x,n y,n z T on the boundary making use of a variabe rotation. We use the exact Riemann sover as numerica ßux in norma direction between two eements T (m and T (kj : F h p ( W q,w + q n = [Ťpr (Ǎrs + Ǎ rs (Ťsq W q + Ť pr (Ǎrs Ǎ rs (Ťsq W + q ], (47

11 Arbitrary High Order Finite Voume Schemes for Seismic Wave Propagation where Wq = ŵ (m q Ψ (m and W q + = ŵ (k j q Ψ (k j are the boundary extrapoated vaues of the reconstructed numerica soution W h from eement (m and the j-th side neighbour (k j, respectivey, since both eements adjacent to a boundary contribute to the numerica ßux. For the case of non-conservative inear systems with piecewise ( constant varying coefþcients, the ßux has to be evauated in each eement with the corresponding coefþcient matrix Ǎpq = Ǎ(m pq = Ǎpq T (m as function of the eement T (m. Inserting (47 into (4 and spitting the boundary integra into the contributions of each face j N E of the eement T (m, we obtain tū(m p N E dv + T (m j= + N E j= (Ǎ(m Ť pr rs + Ǎ(m rs Ť pr (Ǎ(m rs Ǎ(m rs (Ťsq ŵ(m q (Ťsq ŵ(k j q (T (m j (T (m j (m Ψ ds + (k Ψ j ds = Ě(m pq Equation (48 is written in the physica x y z system, but if we transform each physica eement T (m to a canonica reference eement T E in a ξ η ζ reference system (see Figure, the method can be impemented much more efþcienty since a integras can be precomputed beforehand in the reference system. After integration in the reference system and taking into account the orthogonaity of the basis functions for the source term integra on the right hand side, the semi-discrete formuation in D and 3D then reads as tū(m p N T (m E + + (Ǎ(m Ť pr rs + Ǎ(m rs j= N E (Ǎ(m Ť pr rs + Ǎ(m rs j= (Ťsq Sj F,j (Ťsq Sj F +,i,h ŵ (m q + ŵ (k j q = Ě(m pq ū (m q ŵ (m q T (m, where S j denotes the surface of face j in 3D and the edge ength of edge j in D. In (49 we use ßux matrices acting on the degrees of freedom of the reconstructed poynomias simiar to the ßux matrices for ADER-DG schemes introduced in (Dumbser 5; Käser & Dumbser a; Dumbser & Käser a; Käser, Dumbser, de a Puente & Ige, which act on the degrees of freedom of the DG basis poynomias. The ßux matrices can be cacuated anayticay once on the reference eement and then be stored. In the foowing, we give the detais of cacuating those ßux matrices on trianges and tetrahedrons in two and three space dimensions. First, we deþne the oca faces with their oca vertex ordering according to tabe, where the vertex numbering is stricty counter-cockwise in D as we as in 3D. Then, the vector of voume coordinates ξ is given on the faces via mapping functions from the face parameters χ and τ, see tabes and 3. Last but not east, for ßux computation over the face, we have to integrate aong the face inside the eement as we as in the neighbour. This is done consistenty by the transformation from the face parameters χ and τ inside the eement to the corresponding face parameters χ and τ in the neighbour face. Whereas in D this transformation is aways χ = χ, in 3D the transformation depends on the orientation of the neighbour face respect to the oca face of the considered eement, since via rotation of the trianguar faces there may be three possibe orientations. The corresponding mappings are given in tabe 4. In two space dimensions, a possibe ßux matrices are F,j = F +,i,h = (T E j (T E j Ψ Ψ T (m Ψ (m dv. (48 (49 ( ξ (j (χ dχ, j 3, (5 ( ξ (i ( χ dχ, i 3. (5 Index h is not used in D. In three dimensions, a possibe ßux matrices are ( F,j = Ψ ξ (j (χ, τ dχdτ, j 4, (5 F +,i,h = Ψ (T E j ( ξ (i ( χ (h, τ (h dχdτ, i 4, h 3. (53 (T E j The eft state ßux matrix (superscript - F,j accounts for the contribution of the eement (m itsef to the ßuxes over face j and the right state ßux matrix (superscript + F +,i,h accounts for the contribution of the eement s direct side neighbours (k j to the ßuxes over the face j. Index i N E indicates the oca number of the common face as it is seen from neighbour (k j and depends on the mesh generator. Index h 3 accounts for the three possibe orientations of the face due to rotation and denotes the number of the oca node

12 Michae Dumbser, Martin Käser, Josep de a Puente Tabe. Face deþnition on trianges and tetrahedrons Trianges (D Tetrahedrons (3D Face Points Face Points Tabe. D voume coordinates ξ (j in function of the edge parameter χ j 3 ξ (j (χ χ χ η (j (χ χ χ in the neighbour s face which ies on the oca vertex of face j in tetrahedron number (m. Index h aso depends on the mesh generator. On a given tetrahedra mesh, where indices i and h are known, ony four of the possibe matrices F +,i,h are used per eement. 4. The Fuy Discrete Formuation of the ADER-FV Scheme In this section we show how the ADER approach (Toro & Titarev ; Titarev & Toro ; Titarev & Toro 5 can be used for high order time integration of the Finite Voume method on unstructured meshes, caed ADER-FV method in the foowing, for genera inear hyperboic systems. For inear systems, a particuar simpiþcation can be introduced: time-integration and ßux computation can be exchanged, i.e. instead of soving the Riemann Probems for a spatia derivatives on the interface and doing then the Cauchy-Kovaewski procedure with the obtained derivatives, we can integrate the reconstructed soution in time separatey in each eement using the Cauchy- Kovaewski procedure on the reconstructed soution and then pug the time-integrated vaues on the boundaries into the numerica ßux function, which then takes correcty into account the discontinuity at the interface. We emphasize that the pure appication of the Cauchy- Kovaewski procedure requires the soution to be anaytic, whereas the ADER approach uses the soution of Generaized Riemann Probems with piecewise poynomia initia data. This requires ony that the soution is piecewise anaytic on both side of the eement interfaces. Note that the GRPs are aways soved aong the face-norma direction. As in (Käser & Dumbser a; Dumbser & Käser a we Þrst write the governing PDE (3 in the reference system as with Ǎ ξ pq = Ǎpq x + ˇB ξ pq y + ξ Čpq z, ˇB pq η = Ǎpq x + ˇB η pq y + η Čpq z, U p t U q + Ǎ pq ξ + ˇB pq U q η + U q Č pq Ěpq Uq =, (54 ζ Čpq ζ = Ǎpq x + ˇB ζ pq y + ζ Čpq z. (57 The k-th time derivative of the entire state vector U p is obtained via the Cauchy-Kovaewski procedure appied to the governing equation (3 in the reference system (54, and reads as ( k k t k Up = Ǎ pq ξ + ˇB pq η + Č pq ζ Ěpq U q, (58 which can be proven by compete induction. (55 (5 Tabe 3. 3D voume coordinates ξ (j in function of the face parameters χ and τ j 3 4 ξ (j (χ, τ τ χ χ τ η (j (χ, τ χ τ χ ζ (j (χ, τ τ χ τ

13 Arbitrary High Order Finite Voume Schemes for Seismic Wave Propagation 3 Tabe 4. Transformation of the face parameters χ and τ of the tetrahedron s face to the face parameters χ and τ in the neighbour tetrahedron according to the three possibe orientations (h of the neighbour face h 3 χ (h (χ, τ τ χ τ χ τ (h (χ, τ χ τ χ τ We expand the reconstructed soution of (54 in a Tayor series in time about the current time eve t n up to degree M, M (t t n k k W p(ξ, η, ζ, t = k! t k Wp(ξ,η,ζ,tn, (59 k= and repace time derivatives by space derivatives, using eqn. (58: M (t t n k k W p(ξ, η, ζ, t = ( (Ǎ k pq k! ξ + ˇB pq η + Č pq ζ Ěpq W q(ξ,η,ζ,t n. ( k= We now introduce the approximation (33 and obtain M (t t n k k W p(ξ, η, ζ, t = ( (Ǎ k pq k! ξ + ˇB pq η + ( Č pq ζ Ěpq Ψ ξ ŵ q (t n. ( k= This approximation can now be projected onto the reconstruction basis functions Ψ k in order to get an approximation of the evoution of the reconstructed degrees of freedom during one time step from time eve t n to time eve t n+. We obtain Ψ M ( (t t n, n k ( k Ǎ k! pq + ˇB k ξ pq + η Č pq ζ Ěpq Ψm k= ŵ p (t = ŵ qm(t n, ( Ψ n, Ψ where.,. denotes the inner product over the reference eement T E and the division by Ψ n, Ψ denotes the mutipication with the inverse of the mass matrix. This reduces indeed to division by its diagona entries since the mass matrix is diagona due to the supposed orthogonaity of the basis functions. Equation ( can be integrated anayticay in time from the current time eve t n to the next time eve t n+ = t n + t. We obtain t n + t ŵ p(tdt = t n With the deþnition equation (3 becomes simpy I pqm ( t = Ψ n, M k= Ψ n, M t (k+ (k+! k= t (k+ (k+! t n + t ( k ( Ǎ pq ξ + ˇB pq η + Č pq ζ Ěpq k Ψm Ψ n, Ψ ( k ( Ǎ pq ξ + ˇB pq η + Č pq ζ Ěpq k Ψm Ψ n, Ψ ŵ qm(t n. (3 t n ŵ p(tdt = I pqm( tŵ qm(t n. (5 For efþcient agorithms to do the Cauchy-Kovaewski procedure, we refer to (Käser & Dumbser a; Dumbser & Käser a; Käser, Dumbser, de a Puente & Ige. We Þnay obtain the fuy discrete ADER-FV scheme by integration of (49 in time, where t n and t n+ denote the current and the successive time eve: (4 + N E + j= (Ǎ(m Ť pr rs + Ǎ(m rs N E (Ǎ(m Ť pr rs + Ǎ(m rs j= [ ū (m p (Ťsq Sj F,j (Ťsq Sj F +,i,h ( t n+ ū (m p (t n ] T (m + I qnm ( t ŵ (m nm (t n + I qnm ( t ŵ (k j nm (t n = Ě(m pq I qnm( t ŵ nm (m (t n T (m. ( From the structure of eqn. ( we see that the space-time-integrated vaues on the boundaries enter the exact Riemann sover in order to give the space-time integra of the soution of the GRP at the interface. We emphasize that this can ony be done for inear systems. The reconstructed degrees of freedom ŵ nm (m (t n at time eve t n are obtained for each eement at the beginning of a time step using

14 4 Michae Dumbser, Martin Käser, Josep de a Puente the reconstruction operator described in Section 3. The proposed Finite Voume scheme is quadrature-free since no Gaussian integration is used in space and time. It performs high order time-integration from t n to t n+ in one singe step. It thus needs the same memory as a Þrst order expicit Euer time stepping scheme. The scheme ooks amost the same as the ADER-DG scheme presented in (Käser, Dumbser, de a Puente & Ige, except of the foowing two differences: First, for Finite Voume schemes ony the ce averages ū p have to be evoved in time, whereas for DG schemes a degrees of freedom û p must be updated. Second, the ßuxes of ADER-DG schemes are computed directy with the degrees of freedom û p, whereas the ßuxes for ADER-FV schemes are computed using the reconstructed degrees of freedom ŵ p, which are obtained from the ce averages ū p in the separate reconstruction step. 5 BOUNDARY CONDITIONS There is a variety of physicay meaningfu boundary conditions of an eastic medium. However, the two most important types of boundaries are absorbing and free surface boundaries, which wi be discussed in the framework of the ADER-FV method in the foowing. An important difþcuty in the context of Finite Voume schemes in contrast to Discontinuous Gaerkin methods is the generation of appropriate reconstruction stencis at the boundary of the computationa domain. In this artice, we choose one-sided stencis, i.e. stencis that ie competey in the computationa domain. Without changing the stenci search agorithm described in Section 3 the stencis at the boundary are simpy generated by adding to the stenci recursivey the direct Neumann neighbours of the eements aready in the stenci unti the required number of eements n e is reached, starting aways with the centra eement for which reconstruction is to be performed. Since at the boundary severa eements do not have a fu set of direct Neumann neighbours, ony the existing neighbours can be added. For the ßux computation, we then sove inverse Riemann Probems, as for ADER-DG methods. 5. Absorbing Boundaries At absorbing boundaries, no waves are supposed to enter the computationa domain and the waves traveing outward shoud pass the boundary without reßections. In this section we present a very simpe approach, that so far yieded satisfactory resuts, at east for our purposes. The numerica ßux (47 is based on the soution of the Riemann Probem given by the jump across the eement interface. It is a strict upwind method, i.e. outgoing waves at an eement interface are ony inßuenced by the state in the inside eement itsef. In contrast, the ßux contribution of incoming waves is purey due to the state in the neighbour eement. Thus, a simpe impementation of absorbing boundary conditions is to use the foowing numerica ßux in ( at a those tetrahedra faces that coincide with an absorbing boundary: F AbsorbBC p = Ťpq (Ǎ(m qr + Ǎ(m qr (Ťrs ŵ (m s Ψ (m. (7 The ßux function (7 aows ony for outgoing waves, which are merey deþned by the state in the eement due to upwinding. Since incoming waves are not aowed, the respective ßux contribution must vanish, i.e. it is set to zero in the impementation of the method. We are aware that these absorbing boundary conditions have some probems at corners or for grazing incidence of waves. Therefore, in future work, approaches ike the Perfecty Matched Layer (PML technique, as introduced in (Bérenger994 and appied in (Coino & Tsogka ; Komatitsch & Tromp 3 shoud be incorporated to improve the performance of the proposed scheme for such boundaries. 5. Free Surface Boundaries On the free surface of an eastic medium, the norma stress and the shear stresses with respect to the boundary are determined by physica constraints. Outside the eastic medium, there are no externa forces that retract the partices into their origina position. Therefore, the norma stress and the shear stress vaues at the free surface have to be zero. In contrast to cassica continuous Finite Eement methods such as the Spectra Eement method (SEM, we have no direct contro on the vaues at the boundaries within the Finite Voume framework. However, the boundary conditions can be imposed correcty via the numerica ßux. Considering that the numerica ßux is based on the soution of a Riemann Probem at an eement interface and given some boundary extrapoated vaues from inside the computationa domain on a free surface, we must sove a so-caed inverse Riemann Probem such that its soution yieds exacty the free-surface boundary conditions at the domain boundary. In the particuar case of the free surface, the soution of the inverse Riemann Probem can be obtained via symmetry considerations. For those components of the state vector U p, that we want to be zero at the domain boundary, we prescribe a virtua boundary extrapoated component on the outside of the interface that has the same magnitude but opposite sign. For the other components we just copy the inside vaues to the outside. For the free-surface boundary condition the resuting numerica ßux function in ( can then be formuated as foows, F FreeBC p = Ťpq (Ǎ(m qr + + Ťpq (Ǎ(m qr Ǎ(m qr Ǎ(m qr (Ťrs ŵ (m s Ψ (m + Γ rs (Ť st ŵ (m t Ψ (m, where the matrix Γ rs = diag (,,,,,,,,,,..., accounts for the mirroring of norma and shear stresses with respect to the face-norma direction. The viso-eastic memory variabes do not enter the ßux. We remark, that the soution of the inverse Riemann Probem is not equivaent to the FD approach of adding Þctitious ghost points, but comes out naturay from the FV framework and provides the (8

15 Arbitrary High Order Finite Voume Schemes for Seismic Wave Propagation 5 z z z y x y x y x Figure 4. Sequence of discretisations of the computationa domain Ω via reguary reþned tetrahedra meshes used for the numerica convergence anaysis. exact vaues of the norma and shear stresses as required by the free surface boundary condition. Numerica tests such as Lamb s probem in two space dimensions and the LOH test cases in three space dimensions conþrm the performance and accuracy of this approach, especiay compared to conventiona Finite Difference schemes, as shown in Section 8.. CONVERGENCE ANALYSIS In this section we present the resuts of a numerica convergence anaysis to demonstrate the very high accuracy that can be obtained with the proposed ADER-FV method on unstructured tetrahedra meshes considering viscoeastic attenuation. We show resuts from second order to sixth order ADER-FV schemes, which are denoted by ADER-FV O to ADER-FV O respectivey. Note, that the same order for time and space accuracy is automaticay obtained. To determine the convergence orders we sove the three-dimensiona seismic wave equations with viscoeastic attenuation in ( on the unit-cube as sketched in Figure 4, i.e. on a computationa domain Ω=[, ] [, ] [, ] R 3 with periodic boundary conditions. The homogeneous materia parameters are set to λ =, µ =, ρ =, Q P =, Q S =, (9 throughout the computationa domain Ω. The Q-factors are assumed to be frequency independent over the frequency band [., ] Hz. To this end, we are using 5 mechanisms as outined in Section. For the quaity of the approximation of the frequency independent Q factors and the associated computationa effort in function of the number of mechanisms used see (Käser, Dumbser, de a Puente & Ige. Approximatey the same CPU time growth as shown there for ADER-DG schemes aso hods for the ADER-FV method presented in this artice. These attenuation properties introduce damping and dispersion of the P- and S-waves. We know, e.g. from (Stein & Wysession 3, that a space-time harmonic soution to this probem can be found under the form U p(x, y, z, t =Up e i (ωt kxx kyy kzz, p =,..., n v (7 where Up is the initia ampitude vector, ω the wave frequencies to determine, and k =(kx,k y,k z T =(π, π, π T. (7 is the wave number vector eading to a periodic, pane sinusoida wave in the unit-cube with the wave front perpendicuar to the cube s space diagona. In the foowing, we brießy describe how we determine the wave frequencies ω: With the assumption, that equation (7 is the anaytic soution of the governing equation (3, we cacuate the Þrst time and space derivatives of equation (7 anayticay and pug them into equation (3. From there, we can derive the so-caed dispersion reation, which is the foowing eigenprobem (Ǎ pqk x + ˇB pqk y + Č pqk z i Ě pq U q = ω U q, p,q =,..., n v, (7 with i =. Soving the eigenprobem (7 gives us the matrix R pq of right eigenvectors R p,..., R pnv and the associated eigenvaues ω p. Recaing, e.g. from (Toro 999, that each soution of the inear hyperboic system (3 is given by a inear combination of the right

16 Michae Dumbser, Martin Käser, Josep de a Puente Tabe 5. Convergence rates of veocity component v for ADER-FV O to ADER-FV O schemes. Viscoeastic attenuation is modeed using Þve mechanisms. h L O L L O L I CPU [s] eigenvectors, i.e. U p = R pqν q, we can compute the coefþcients as ν p = Rpq Uq via the initia ampitude vector. Now, we can synthesize the exact soution of the attenuated pane wave in the form U p(x, y, z, t =R pqν q e i (ωqt kxx kyy kzz. (73 In the convergence test, we use a pane P-wave and a pane S-wave traveing in opposite directions aong the space diagona n =(,, T of the domain Ω as aready shown in (Dumbser & Käser a. Therefore, the initia condition at t =is given by (73 using the combination of ony two right eigenvectors (R p,..., R p9 with the coefþcients ν = ν 9 =and zero otherwise. The tota simuation time T is set to T =.s. The CFL number is set in a computations to.5. For a thorough investigation of the inear stabiity properties of ADER-FV schemes via differentia approximation and via von Neumann stabiity anaysis see (Dumbser, Schwartzkopff & Munz. The numerica anaysis to determine the convergence orders is performed on a sequence of tetrahedra meshes as shown in Figure 4. The mesh sequence is obtained by dividing the computationa domain Ω into a number of sub-cubes, which are then subdivided into Þve tetrahedrons as shown in Figure 4. This way, the reþnement is controed by changing the number of sub-cubes in each space dimension. We can arbitrariy pick one of the variabes of the system of the seismic wave equations (3 to numericay determine the convergence order of the used ADER-FV schemes. In Tabe 5 we show the error for the veocity component v. The error of the reconstructed numerica soution W h with respect to the exact soution U e is measured in the L -norm and the continuous L -norm ( W h U e L (Ω = W h U e dv, (74 where the integration is approximated by Gaussian integration which is exact for a poynomia degree twice that of the basis functions of the numerica scheme. The L -norm is approximated by the maximum error arising at any of these Gaussian integration points. The Þrst coumn in Tabe 5 shows the mesh spacing h, represented by the maximum diameter of the circumscribed spheres of the tetrahedrons. The foowing four coumns show the L and L errors with the corresponding convergence orders O L and O L determined by successivey reþned meshes. In the ast two coumns we give the number I of iterations and the CPU times in seconds needed to reach the simuation time T =.s on one Pentium Xeon 3. GHz processor with 4GB of RAM. Ω 7 COMPARISON OF ADER-FV AND ADER-DG SCHEMES In this section we provide a thorough comparison of the ADER-FV schemes presented in this paper for the aneastic wave equations and the ADER-DG method proposed for the same equations previousy in (Käser, Dumbser, de a Puente & Ige. First, we reca the fuy discrete versions of both schemes in equations (75 and (7, respectivey.

17 Arbitrary High Order Finite Voume Schemes for Seismic Wave Propagation 7 Tabe. Convergence rates of veocity component v for the ADER-DG O4 scheme. Viscoeastic attenuation is modeed using Þve mechanisms. h L O L L O L I CPU [s] The ADER Finite Voume scheme for the aneastic wave equations deveoped in this artice reads as [ T (m ( ū ] (m p t n+ ū (m p (t n + + N E Ť pr rs + rs (Ťsq Sj F,j I qnm ( t ŵ nm (m (t n j= + + N E Ť pr rs + rs (Ťsq Sj F +,i,h I qnm ( t ŵ (k j nm (t n = T (m Ě pq (m I qnm( t ŵ nm (m (t n. j= (75 We reca that the ADER Discontinuous Gaerkin method for the aneastic wave equations presented in (Käser, Dumbser, de a Puente & Ige has the foowing form: [ J M k û (m ( ] p t n+ û (m p (t n + + N E Ť j pr rs + rs (Ť sq j S j F,j k I qnm ( t û (m nm (t n + + j= N E Ťpr j j= (Ǎ(m rs Ǎ(m rs (Ť j sq S j F +,j,i,h k I qnm ( t û (k j nm (t n Ǎ pq J K ξ k I qnm( t û (m nm (t n ˇB pq J K η k I qnm( t û (m nm (t n Č pq J K ζ k I qnm( t û (m nm (t n = J M k Ě (m pq I qmn ( t û (m mn (t n. (7 From the fuy discrete version of both schemes we can immediatey deduce those features that both schemes have in common as we as their differences, which wi have an important impact on CPU time and memory requirements of both methods. Both schemes are one-step methods, i.e. they directy integrate the governing equation (3 from time eve t n to time eve t n+ without any intermediate stages. This is possibe thanks to the Cauchy-Kovaewski procedure that is identica in both methods. However, in the ADER-FV scheme (75 the Cauchy-Kovaewski procedure is appied to the reconstructed degrees of freedom ŵ nm whereas in the ADER-DG scheme (7 this procedure can be directy appied to the degrees of freedom û nm aready given by the spatia DG discretisation. Except of this difference, the method for carrying out the Cauchy-Kovaewski procedure is the same in both schemes, see the tensor I qnm ( t appearing in both methods. Furthermore, both methods are competey quadrature-free since a spatia integras are computed anayticay and then stored in the ßux matrices. However, the ßux matrices do not have the same size for ADER-DG and ADER-FV schemes. This is due to a very important difference that distinguishes both approaches: whereas in the ADER-DG scheme a poynomia coefþcients û p (matrix of size n v L are evoved in time, the ADER-FV scheme ony advances the ce averages ū p (vector of ength n v, see (75 and (7. We repeat that n v is the number of variabes in the system of governing equations and L = (M +(M +and L = (M +(M +(M +3are the number of degrees of freedom in two and three space dimensions, respectivey. This means, that the fuy discrete system for the ADER-DG scheme is by a factor of L arger than the corresponding system for ADER-FV schemes. This is aso reßected in the size of the ßux matrices, which are simpe vectors of ength L in the case of ADER-FV schemes and matrices of size L L for ADER-DG schemes. The arge ßux matrices are the Þrst key factor eading to the much arger CPU time observed for ADER-DG in comparison to ADER-FV. As a side note we remark that due to the representation of the reconstructed soution W of the Finite Voume scheme in terms of the same basis functions as for the Discontinuous Gaerkin schemes, the ßux matrix of ADER-FV schemes is identica to the Þrst row of the corresponding ADER- DG ßux matrix. In addition to the arger ßux matrices, in the ADER-DG agorithm aso stiffness matrices appear due to the non-vanishing voume integra after the integration by parts operation in the derivation of the scheme, see e.g. (Cockburn, Karniadakis & Shu ; Käser & Dumbser a; Dumbser & Käser a. We see that the stiffness matrices K ξ k, Kη k and Kζ k appear in (7 but not in (75. This adds an additiona CPU effort for the ADER-DG scheme, however, it is very sma compared to the ßux computation since the stiffness matrices are very sparse and therefore can be mutipied very efþcienty. The second and even more important key factor eading to the much arger CPU times of ADER-DG compared to ADER-FV is the unfavourabe time step restriction that comes out of a von Neumann stabiity anaysis, see e.g. (Dumbser 5. In genera, one can roughy M+ say that the ADER-DG time step imit decreases proportiona to, where M is the degree of the DG basis poynomias. For the Finite Voume scheme, the time step imit is independent of the degree M of the reconstruction basis poynomias, or even becomes arger, at east

18 8 Michae Dumbser, Martin Käser, Josep de a Puente on Cartesian grids in mutipe space dimensions. For detais on this topic see (Dumbser, Schwartzkopff & Munz. This means that for exampe a Þfth order ADER-FV scheme has a time step imit that is eeven times arger than the time step imit of a Þfth order ADER-DG scheme on the same mesh. Athough this disadvantage seems to be so heavy that ADER-DG schemes may suddeny appear very unattractive from this point of view, we wi expain ater in this section that ADER-DG sti has very strong advantages over ADER-FV which makes them at east competitive against ADER-FV, if not superior. Before discussing the inconveniences of ADER-FV schemes, we woud ike to make some remarks on the MPI paraeization of both schemes on modern massivey parae systems. Since both methods are one-step schemes, the tota communication overhead is consideraby ow compared to methods using high order Runge-Kutta time integration since in the ADER approach data has to be exchanged ony at the beginning of each time step. Then, each subdomain can evove the soution independenty of its neighbour domains. Unstructured mesh partitioning is done for both schemes with the free METIS software package described in (Karypis & Kumar 998. Whereas ADER-DG schemes must exchange ony the degrees of freedom û p of the direct neighbours of a subdomain boundary, ADER-FV schemes must exchange aso a the ce averages ū p necessary for the reconstruction procedure. In our impementation, we decided to spit the MPI communications for ADER-FV into two parts. First, a the necessary ce averages ū p needed for reconstruction in each subdomain are exchanged. Then, each subdomain performs the reconstruction and Þnay a subdomains exchange the reconstructed degrees of freedom ŵ p ony at the direct neighbours of a subdomain boundary, exacty as in the ADER-DG case. Therefore, the communication overhead is higher for ADER-FV, but since many of the reconstruction stencis are overapping, the overhead is not arger than a factor of two. ProÞing has shown that both parae codes scae quite we with the number of processors. Typica runs of the parae ADER-FV and ADER-DG codes on the SGI Atix custer of the LRZ supercomputing center in München and on the NEC Linux custer of the HLRS supercomputing center in Stuttgart use between 4 and 8 processors. The main inconvenience of ADER-FV schemes is the necessary reconstruction procedure in order to provide high order accuracy in space from the given ce averages. This step is not necessary for ADER-DG schemes since they directy evove a poynomia coefþcients in time. Athough reconstruction can become quite cumbersome on unstructured tetrahedra meshes in three space dimensions, the systematic framework presented in (Dumbser & Käser b and outined aso in this artice is sti quite easy to impement due to the use of hierarchica orthogona reconstruction basis functions Ψ and the transformation of the reconstruction stencis to a reference coordinate system aigned with the centra eement. The resuting agorithm is robust and in particuar it is aso cost-efþcient. As we wi see ater when anayzing in detai the CPU times of a steps required by ADER-DG and ADER-FV, the reconstruction operator is sti ess expensive than the ADER-DG ßux computation. To increase computationa efþciency, we store the inverse of the reconstruction matrix in (45 for each eement and then mutipy the vector of ce averages in the stenci S (m with this inverse matrix. Athough this speeds up reconstruction consideraby, the associated memory oad is quite high. Most of the memory requirements of our proposed ADER-FV scheme are due to the storage of these inverse reconstruction matrices. Due to the east squares reconstruction approach using twice the number of necessary eements in 3D, the inverse reconstruction matrices have the size L L. Fortunatey, they have to be stored ony once for each eement, independent of the number of variabes n v of the system (3. After this genera discussion, we woud ike to show the reader in more detai the difference in CPU time requirements of ADER-DG and ADER-FV. On this behaf we run the same convergence test as shown in the previous Section on the same mesh once with an ADER-DG scheme and once with an ADER-FV scheme. The computer system is in both cases the same Inte Xeon workstation with 3. GHz and 4 GB of RAM. Both codes run in their seria version, without making use of the sparsity of the fux matrices. We then take the detaied function timing proþe generated by the compier after the run. The CPU times needed for the reconstruction, the Cauchy-Kovaewski procedure and the ßux computation are presented in Tabes 7 and 8 per eement and time step. We note that for ADER-DG no reconstruction is necessary and that the times for ßux computation aso incude the stiffness matrices and the evauation of the source term on the right hand side of (7. A the points mentioned in the previous genera discussion are conþrmed by the resuts presented in Tabes 7 and 8. The reconstruction is ony used for ADER-FV schemes and the Cauchy-Kovaewski procedure is the same in both methods and therefore aso needs roughy the same CPU time. Furthermore, the ßux computation of ADER-DG is much more expensive than the one for the ADER-FV scheme due to the arger ßux matrices and the presence of the stiffness terms. However, we woud ike to note that for ADER-DG schemes about 5 % of the CPU time used for ßux computations can be saved making use of the sparsity of the ßux matrices. The CPU times presented in (Dumbser & Käser a; Käser, Dumbser, de a Puente & Ige and in Tabe use this property in order to speed up the code. We Þnay have to consider the difference in the time step imit, which expains the different number of iterations and the arge discrepancy in CPU times shown in Tabes 5 and. However, especiay having a ook at the error norms presented in Tabes 5 and we have to emphasize that ADER-DG schemes are much more accurate than ADER-FV methods. This aows consideraby coarser meshes for ADER- DG to reach the same precision as ADER-FV which at the same time reduces computationa effort due to the reduced number of eements and due to the arger time step induced by the coarser mesh. Comparing the CPU times given in this artice for ADER-FV schemes, see Tabe 5 and comparing with the CPU times obtained for the same test probem using ADER-DG schemes, see (Käser, Dumbser, de a Puente & Ige and Tabe, we note that on the same mesh the fourth order Finite Voume schemes are about ten times faster than the corresponding ADER-DG schemes. However, the ADER-DG method is aso about ten times more accurate. At the end, the considerabe advantage in accuracy makes the ADER-DG scheme superior to ADER-FV schemes comparing the CPU time needed by both methods at the same eve of accuracy. However, there may be important reaistic appications where the coarse meshes needed by ADER-DG to be competitive with ADER-FV are not reaizabe. This is the case, for exampe, when sma geometrica features have to be resoved by the mesh, such as compex surface topography, compex ayered sediment

19 Arbitrary High Order Finite Voume Schemes for Seismic Wave Propagation 9 Tabe 7. Detaied CPU time comparison for the individua steps necessary for ADER-FV and ADER-DG schemes in 3D per eement and time step for the purey eastic wave equations, without attenuation. ADER-FV O4 ( t FV =.4E- ADER-DG O4 ( t DG =3.77E-3 CPU time percentage CPU time percentage Reconstruction. ms % ms % Cauchy-Kovaewski procedure. ms 4 %. ms % Fux & stiffness computation.7 ms 5 %.3 ms 9 % Tota.48 ms %.4 ms % ADER-FV O ( t FV =.4E- ADER-DG O ( t DG =.4E-3 CPU time percentage CPU time percentage Reconstruction.53 ms 35 % ms % Cauchy-Kovaewski procedure.53 ms 35 %.5 ms % Fux & stiffness computation.4 ms 3 % 3.99 ms 88 % Tota.5 ms % 4.54 ms % Tabe 8. Detaied CPU time comparison for the individua steps necessary for ADER-FV and ADER-DG schemes in 3D per eement and time step for the aneastic wave equations using 5 attenuation mechanisms. ADER-FV O4 ( t FV =.4E- ADER-DG O4 ( t DG =3.77E-3 CPU time percentage CPU time percentage Reconstruction.34 ms 5 % ms % Cauchy-Kovaewski procedure.55 ms 4 %.5 ms 7 % Fux & stiffness computation.48 ms 35 %.47 ms 73 % Tota.3 ms %.3 ms % ADER-FV O ( t FV =.4E- ADER-DG O ( t DG =.4E-3 CPU time percentage CPU time percentage Reconstruction. ms 8 % ms % Cauchy-Kovaewski procedure 4.9 ms 57 % 4.3 ms 39 % Fux & stiffness computation.7 ms 5 %.3 ms % Tota 7. ms %.49 ms % structures embedded in the mode or aso thin ayers with different materia properties. In a these cases, the Þna mesh resoution is more or ess given aready at the beginning of the simuation due to the requirement of resoving a the sma features. Since ADER-FV is much faster than ADER-DG on the same mesh, though ess accurate, it may be the preferabe method of choice in such cases. Since both methods are abe to treat unstructured meshes and since both schemes have many common parts, we are running the ADER-FV and ADER-DG schemes in the same software package in order to be ßexibe to opt either for ADER-DG or ADER-FV, depending on the requirements of the test case. In the foowing section, we present the resuts obtained with ADER-FV for severa standard benchmark probems. 8 APPLICATION EXAMPLES We appy the proposed ADER-FV method on we-deþned two- and three-dimensiona test probems for which aso anaytic reference soutions are avaiabe. The D benchmark is the we-known Lamb s probem (Lamb 94 in the same setup as given in (Komatitsch & Viotte 998 and (Käser & Dumbser a to verify the accuracy of the scheme at the free-surface boundary. The two 3D benchmark probems LOH. and LOH.3 were pubished in the Þna report of the LIFELINES PROGRAM TASK A (Day, Bieak, Dreger, Graves, Larsen, Osen & Pitarka 3 of the PaciÞc Earthquake Engineering Research Center and are part of a muti-institutiona code vaidation project of a series of different numerica methods empoyed in numerica modeing of earthquake ground motion in three-dimensiona earth modes. Therefore, besides a quasi-anaytic soution, simuation resuts from four different we-estabished codes exist and serve as additiona reference soutions. Furthermore, reference soutions are provided by the ADER-DG scheme proposed by the authors in (Käser & Dumbser a; Dumbser & Käser a; Käser, Dumbser, de a Puente & Ige. Both LOH test cases contain a heterogeneous ayered medium with a free surface boundary condition. Whereas LOH. soves ony the purey eastic wave equations without attenuation, the LOH.3 benchmark aso incudes viscoeastic behaviour with its associated attenuation and dispersion mechanisms. The resuts of the four

20 Michae Dumbser, Martin Käser, Josep de a Puente reference codes given in (Day, Bieak, Dreger, Graves, Larsen, Osen & Pitarka 3 are denoted by four-character abbreviations indicating the respective institutions: - UCBL (Doug Dreger and Shawn Larsen, University of Caifornia, Berkeey/Lawrence Livermore Nationa Laboratory, - UCSB (Kim Osen, University of Caifornia, Santa Barbara, - WCC (Arben Pitarka, URS Corporation, and - CMUN (Jacobo Bieak, Carnegie-Meon University. The Þrst three codes use Finite Differences on uniform structured grids with staggered ocations of the veocity and stress components and fourth order accuracy in space. The CMUN code uses piecewise inear interpoation on unstructured tetrahedra Finite Eements. The quasianaytic soution is a frequency-wavenumber soution obtained by a modiþcation of the method presented in (Luco & Apse 983; Apse & Luco 983 and is compared to a numerica soutions to evauate their accuracy. 8. Lamb s Probem in Two Space Dimensions A cassica test case to vaidate the impementation of free surface boundary conditions and point sources is Lamb s probem (Lamb 94, consisting in a vertica (with respect to the surface point force acting on the free surface. The soution of Lamb s probem for a pane surface can be computed anayticay, see e.g. (Piant 979 and can hence be used to assess the quaity of numerica methods. In this paper we use the FORTRAN code EXDDIR of Berg and If (994 to compute the exact two-dimensiona soution of the seismic response from a vertica directiona point source in an eastic haf space with a free surface. The code EXDDIR is based on the Cagniard-de Hoop technique (de Hoop 9 and aows the use of an arbitrary source time function for dispacements or veocities. Considering the accuracy of a numerica method with respect to the correct treatment of sources and the free-surface boundary condition, Lamb s probem poses a chaenging test case in particuar because of the Rayeigh waves propagating aong the free surface. The setup of the physica probem is chosen as in the paper of Komatitsch and Viotte (998, who soved this probem using the Spectra Eement method, see e.g. (Komatitsch & Tromp 999; Komatitsch & Tromp. Furthermore, this probem was soved in (Käser & Dumbser a on very coarse trianguar meshes using a tenth order ADER-DG scheme. We use a homogeneous eastic medium with a P-wave veocity of c p = 3 ms, an S-wave veocity of c s = ms and a mass density of ρ = kg m 3. The numerica mode with origin (, at the eft bottom corner is 4 m wide and has a height of m on the eft boundary. The tit ange of the free surface is φ =. The directiona point source, acting as a force perpendicuar to this tited surface, is ocated at the free surface at x s 7., 33.8 T. The two receivers are ocated at (94.9, and (34.8, such that their distances from the source aong the surface are 99 m and 7 m, respectivey. On the eft, right and bottom boundaries of the mode we use then absorbing boundary conditions as described in Section 5. We use a sixth order ADER-FV scheme on a trianguar mesh buit in such a way that the free surface boundary at the top is resoved with 8 trianges, the eft and right boundaries of the mode are discretised using 75 trianges reþned towards the surface, and at the bottom 5 eements are used. The resuting mesh consists of 943 trianges, which is about 57 times more than the mesh used in (Käser & Dumbser a for the tenth order ADER- DG scheme using ony 34 trianges. In order to avoid undesired effects of possiby reßected wave energy at the right mode boundary, we extended the mesh up to a width of 47 m for the numerica computations. The source time function that speciþes the tempora variation of the point source is a Ricker waveet given by S T ( (t =a.5+a(t t D e a (t t D, (77 where t D =.8 s is the source deay time and a = kg m s and a = (πf c are constants determining the ampitude and frequency of the Ricker waveet of centra frequency f c =4.5 Hz. The wave propagation is simuated unti time T end =.3 s when a waves have aready passed the two receivers. We repeat that for the resuts shown in this paper, a sixth order ADER-FV O scheme with a Courant number of CFL =.5 is used. In order to reach the Þna simuation time T end =.3 we need 333 time steps. In Figure 5 we present the snapshots of the absoute vaue of the veocity vector of the seismic wave Þed at t =.48s. In Figure we present the unscaed seismograms obtained from our numerica simuations, as recorded by receiver and, respectivey, together with the anaytic soution provided by EXDDIR. The anaytic and numerica soutions match very we, such that the ines are hardy distinguishabe on this scae. Therefore, the difference between anaytic and numerica soution is aso potted. The maximum reative error on the reguar mesh remains aways ess than 5%. We concude from this exampe that the accurate soution of Lamb s probem with the ADER-FV method proposed in this artice conþrms that the impementation of free surface boundary conditions as suggested in Section 5. eads to the correct physica behaviour of eastic surface waves. We emphasize that the tenth order ADER-DG scheme presented in (Käser & Dumbser a was abe to produce errors of ess than % on the much coarser mesh for the same test probem. This underines the important trade-off between speed and accuracy of ADER-FV and ADER-DG schemes, respectivey, that aways has to be taken into account. 8. Layer Over Hafspace Test Cases LOH. and LOH.3 in Three Space Dimensions The setup of the test probems LOH. and LOH.3 (Layer Over Hafspace is shown in Figure 7(a, where for carity ony one of four symmetrica quarters of the compete computationa domain Ω=[ 5m, 5m] [ 5m, 5m] [m, 7m] is potted.