Analysis of a high-order space and time discontinuous Galerkin method for elastodynamic equations. Application to 3D wave propagation

Delcourte, Sarah; Glinsky, Nathalie

doi:10.1051/m2an/2015001

Delcourte, Sarah ^1,² ; Glinsky, Nathalie ^3,⁴

¹ Universitéde Lyon, CNRS UMR5208, Université Lyon 1, Institut Camille Jordan, 43 blvd du 11 novembre 1918, 69622 Villeurbanne cedex, France.
² UCBL/INRIA Grenoble Rhône-Alpes/INSMI − KALIFFE, France.
³ IFSTTAR/CEREMA, DTer Méd., 56 boulevard Stalingrad, 06359 Nice cedex 4, France.
⁴ INRIA Sophia Antipolis Méditerranée, team Nachos, 2004 route des Lucioles, 06902 Sophia Antipolis cedex, France.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 1085-1126.

Résumé

In this paper, we introduce a high-order discontinuous Galerkin method, based on centered fluxes and a family of high-order leap-frog time schemes, for the solution of the 3D elastodynamic equations written in velocity-stress formulation. We prove that this explicit scheme is stable under a CFL type condition obtained from a discrete energy which is preserved in domains with free surface or decreasing in domains with absorbing boundary conditions. Moreover, we study the convergence of the method for both the semi-discrete and the fully discrete schemes, and we illustrate the convergence results by the propagation of an eigenmode. We also propose a series of absorbing conditions which allow improving the convergence of the global scheme. Finally, several numerical applications of wave propagation, using a 3D solver, help illustrating the various properties of the method.

Reçu le : 2012-07-11

MR Zbl

DOI : 10.1051/m2an/2015001

Classification : 35L50, 35F10, 35F15, 35L05, 35Q99
Mots clés : Discontinuous Galerkin method, centered flux, leap-frog scheme, elastodynamic equation

Affiliations des auteurs :

Delcourte, Sarah ^{1, 2} ; Glinsky, Nathalie ^{3, 4}

@article{M2AN_2015__49_4_1085_0,
     author = {Delcourte, Sarah and Glinsky, Nathalie},
     title = {Analysis of a high-order space and time discontinuous {Galerkin} method for elastodynamic equations. {Application} to {3D} wave propagation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1085--1126},
     publisher = {EDP-Sciences},
     volume = {49},
     number = {4},
     year = {2015},
     doi = {10.1051/m2an/2015001},
     mrnumber = {3371905},
     zbl = {1320.74101},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2015001/}
}

TY  - JOUR
AU  - Delcourte, Sarah
AU  - Glinsky, Nathalie
TI  - Analysis of a high-order space and time discontinuous Galerkin method for elastodynamic equations. Application to 3D wave propagation
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2015
SP  - 1085
EP  - 1126
VL  - 49
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2015001/
DO  - 10.1051/m2an/2015001
LA  - en
ID  - M2AN_2015__49_4_1085_0
ER  -

%0 Journal Article
%A Delcourte, Sarah
%A Glinsky, Nathalie
%T Analysis of a high-order space and time discontinuous Galerkin method for elastodynamic equations. Application to 3D wave propagation
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2015
%P 1085-1126
%V 49
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2015001/
%R 10.1051/m2an/2015001
%G en
%F M2AN_2015__49_4_1085_0

Delcourte, Sarah; Glinsky, Nathalie. Analysis of a high-order space and time discontinuous Galerkin method for elastodynamic equations. Application to 3D wave propagation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 1085-1126. doi : 10.1051/m2an/2015001. http://www.numdam.org/articles/10.1051/m2an/2015001/

Bibliographie
Cité par

S. Adjerid, K. Devine, J. Flaherty and L. Krivodonova, A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. Comput. Methods Appl. Mech. Engrg. 191 (2002) 1097–1112. | DOI | MR | Zbl

C. Agut, J. Diaz and A. Ezziani, High-order schemes combining the modified equation approach and discontinuous Galerkin approximations for the wave equation. Commun. Comput. Phys. 11 (2012) 691–708. | DOI | MR | Zbl

M. Ainsworth, Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. J. Comput. Phys. 19 (2004) 106–130. | DOI | MR | Zbl

M. Ainsworth, P. Monk and W. Muniz, Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation. J. Sci. Comput. 27 (2006) 5–40. | DOI | MR | Zbl

P.F. Antonietti, I. Mazzieri, A. Quarteroni and F. Rapetti, Non-conforming high order approximations of the elastodynamics equation. Comput. Methods Appl. Mech. Engrg. 209–212 (2012) 212–238. | DOI | MR | Zbl

S. Aoi and H. Fujiwara, 3D Finite-difference method using discontinuous grids. Bull. Seism. Soc. Am. 89 (1999) 918–930. | DOI

Z. Alterman and F.C. Karal, Propagation of elastic waves in layered media by finite-difference methods. Bull. Seism. Soc. Am. 58 (1968) 367–398.

H. Bao, J. Bielak, O. Ghattas, L. Kallivokas, D.R. O’Hallaron, J.R. Schewchuk and J. Xu, Large-scale simulation of elastic wave propagation in heterogeneous media on parallel computers. Comput. Methods Appl. Mech. Engrg. 152 (1998) 85–102. | DOI | Zbl

A. Bayliss, K.E. Jordan, B.J. Lemesurier and E. Turkel, A fourth-order accurate finite-difference scheme for the computation of elastic waves. Bull. Seism. Soc. Am. 76 (1986) 1115–1132. | DOI

M. Benjemaa, Étude et simulation numérique de la rupture dynamique des séismes par des méthodes d’éléments finis discontinus. Ph.D. thesis, Nice-Sophia Antipolis University (2007).

M. Benjemaa, S. Piperno and N. Glinsky-Olivier, Étude de stabilité d’un schéma volumes finis pour les équations de l’élasto-dynamique en maillages non structurés, INRIA report 5817 (2006).

M. Bouchon and K. Aki, Discrete wave-number representation of seismic-source wave fields. Bull. Seism. Soc. Am. 67 (1977) 259–277. | DOI

T. Bui-Thanh and O. Ghattas, Analysis of an hp-nonconforming discontinuous Galerkin spectral element method for wave propagation. SIAM J. Numer. Anal. 50 (2012) 1801–1826. | DOI | MR | Zbl

E. Chaljub, Y. Capdeville and J.P. Vilotte, Solving elastodynamics in a fluid–solid heterogeneous sphere: a parallel spectral element approximation on non–conforming grids. J. Comput. Phys. 187 (2003) 457–491. | DOI | MR | Zbl

Y. Cheng and C.-W. Shu, Superconvergence and time evolution of discontinuous Galerkin finite element solutions. J. Comput. Phys. 227 (2008) 9612–9627. | DOI | MR | Zbl

N. Chevaugeon, K. Hillewaert, X. Gallez, P. Ploumhans and J.-F. Remacle, Optimal numerical parameterization of discontinuous Galerkin method applied to wave propagation problems. J. Comput. Phys. 223 (2007) 188–207. | DOI | MR | Zbl

P. Ciarlet, The finite element method for elliptic problems. North Holland-Elsevier science publishers, Amsterdam, New York, Oxford (1978). | MR | Zbl

B. Cockburn and C.W. Shu, TVB Runge−Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52 (1989) 411–435. | MR | Zbl

B. Cockburn and C.W. Shu, The Runge−Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141 (1998) 199–224. | DOI | MR | Zbl

B. Cockburn, S.Y. Lin and C.W. Shu, TVB Runge−Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. J. Comput. Phys. 84 (1989) 90–113. | DOI | MR | Zbl

B. Cockburn, S. Hou and C.W. Shu, The Runge−Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comput. 54 (1990) 545–581. | MR | Zbl

B. Cockburn, G.E. Karnadiakis and C.W. Shu, Discontinuous Galerkin Methods: Theory, Computation and Application. Lect. Notes Comput. Sci. Engrg. Springer, Berlin (2000). | MR | Zbl

J.D. De Basabe, M.K. Sen and M. Wheeler, The interior penalty discontinuous Galerkin method for elastic wave propagation: grid dispersion. Geophys. J. Int. 175 (2008) 83–93. | DOI

L. Demkowicz and J. Kurtz, Projection-based interpolation and automatic hp-adaptivity for finite element discretizations of elliptic and Maxwell problems. ESAIM: Proc. 21 (2007) 1–15. | DOI | MR | Zbl

S. Delcourte, L. Fezoui and N. Glinsky-Olivier, A high-order Discontinuous Galerkin method for the seismic wave propagation. ESAIM Proc. 27 (2009) 70–89. | DOI | MR | Zbl

M. Dumbser and M. Käser, An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes II: the three-dimensional isotropic case. Geophys. J. Int. 167 (2006) 319–336. | DOI

M. Dumbser, M. Käser and J. De La Puente, Arbitrary high order finite volume schemes for seismic wave propagation on unstructured meshes in 2D and 3D. Geophys. J. Int. 171 (2007) 665–694. | DOI

V. Etienne, E. Chaljub, J. Virieux and N. Glinsky, An hp-adaptive discontinuous Galerkin finite-element method for 3-D elastic wave modelling. Geophys. J. Int. 183 (2010) 941–962. | DOI

H. Fahs, High-order Leap-Frog based discontinuous Galerkin method for the time-domain Maxwell equations on non-conforming simplicial meshes. Numer. Math. Theor. Methods Appl. 2 (2009) 275–300. | DOI | MR | Zbl

L. Fezoui, S. Lanteri, S. Lohrengel and S. Piperno, Convergence and stability of a Discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes. ESAIM: M2AN 39 (2005) 1149–1176. | DOI | Numdam | MR | Zbl

J.S. Hesthaven and T. Warburton, Nodal discontinuous Galerkin methods: algorithms, analysis and applications. Texts Appl. Math. 54 (2008). | DOI | MR | Zbl

F.Q. Hu, M.Y. Hussaini and P. Rasetarina, An analysis of the discontinuous Galerkin method for wave propagation problems. J. Comput. Phys. 151 (1999) 921–946. | DOI | Zbl

M. Galis, P. Moczo and J. Kristek, A 3-D hybrid finite-difference–finite-element viscoelastic modelling of seismic wave motion. Geophys. J. Int. 175 (2008) 153–184. | DOI

D. Givoli, High-order local non-reflecting boundary conditions: a review. Wave Motion 39 (2004) 319–326. | DOI | MR | Zbl

N. Glinsky, S. Moto Mpong and S. Delcourte, A high-order discontinuous Galerkin scheme for elastic wave propagation. INRIA report No. 7476 (2010).

C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equations. Math. Comput. 46 (1986) 1–26. | DOI | MR | Zbl

C. Johnson, U. Nävert and J. Pitkäranta, Finite element methods for linear hyperbolic equations. Comput. Methods Appl. Mech. Engrg. 45 (1984) 285–312,. | DOI | MR | Zbl

M. Käser and M. Dumbser, An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes I: the two-dimensional isotropic case with external source term. Geophys. J. Int. 162 (2006) 855–877. | DOI

M. Käser, M. Dumbser, J. De La Puente and H. Igel, An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes III: viscoelastic attenuation. Geophys. J. Int. 168 (2007) 224–242. | DOI

M. Käser, V. Hermann and J. De La Puente, Quantitative accuracy analysis of the discontinuous Galerkin method for seismic wave propagation. Geophys. J. Int. 173 (2008) 990–999,. | DOI

K.R. Kelly, R.W. Ward, S. Treitel and R.M. Alford, Synthetic seismograms: a finite-difference approach. Geophys. 41 (1976) 2–27. | DOI

D. Komatitsch and J.P. Vilotte, The spectral-element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures. Bull. Seism. Soc. Am. 88 (1998) 368–392.

T. Lähivaara and T. Huttunen, A non-uniform basis order for the discontinuous Galerkin method of the acoustic and elastic wave equations. Appl. Numer. Math. 61 (2011) 473–486. | DOI | MR | Zbl

P. Lesaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical Aspects of Finite Element Methods in Partial Differential Equations, edited by C.A. deBoor. Academic Press, New York (1974) 89–123. | MR | Zbl

A.R. Levander, Fourth-order finite-difference P-SV seismograms. Geophys. 53 (1988) 1425–1436. | DOI

J. Lysmer and L.A. Drake, A finite element method for seismology, in Methods of Computational Physics, edited by B.A. Bolt. Academic Press, New York 11 (1972) 181–216.

R. Madariaga, Dynamics of an expanding circular fault. Bull. Seis. Soc. Am. 66 (1976) 639–666. | DOI

K.J. Marfurt, Accuracy of finite-difference and finite-element modelling of the scalar and elastic wave equations. Geophys. 49 (1984) 533–549. | DOI

I. Mazzieri and F. Rapetti, Dispersion analysis of triangle-based spectral elements methods for elastic wave propagation. Numer. Algorithms 60 (2012) 631–650. | DOI | MR | Zbl

E.D. Mercerat, J.-P. Vilotte and F. Sanchez-Sesma, Triangular spectral element simulation of 2D elastic wave propagation using unstructured triangular grids. Geophys. J. Int. 166 (2006) 679–698. | DOI

P. Moczo, E. Bystrický, J. Kristek, J.M. Carcione and M. Bouchon, Hybrid modeling of P-SV seismic motion at inhomogeneous viscoelastic topographic structures. Bull. Seism. Soc. Am. 87 (1997) 1305–1323.

C. Pelties, J. De La Puente, J.P. Ampuero, G. Brietzke and M. Käser, Three-dimensional dynamic rupture simulations with a high-order discontinuous Galerkin method on unstructured tetrahedral meshes. J. Geophys. Res. 117 (2012) B02309.

F. Peyrusse, N. Glinsky, C. Gélis and S. Lanteri, A nodal discontinuous Galerkin method for site effects assessment in viscoelastic media. Verification and validation in the Nice basin. Geophys. J. Int. (2014) 315–334.

A. Pitarka, 3D elastic finite-difference modeling of seismic motion using staggered-grids with nonuniform spacing. Bull. Seism. Soc. Am. 89 (1999) 85–106. | DOI | MR

W. Reed and T. Hill, Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, NM (1973).

C. Scheid and S. Lanteri, Convergence of a Discontinuous Galerkin scheme for the mixed time domain Maxwell’s equations in dispersive media. INRIA Report 7634 (2011).

E.H. Saenger, N. Gold and S.A. Shapiro, Modeling the propagation of elastic waves using a modified finite-difference grid. Wave Motion 31 (2000) 77–92. | DOI | MR | Zbl

S. Sherwin, Dispersion analysis of the continuous and discontinuous Galerkin formulation. Lect. Notes Comput Sci. Engrg. 11 (2000) 425–432. | DOI | MR | Zbl

E. Süli, C. Schwab and P. Houston, hp-DGFEM for partial differential equations with non-negative characteristics form, in Discontinuous Galerkin Methods Theory. Computation and Applications, edited by B. Cockburn, G.E. Karnadiakis and C.W. Shu. In vol. 11 of Lect. Notes Comput. Sci. Eng. Springer, Berlin (2000) 221–230. | MR | Zbl

J. Tago, V.M. Cruz-Atienza, J. Virieux, V. Etienne and F.J. Sanchez-Sesma, A 3D hp-adaptive discontinuous Galerkin method for modelling earthquake dynamics. J. Geophys. Research: Solid Earth 117 (2012).

J. Virieux, P-SV wave propagation in heterogeneous media, velocity-stress finite difference method. Geophys. 51 (1986) 889–901. | DOI

J.L. Young, High-order, leapfrog methodology for the temporally dependent Maxwell’s equations. Radio Sci. 36 (2001) 9–17. | DOI

L.C. Wilcox, G. Stadler, C. Burstedde and O. Ghattas, A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media. J. Comput. Phys. 229 (2010) 9373–9396. | DOI | MR | Zbl

X. Zhong and C.-W. Shu, Numerical resolution of discontinuous Galerkin methods for time dependent wave equations. Comput. Methods Appl. Mech. Engrg. 200 (2011) 2814–2827. | DOI | MR | Zbl

Cité par Sources :