Geodesic
Analysis of a high-order space and time discontinuous Galerkin method for elastodynamic equations. Application to 3D wave propagation
Delcourte, Sarah1, 2 ; Glinsky, Nathalie3, 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

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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 :
DOI : 10.1051/m2an/2015001
Classification : 35L50, 35F10, 35F15, 35L05, 35Q99
Keywords: Discontinuous Galerkin method, centered flux, leap-frog scheme, elastodynamic equation

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. 
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     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},
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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

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