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High-Order Schemes Combining the Modified Equation Approach and Discontinuous Galerkin Approximations for the Wave Equation

Published online by Cambridge University Press:  20 August 2015

Cyril Agut ,
Julien Diaz  and
Abdelaâziz Ezziani
Cyril Agut*
Affiliation:
LMA, CNRS UMR 5142, Université de Pau, France INRIA Bordeaux Research Center, Project Team Magique3D, France
Julien Diaz*
Affiliation:
LMA, CNRS UMR 5142, Université de Pau, France INRIA Bordeaux Research Center, Project Team Magique3D, France
Abdelaâziz Ezziani
Affiliation:
LMA, CNRS UMR 5142, Université de Pau, France INRIA Bordeaux Research Center, Project Team Magique3D, France
*
Corresponding author.Email:cyril.agut@univ-pau.fr
Email address:julien.diaz@inria.fr
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Abstract

We present a new high order method in space and time for solving the wave equation, based on a new interpretation of the “Modified Equation” technique. Indeed, contrary to most of the works, we consider the time discretization before the space discretization. After the time discretization, an additional biharmonic operator appears, which can not be discretized by classical finite elements. We propose a new Discontinuous Galerkin method for the discretization of this operator, and we provide numerical experiments proving that the new method is more accurate than the classical Modified Equation technique with a lower computational burden.

Keywords

65M1265M6035L05High order schemesdiscontinuous Galerkin methodacoustic wave equation
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 2 , February 2012 , pp. 691 - 708
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Agut, C., Diaz, J., New high order schemes based on the modified equation technique for solving the wave equation, Technical Report 7331, INRIA, 2010.Google Scholar
[2]Ainsworth, M. and Monk, P. and Muniz, W., Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation, J. Sci. Comput., 27(1-3), 2006.Google Scholar
[3]Anné, L. and Joly, P. and Tran, Q. H., Construction and analysis of higher order finite difference schemes for the 1D wave equation, Computational Geosciences 4, 207–249, 2000.CrossRefGoogle Scholar
[4]Arnold, D. N. and Brezzi, F. and Cockburn, B. and Marini, L. D., Unified analysis of disconitn-uous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., Vol. 39, pp. 1749–1779, 2002.Google Scholar
[5]Cohen, G., Higher-Order Numerical Methods for Transient Wave Equations, Springer-Verlag, 2000.Google Scholar
[6]Dablain, M. A., The application of high order differencing for the scalar wave equation. Geophysics, 51:54–56, 1, 1986.CrossRefGoogle Scholar
[7]Gilbert, J. C., and Joly, P., Higher order time stepping for second order hyperbolic problems and optimal cfl conditions. SIAM Numerical Analysis and Scientific Computing for PDE’s and Their Challenging Applications, 2006.Google Scholar
[8]Grote, M. J., Schneebeli, A., and Schötzau, D.. Discontinuous galerkin finite element method for the wave equation. SIAM J. Numer. Anal., 44:2408–2431, 2006.Google Scholar
[9]Grote, M. J. and Schötzau, D.. Convergence analysis of a fully dicrete Dicontinuous Galerkin method for the wave equation. Preprint No. 2008-04, 2008.Google Scholar
[10]Mozolevski, I. and Suli, E., hp-version interior penalty DGFEMs for the biharmonic equation, Comput. Meth. Appl. Engrg., 196, 13-16(2007), 1851–1863.Google Scholar
[11]Shubin, G. R. and Bell, J. B., A modified equation approachto constructing fourth-order methods for acoustic wave propagation, SIAM J. Sci. Stat. Comput., 8 (1987), pp. 135–151.Google Scholar