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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Late-time/stiff-relaxation asymptotic-preserving approximations of hyperbolic equations


Authors: Christophe Berthon, Philippe G. LeFloch and Rodolphe Turpault
Journal: Math. Comp. 82 (2013), 831-860
MSC (2010): Primary 35L65, 65M99
Published electronically: December 13, 2012
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Abstract: We investigate the late-time asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws containing stiff relaxation terms. First, we introduce a Chapman-Enskog-type asymptotic expansion and derive an effective system of equations describing the late-time/stiff-relaxation singular limit. The structure of this new system is discussed and the role of a mathematical entropy is emphasized. Second, we propose a new finite volume discretization which, in late-time asymptotics, allows us to recover a discrete version of the same effective asymptotic system. This is achieved provided we suitably discretize the relaxation term in a way that depends on a matrix-valued free-parameter, chosen so that the desired asymptotic behavior is obtained. Our results are illustrated with several models of interest in continuum physics, and numerical experiments demonstrate the relevance of the proposed theory and numerical strategy.


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Additional Information

Christophe Berthon
Affiliation: Université de Nantes, Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629, 2 rue de la Houssinière, BP 92208, 44322 Nantes, France
Email: Christophe.Berthon@math.univ-nantes.fr

Philippe G. LeFloch
Affiliation: Laboratoire Jacques-Louis Lions, Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6), 4 Place Jussieu, 75252 Paris, France
Email: contact@philippelefloch.org

Rodolphe Turpault
Affiliation: Université de Nantes, Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629, 2 rue de la Houssinière, BP 92208, 44322 Nantes, France
Email: Rodolphe.Turpault@univ-nantes.fr

DOI: http://dx.doi.org/10.1090/S0025-5718-2012-02666-4
PII: S 0025-5718(2012)02666-4
Keywords: Nonlinear hyperbolic system, stiff source term, late-time behavior, diffusive regime, finite volume scheme, asymptotic preserving.
Received by editor(s): November 15, 2010
Received by editor(s) in revised form: March 10, 2011, and September 18, 2011
Published electronically: December 13, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.