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Coupling techniques for nonlinear hyperbolic equations. IV. Well-balanced schemes for scalar multi-dimensional and multi-component laws


Authors: Benjamin Boutin, Frédéric Coquel and Philippe G. LeFloch
Journal: Math. Comp. 84 (2015), 1663-1702
MSC (2010): Primary 65M08; Secondary 35L60
DOI: https://doi.org/10.1090/S0025-5718-2015-02933-0
Published electronically: January 9, 2015
MathSciNet review: 3335887
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Abstract: This series of papers is devoted to the formulation and the approximation of coupling problems for nonlinear hyperbolic equations. The coupling across an interface in the physical space is formulated in terms of an augmented system of partial differential equations. In an earlier work, this strategy allowed us to develop a regularization method based on a thick interface model in one space variable for coupling scalar equations. In the present paper, we significantly extend this framework and, in addition, encompass equations in several space variables. This new formulation includes the coupling of several distinct scalar conservation laws and allows for a possible covering in space. Our main contributions are, on one hand, the design and analysis of a well-balanced finite volume method on general triangulations and, on the other hand, a proof of convergence of this method toward entropy solutions, extending Coquel, Cockburn, and LeFloch's theory (restricted to a single conservation law without coupling). The core of our analysis is, first, the derivation of entropy inequalities as well as a discrete entropy dissipation estimate and, second, a proof of convergence toward the entropy solution of the coupling problem.


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Additional Information

Benjamin Boutin
Affiliation: Institut de Recherche Mathématiques de Rennes, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes, France
Email: benjamin.boutin@univ-rennes1.fr

Frédéric Coquel
Affiliation: Centre de Mathématiques Appliquées & Centre National de la Recherche Scientifique, Ecole Polytechnique, 91128 Palaiseau, France
Email: coquel@cmap.polytechnique.fr

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

DOI: https://doi.org/10.1090/S0025-5718-2015-02933-0
Received by editor(s): June 1, 2012
Received by editor(s) in revised form: October 31, 2013
Published electronically: January 9, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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