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Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system


Authors: Emmanuel Audusse, François Bouchut, Marie-Odile Bristeau and Jacques Sainte-Marie
Journal: Math. Comp. 85 (2016), 2815-2837
MSC (2010): Primary 65M12, 74S10, 76M12, 35L65
DOI: https://doi.org/10.1090/mcom/3099
Published electronically: March 24, 2016
MathSciNet review: 3522971
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Abstract: A lot of well-balanced schemes have been proposed for discretizing the classical Saint-Venant system for shallow water flows with nonflat bottom. Among them, the hydrostatic reconstruction scheme is a simple and efficient one. It involves the knowledge of an arbitrary solver for the homogeneous problem (for example, Godunov, Roe, kinetic, etc.). If this solver is entropy satisfying, then the hydrostatic reconstruction scheme satisfies a semi-discrete entropy inequality. In this paper we prove that, when used with the classical kinetic solver, the hydrostatic reconstruction scheme also satisfies a fully discrete entropy inequality, but with an error term. This error term tends to zero strongly when the space step tends to zero, including solutions with shocks. We also prove that the hydrostatic reconstruction scheme does not satisfy the entropy inequality without error term.


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Emmanuel Audusse
Affiliation: Université Paris 13, Laboratoire d’Analyse, Géométrie et Applications, 99 av. J.-B. Clément, F-93430 Villetaneuse, France; Inria, ANGE project-team, Paris – 2 rue Simone Iff, F75012 Paris, France; CEREMA, ANGE project-team, 134 rue de Beauvais, F-60280 Margny-Lès-Compiègne, France; Sorbonne University, UPMC University Paris VI, ANGE project-team, UMR 7958 LJLL, F-75005 Paris, France
Email: audusse@math.univ-paris13.fr

François Bouchut
Affiliation: Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), CNRS, UPEM, UPEC, F-77454, Marne-la-Vallée, France
Email: Francois.Bouchut@u-pem.fr

Marie-Odile Bristeau
Affiliation: Inria, ANGE project-team, Paris – 2 rue Simone Iff, F75012 Paris, France; CEREMA, ANGE project-team, 134 rue de Beauvais, F-60280 Margny-Lès-Compiègne, France; Sorbonne University, UPMC University Paris VI, ANGE project-team, UMR 7958 LJLL, F-75005 Paris, France
Email: Marie-Odile.Bristeau@inria.fr

Jacques Sainte-Marie
Affiliation: Inria, ANGE project-team, Paris – 2 rue Simone Iff, F75012 Paris, France; CEREMA, ANGE project-team, 134 rue de Beauvais, F-60280 Margny-Lès-Compiègne, France; Sorbonne University, UPMC University Paris VI, ANGE project-team, UMR 7958 LJLL, F-75005 Paris, France
Email: Jacques.Sainte-Marie@inria.fr

DOI: https://doi.org/10.1090/mcom/3099
Keywords: Shallow water equations, well-balanced schemes, hydrostatic reconstruction, kinetic solver, fully discrete entropy inequality
Received by editor(s): September 12, 2014
Received by editor(s) in revised form: May 29, 2015
Published electronically: March 24, 2016
Article copyright: © Copyright 2016 American Mathematical Society