Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system
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- by Emmanuel Audusse, François Bouchut, Marie-Odile Bristeau and Jacques Sainte-Marie PDF
- Math. Comp. 85 (2016), 2815-2837 Request permission
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.References
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Additional Information
- 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
- MR Author ID: 717952
- 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
- MR Author ID: 314037
- ORCID: 0000-0002-2545-1655
- 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
- MR Author ID: 41730
- 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
- Received by editor(s): September 12, 2014
- Received by editor(s) in revised form: May 29, 2015
- Published electronically: March 24, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Math. Comp. 85 (2016), 2815-2837
- MSC (2010): Primary 65M12, 74S10, 76M12, 35L65
- DOI: https://doi.org/10.1090/mcom/3099
- MathSciNet review: 3522971