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Well-balanced schemes to capture non-explicit steady states: Ripa model


Authors: Vivien Desveaux, Markus Zenk, Christophe Berthon and Christian Klingenberg
Journal: Math. Comp. 85 (2016), 1571-1602
MSC (2010): Primary 65M60, 65M12
DOI: https://doi.org/10.1090/mcom/3069
Published electronically: January 5, 2016
MathSciNet review: 3471101
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Abstract: The present paper concerns the derivation of numerical schemes to approximate the weak solutions of the Ripa model, which is an extension of the shallow-water model where a gradient of temperature is considered. Here, the main motivation lies in the exact capture of the steady states involved in the model. Because of the temperature gradient, the steady states at rest, of prime importance from the physical point of view, turn out to be very nonlinear and their exact capture by a numerical scheme is very challenging. We propose a relaxation technique to derive the required scheme. In fact, we exhibit an approximate Riemann solver that satisfies all the needed properties (robustness and well-balancing). We show three relaxation strategies to get a suitable interpretation of this adopted approximate Riemann solver. The resulting relaxation scheme is proved to be positive preserving, entropy satisfying and to exactly capture the nonlinear steady states at rest. Several numerical experiments illustrate the relevance of the method.


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Vivien Desveaux
Affiliation: Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, CNRS UMR 7352, Université de Picardie Jules Verne, 80039 Amiens, France
Email: vivien.desveaux@u-picardie.fr

Markus Zenk
Affiliation: Universität Würzburg, Campus Hubland Nord, Emil-Fischer-Strasse 40, 97074 Würzburg, Germany
Email: markus.zenk@gmx.de

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@univ-nantes.fr

Christian Klingenberg
Affiliation: Universität Würzburg, Campus Hubland Nord, Emil-Fischer-Strasse 40, 97074 Würzburg, Germany
Email: klingenberg@mathematik.uni-wuerzburg.de

DOI: https://doi.org/10.1090/mcom/3069
Keywords: Hyperbolic system, Ripa model, source terms, steady states, relaxation schemes, well-balanced schemes
Received by editor(s): February 28, 2014
Received by editor(s) in revised form: January 25, 2015
Published electronically: January 5, 2016
Article copyright: © Copyright 2016 American Mathematical Society