Communications in Mathematical Sciences

Volume 13 (2015)

Number 5

A simple well-balanced and positive numerical scheme for the shallow-water system

Pages: 1317 – 1332

(Fast Communication)

DOI: http://dx.doi.org/10.4310/CMS.2015.v13.n5.a11

Authors

Emmanuel Audusse (LAGA, UMR CNRS 7539, Université Paris XIII, Villetaneuse, France; INRIA, ANGE Project-Team, Rocquencourt, Le Chesnay, France; CEREMA, ANGE Project-Team, Margny-Lès-Compiègne, France; and LJLL, ANGE Project-Team, UPMC Université Paris VI, Paris, France)

Christophe Chalons (LMV, UMR CNRS 8100, Universit´e de Versailles-Saint-Quentin-en-Yvelines, Versailles, France)

Philippe Ung (MAPMO, UMR CNRS 7349, Université d’Orléans, France; INRIA, ANGE Project-Team, Rocquencourt, Le Chesnay, France; CEREMA, ANGE Project-Team, Margny-Lès-Compiègne, France; and LJLL, ANGE Project-Team, UPMC Université Paris, France)

Abstract

This work considers the numerical approximation of the shallow-water equations. In this context, one faces three important issues related to the well-balanced, positivity and entropy-preserving properties, as well as the ability to consider vacuum states. We propose a Godunov-type method based on the design of a three-wave Approximate Riemann Solver (ARS) which satisfies the first two properties and a weak form of the last one together. Regarding the entropy, the solver satisfies a discrete non-conservative entropy inequality. From a numerical point of view, we also investigate the validity of a conservative entropy inequality.

Keywords

shallow-water equations, approximate Riemann solver, finite volume method, positivity preserving, well-balanced scheme

2010 Mathematics Subject Classification

35L40, 35Q35, 76M12

Full Text (PDF format)

Published 22 April 2015