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An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment

Published online by Cambridge University Press:  05 June 2008

François Bouchut  and
Tomás Morales de Luna
François Bouchut
Affiliation:
Département de Mathématiques et Applications, École Normale Supérieure & CNRS, 45 rue d'Ulm, 75230 Paris cedex 05, France. francois.bouchut@ens.fr
Tomás Morales de Luna
Affiliation:
Departamento de Anlálisis Matemático, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinons s/n, 29071 Málaga, Spain. morales@anamat.cie.uma.es
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Abstract

We consider the system of partial differential equations governingthe one-dimensional flow of two superposed immiscible layers of shallow water. The difficulty in this system comes from the coupling terms involving some derivatives of the unknowns that make the system nonconservative, and eventually nonhyperbolic. Due to these terms, a numerical scheme obtained by performing an arbitrary scheme to each layer, and using time-splitting or other similar techniques leads to instabilities in general. Here we use entropy inequalities in order to control the stability. We introduce a stable well-balanced time-splitting scheme for the two-layer shallow water system that satisfies a fully discrete entropy inequality. In contrast with Roe type solvers, it does not need the computation of eigenvalues, which is not simple for the two-layer shallow water system. The solver has the property to keep the water heights nonnegative, and to be able to treat vanishing values.

Keywords

Two-layer shallow waternonconservative systemcomplex eigenvaluestime-splittingentropy inequalitywell-balanced schemenonnegativity.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 4 , July 2008 , pp. 683 - 698
Copyright
© EDP Sciences, SMAI, 2008

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