We consider the system of partial differential equations governing the 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.
Mots-clés : two-layer shallow water, nonconservative system, complex eigenvalues, time-splitting, entropy inequality, well-balanced scheme, nonnegativity
@article{M2AN_2008__42_4_683_0,
author = {Bouchut, Fran\c{c}ois and Tom\'as Morales de Luna},
title = {An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {683--698},
publisher = {EDP-Sciences},
volume = {42},
number = {4},
year = {2008},
doi = {10.1051/m2an:2008019},
mrnumber = {2437779},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2008019/}
}
TY - JOUR AU - Bouchut, François AU - Tomás Morales de Luna TI - An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2008 SP - 683 EP - 698 VL - 42 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2008019/ DO - 10.1051/m2an:2008019 LA - en ID - M2AN_2008__42_4_683_0 ER -
%0 Journal Article %A Bouchut, François %A Tomás Morales de Luna %T An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment %J ESAIM: Modélisation mathématique et analyse numérique %D 2008 %P 683-698 %V 42 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2008019/ %R 10.1051/m2an:2008019 %G en %F M2AN_2008__42_4_683_0
Bouchut, François; Tomás Morales de Luna. An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 4, pp. 683-698. doi : 10.1051/m2an:2008019. http://www.numdam.org/articles/10.1051/m2an:2008019/
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