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

François Bouchut; Tomás Morales de Luna

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 4, page 683-698
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Bouchut, François, and Morales de Luna, Tomás. "An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment." ESAIM: Mathematical Modelling and Numerical Analysis 42.4 (2008): 683-698. <http://eudml.org/doc/250346>.

@article{Bouchut2008,
abstract = { 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. },
author = {Bouchut, François, Morales de Luna, Tomás},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Two-layer shallow water; nonconservative system; complex eigenvalues; time-splitting; entropy inequality; well-balanced scheme; nonnegativity.; two-layer shallow water; time-splitting; nonnegativity},
language = {eng},
month = {6},
number = {4},
pages = {683-698},
publisher = {EDP Sciences},
title = {An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment},
url = {http://eudml.org/doc/250346},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Bouchut, François
AU - Morales de Luna, Tomás
TI - An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/6//
PB - EDP Sciences
VL - 42
IS - 4
SP - 683
EP - 698
AB - 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.
LA - eng
KW - Two-layer shallow water; nonconservative system; complex eigenvalues; time-splitting; entropy inequality; well-balanced scheme; nonnegativity.; two-layer shallow water; time-splitting; nonnegativity
UR - http://eudml.org/doc/250346
ER -

References

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