# A $Q$-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shadow water system

Manuel Castro; Jorge Macías; Carlos Parés

- Volume: 35, Issue: 1, page 107-127
- ISSN: 0764-583X

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topCastro, Manuel, Macías, Jorge, and Parés, Carlos. "A $Q$-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shadow water system." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.1 (2001): 107-127. <http://eudml.org/doc/194038>.

@article{Castro2001,

abstract = {The goal of this paper is to construct a first-order upwind scheme for solving the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water fluids. This is done by generalizing a numerical scheme presented by Bermúdez and Vázquez-Cendón [3, 26, 27] for solving one-layer shallow water equations, consisting in a $Q$-scheme with a suitable treatment of the source terms. The difficulty in the two layer system comes from the coupling terms involving some derivatives of the unknowns. Due to these terms, a numerical scheme obtained by performing the upwinding of each layer, independently from the other one, can be unconditionally unstable. In order to define a suitable numerical scheme with global upwinding, we first consider an abstract system that generalizes the problem under study. This system is not a system of conservation laws but, nevertheless, Roe’s method can be applied to obtain an upwind scheme based on Approximate Riemann State Solvers. Following this, we present some numerical tests to validate the resulting schemes and to highlight the fact that, in general, numerical schemes obtained by applying a $Q$-scheme to each separate conservation law of the system do not yield good results. First, a simple system of coupled Burgers’ equations is considered. Then, the $Q$-scheme obtained is applied to the two-layer shallow water system.},

author = {Castro, Manuel, Macías, Jorge, Parés, Carlos},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {Q-schemes; coupled conservation laws; source terms; 1D shallow water equations; two-layer flows; hyperbolic systems; first-order upwind scheme; one-dimensional flow of two superposed immiscible layers},

language = {eng},

number = {1},

pages = {107-127},

publisher = {EDP-Sciences},

title = {A $Q$-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shadow water system},

url = {http://eudml.org/doc/194038},

volume = {35},

year = {2001},

}

TY - JOUR

AU - Castro, Manuel

AU - Macías, Jorge

AU - Parés, Carlos

TI - A $Q$-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shadow water system

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2001

PB - EDP-Sciences

VL - 35

IS - 1

SP - 107

EP - 127

AB - The goal of this paper is to construct a first-order upwind scheme for solving the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water fluids. This is done by generalizing a numerical scheme presented by Bermúdez and Vázquez-Cendón [3, 26, 27] for solving one-layer shallow water equations, consisting in a $Q$-scheme with a suitable treatment of the source terms. The difficulty in the two layer system comes from the coupling terms involving some derivatives of the unknowns. Due to these terms, a numerical scheme obtained by performing the upwinding of each layer, independently from the other one, can be unconditionally unstable. In order to define a suitable numerical scheme with global upwinding, we first consider an abstract system that generalizes the problem under study. This system is not a system of conservation laws but, nevertheless, Roe’s method can be applied to obtain an upwind scheme based on Approximate Riemann State Solvers. Following this, we present some numerical tests to validate the resulting schemes and to highlight the fact that, in general, numerical schemes obtained by applying a $Q$-scheme to each separate conservation law of the system do not yield good results. First, a simple system of coupled Burgers’ equations is considered. Then, the $Q$-scheme obtained is applied to the two-layer shallow water system.

LA - eng

KW - Q-schemes; coupled conservation laws; source terms; 1D shallow water equations; two-layer flows; hyperbolic systems; first-order upwind scheme; one-dimensional flow of two superposed immiscible layers

UR - http://eudml.org/doc/194038

ER -

## References

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