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.

Keywords: Q-schemes, coupled conservation laws, source terms, 1D shallow water equations, two-layer flows, hyperbolic systems

@article{M2AN_2001__35_1_107_0, author = {Castro, Manuel and Mac\'\i as, Jorge and Par\'es, Carlos}, 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}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {35}, number = {1}, year = {2001}, pages = {107-127}, zbl = {1094.76046}, mrnumber = {1811983}, language = {en}, url = {http://www.numdam.org/item/M2AN_2001__35_1_107_0} }

Castro, Manuel; Macías, Jorge; 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, Volume 35 (2001) no. 1, pp. 107-127. http://www.numdam.org/item/M2AN_2001__35_1_107_0/

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