AQ-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system

Abstract

The goal of this paper is to construct a rst-order upwind scheme for solving the system of partial dierential 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 Berm udez and V azquez-Cend on (3, 26, 27) for solving one-layer shallow water equations, consisting in a Q-scheme with a suitable treatment of the source terms. The diculty 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 dene a suitable numerical scheme with global upwinding, we rst 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.