A Q-scheme for a class of systems of coupled conservation laws with sourceterm. Application to a two-layer 1-D shallow water system

The goal of this paper is to construct a first-order upwind scheme for solv ing the system of partial differential equations governing the one-dimensio nal ow of two superposed immiscible layers of shallow water fluids. This is done by generalizing a numerical scheme presented by Bermudez and Vazquez- Cendon [3, 26, 27] for solving one-layer shallow water equations, consistin g in a Q-scheme with a suitable treatment of the source terms. The difficul ty in the two layer system comes from the coupling terms involving some der ivatives of the unknowns. Due to these terms, a numerical scheme obtained b y performing the upwinding of each layer, independently from the other one, can be unconditionally unstable. In order to define a suitable numerical s cheme with global upwinding, we first consider an abstract system that gene ralizes the problem under study. This system is not a system of conservatio n laws but, nevertheless, Roe's method can be applied to obtain an upwind s cheme based on Approximate Riemann State Solvers. Following this, we presen t some numerical tests to validate the resulting schemes and to highlight t he 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.