An Entropy Satisfying Scheme for Two-layer Shallow Water Equations with Uncoupled Treatment

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. 1991 Mathematics Subject Classification. 74S10, 35L60, 74G15. The dates will be set by the publisher. 1. The two-layer shallow water system The two-layer shallow water system of partial differential equations governs the one-dimensional flow of two superposed shallow layers of immiscible fluids with different constant densities