On an one-dimensional bi-layer shallow-water problem Original Research Article

This paper is concerned with the mathematical analysis and the numerical approximation of the system of partial differential equations governing the one-dimensional flow of two superposed shallow layers of immiscible viscous fluid in a channel with variable rectangular cross-section. First, we prove the existence and uniqueness of solution for small data and some smoothness results. Next, a first-order upwind scheme for numerically solving the system is proposed. We apply this scheme to the simulation of some two-layer exchange flows through straits with a sill and a contraction.