Conservative semi-implicit semi-Lagrangian scheme for simulation of shallow flows

A fully conservative semi-Lagrangian (SL) scheme is presented to solve for the shallow-water equations. Existing inherently conservative SL schemes only ensure the conservation of mass while momentum is not fully conserved. The gravity terms, which are mainly responsible for the wave structure in dam break flows, are then discretized by using traditional non-conservative Eulerian schemes. In the presence of large variations in water surface (e.g., dam-break type flows), such an approach leads to incorrect shock speed and highly oscillatory results. Indeed, if the conservation of the gravity terms is forced, the use of existing schemes will be restricted to small time steps. In this paper we present a fully conservative scheme which can accurately simulate the shallow flows with a large time step. In our approach, both convective and gravitational terms are treated in a conservative manner, which ensures an accurate shock speed. The fully conservation property improves considerably the performance of common SL schemes for a wide range of practical applications.