A Godunov-type scheme for the Saint-Venant-Exner equations with a moving steady states

The Saint-Venant-Exner system is widely used in industrial codes to model the transport of bed sediments. In practice, most of the methods used for its numerical resolution suffer from significant stability problems due to the fact that they are mainly guaranteed by separation techniques which allow weak coupling between hydraulic and morphodynamic software. The search for an adequate alternative method to this problem is a focus towards which several approaches converge. Recently, many authors have proposed acceptable methods but they are not so trivial to implement in an industrial context and some do not take into account all aspects such as mobile steady states. In this work, we derive a well-balanced scheme that takes into account all stable states, to approach weak solutions of the sediment transport model in shallow waters. We demonstrate that it captures exactly all regular steady solutions with non-evanescent velocities, retains water level positivity and is able to degenerate to a classical shallow water model when the sedimentary transport flow is null. A number of numerical test cases are also presented to illustrate these properties.