Bedload transport in shallow water models: why splitting (may) fail, how hyperbolicity (can) help

In this paper, we are concerned with models for sedimentation transport consisting of a shallow water system coupled with a so called Exner equation that described the evolution of the topography. We show that, for some model of the bedload transport rate including the well-known Meyer-Peter and Müller model, the system is hyperbolic and, thus, linearly stable, only under some constraint on the velocity. In practical situations, this condition is hopefully fulfilled. The numerical approximations of such system are often based on a splitting method, solving first shallow water equation on a time step and, after updating the topography. It is proved that this strategy can create spurious/unphysical oscillations which are related to the study of hyperbolicity e.g. the sign of some eigenvalue of the coupled system differs from the splitting one. Some numerical results are given to illustrate these problems and the way to overcome them in some cases using an stronger C.F.L. condition.