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

In this paper, we are concerned with sediment transport models consisting of a shallow water system coupled with the so called Exner equation to describe the evolution of the topography. We show that, for some bedload transport models like 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. Numerical approximations of such system are often based on a splitting method, solving first shallow water equation on a time step and, updating afterwards the topography. It is shown that this strategy can create spurious/unphysical oscillations which are related to the study of hyperbolicity. Using an upper bound of the largest eigenvector may improve the results although the instabilities cannot be always avoided, e.g. in supercritical regions.

► Characterization of hyperbolicity for shallow water-Exner equations.
► Common bedload transport models remain hyperbolic in physical situations.
► Hyperbolicity may be lost, for instance with Meyer-Peter and Müller.
► Study of instabilities when using to splitting methods.