An improved numerical scheme for a coupled system to model soil erosion and polydispersed sediments transport

Overland flow and soil erosion play an essential role in water quality and soil degradation. Such processes, involving the interactions between water flow, suspended particles and soil, are classically described by a well-established system of PDE coupling the shallow water equations and the Hairsine-Rose model. The numerical approximation of this coupled system requires advanced methods to preserve some important physical and mathematical properties in particular the steady states, the positivity of both water depth and sediment concentrations. Recently, a well-balanced MUSCL-Hancock scheme has been proposed by Heng et al. [25] in which an additional and artificial limitation on the time step is required to ensure the positivity of sediment concentrations. The main result of this paper is to propose a new approximation for which the CFL condition of shallow water equations is sufficient to preserve the positivity of sediment concentrations. In addition, the use of the hydrostatic reconstruction technique [2] allows to obtain an easy well-balanced scheme, to guarantee the positivity of water depth and to verify the maximum principle for sediment concentrations in the convection step. Moreover, the simulation is robust as one can deal with an arbitrary number of sediment classes.