A faster numerical scheme for a coupled system to model soil erosion and suspended sediment transport
Résumé
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 \citet{Heng2009} in which an additional and artificial limitation on the time step is required to ensure the positivity of sediment concentrations. This artificial condition can lead the computation to be costly when dealing with very shallow flow and wet/dry fronts. The main result of this paper is to propose a new and faster scheme for which only the CFL condition of shallow water equations is sufficient to preserve the positivity of sediment concentrations. In addition, the use of up-to-date numerical methods allows to obtain easily a well-balanced scheme, to guarantee the positivity of water depth and to verify the maximum principle for sediment concentrations in the convective step. The numerical scheme is tested on classical benchmarks, and we also perform a test on a realistic topography to justify again the quality of proposed approach.
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