Parallelization of a relaxation scheme modelling the bedload transport of sediments in shallow water flow

In this work we are interested in numerical simulations for bedload erosion processes. We present a relaxation solver that we apply to moving dunes test cases in one and two dimensions. In particular we retrieve the so-called anti-dune process that is well described in the experiments. In order to be able to run 2D test cases with reasonable CPU time, we also describe and apply a parallelization procedure by using domain decomposition based on the classical MPI library.Comment: 19 page

Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system

We introduce a new second-order central-upwind scheme for the Saint-Venant system of shallow water equations on triangular grids. We prove that the scheme both preserves “lake at rest” steady states and guarantees the positivity of the computed fluid depth. Moreover, it can be applied to models with discontinuous bottom topography and irregular channel widths. We demonstrate these features of the new scheme, as well as its high resolution and robustness in a number of numerical examples

A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime

We investigate the Riemann problem for the shallow water equations with variable and (possibly) discontinuous topography and provide a complete description of the properties of its solutions: existence; uniqueness in the non-resonant regime; multiple solutions in the resonant regime. This analysis leads us to a numerical algorithm that provides one with a Riemann solver. Next, we introduce a Godunov-type scheme based on this Riemann solver, which is well-balanced and of quasi-conservative form. Finally, we present numerical experiments which demonstrate the convergence of the proposed scheme even in the resonance regime, except in the limiting situation when Riemann data precisely belong to the resonance hypersurface.Comment: 39 page