On Lagrange-Projection Schemes for Shallow Water Flows Over Movable Bottom with Suspended and Bedload Transport

In the present work we aim to simulate shallow water flows over movable bottom with suspended and bedload transport.In order to numerically approximate such a system,we proceed step by step.We start by considering shallow water equations with non-constant density of the mixture water-sediment.Then,the Exner equation is included to take into account bedload sediment transport.Finally,source terms for friction,erosion and deposition processes are considered.Indeed,observe that the sediment particle could go in suspension into the water or being deposited on the bottom.For the numerical scheme,we rely on well-balanced Lagrange-projection methods.In particular,since sediment transport is generally a slow process,we aim to develop semi-implicit schemes in order to obtain fast simulations.The Lagrange-projection splitting is well-suited for such a purpose as it entails a decomposition of the(fast)acoustic waves and the(slow)material waves of the model.Hence,in subsonic regimes,an implicit approximation of the acoustic equations allows us to neglect the corresponding CFL condition and to obtain fast numerical schemes with large time step.

AnAll-Regime Lagrange-Projection Like Scheme for the Gas Dynamics Equations on Unstructured Meshes

We propose an all regime Lagrange-Projection like numerical scheme for the gas dynamics equations.By all regime,we mean that the numerical scheme is able to compute accurate approximate solutions with an under-resolved discretization with respect to the Mach number M,i.e.such that the ratio between the Mach number M and the mesh size or the time step is small with respect to 1.The key idea is to decouple acoustic and transport phenomenon and then alter the numerical flux in the acoustic approximation to obtain a uniform truncation error in term of M.This modified scheme is conservative and endowed with good stability properties with respect to the positivity of the density and the internal energy.A discrete entropy inequality under a condition on the modification is obtained thanks to a reinterpretation of the modified scheme in the Harten Lax and van Leer formalism.A natural extension to multi-dimensional problems discretized over unstructured mesh is proposed.Then a simple and efficient semi implicit scheme is also proposed.The resulting scheme is stable under a CFL condition driven by the(slow)material waves and not by the(fast)acoustic waves and so verifies the all regime property.Numerical evidences are proposed and show the ability of the scheme to deal with tests where the flow regime may vary from low to high Mach values.