An entropy stable vectorial BGK scheme for the Saint-Venant system with hydrostatic reconstruction and adaptive time stepping

In this paper, we discretize the Saint-Venant system through kinetic schemes with hydrostatic reconstruction and adaptive time stepping, aiming to satisfy a fully discrete entropy inequality while preserving the water height non-negativity and the hydrostatic equilibrium. This method is based on vectorial BGK models with finitely many velocities, where the contribution of the bathymetry is accounted as a force term involving a shift matrix. Over a flat bathymetry, we obtain a fully discrete entropy inequality for all entropies. When the bathymetry is variable, we quantify precisely the variation of energy of the system. To ensure that the latter is always dissipated, an implicit integrator can be used. More precisely, we consider some local-in-space and dynamical-in-time convex combination between the forward and backward Euler strategies, improving accuracy by limiting unnecessary diffusion and reducing the computational cost. Finally, we explain how this scheme can be implemented by means of a fixed point method, and we evaluate its interest through numerical experiments.