Implicit kinetic schemes for the Saint-Venant system

¹ Centre Inria de Sorbonne université, 2 rue Simone Iff, CS 42112, 75589 Paris Cedex 12, France
² Sorbonne Université, Univ. Paris Diderot, SPC, CNRS, LJLL, F-75005 Paris, France
³ Institut de Mathématiques de Bordeaux, équipe EDP et physique-mathématique, Université de Bordeaux, 351 cours de la Libération, F-33405 Talence, France

^* Corresponding author: mathieu.rigal@math.u-bordeaux.fr

Received: 13 May 2024
Accepted: 16 May 2025

Abstract

Explicit (in time) kinetic schemes applied to the nonlinear shallow water equations have been extensively studied in the past. The novelty of this paper is to investigate an implicit version of such methods in order to improve their stability properties. In the case of a flat bathymetry we obtain a fully implicit kinetic solver satisfying a discrete entropy inequality and keeping the water height non negative without any restriction on the time step. Remarkably, a simplified version of this nonlinear implicit scheme allows to express the update explicitly which we implement in practice. The case of varying bottoms is then dealt with through an iterative solver combined with the hydrostatic reconstruction technique. We show that this scheme preserves the water height non-negativity under a CFL condition and satisfies a discrete entropy inequality without error term, which is an improvement over its explicit version. An extension of the implicit and iterative methods to the two dimensional case is also discussed. Finally we perform some numerical validations underlining the advantages and the computational cost of our strategy.