Numerical approximation of the shallow water equations with coriolis source term

¹ LAGA, Institut Galilée, Université Sorbonne Paris Nord — 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse.
² INRIA Paris — Sorbonne Université — CNRS (LJLL), team ANGE, 2 rue Simone Iff, CS 42112, 75589 Paris cedex 12.
³ Institut Camille Jordan, Université Claude Bernard Lyon 1 — 43, boulevard du 11 novembre 1918, 69622 Villeurbanne.
⁴ Institut Denis Poisson, Université D’Orléans, rue de Chartres, 45067 Orléans.
⁵ Institut de Mathématiques de Marseille, Université d’Aix-Marseille — Technopôle Château Gombert, 13453, Marseille.

Abstract

We investigate in this work a class of numerical schemes dedicated to the non-linear Shallow Water equations with topography and Coriolis force. The proposed algorithms rely on Finite Volume approximations formulated on collocated and staggered meshes, involving appropriate diffusion terms in the numerical fluxes, expressed as discrete versions of the linear geostrophic balance. It follows that, contrary to standard Finite-Volume approaches, the linear versions of the proposed schemes provide a relevant approximation of the geostrophic equilibrium. We also show that the resulting methods ensure semi-discrete energy estimates. Numerical experiments exhibit the efficiency of the approach in the presence of Coriolis force close to the geostrophic balance, especially at low Froude number regimes.