A fully well-balanced scheme for shallow water equations with Coriolis force

TL;DR

This paper develops a fully well-balanced and positive-preserving numerical scheme for shallow water equations with Coriolis force, accurately maintaining steady states including geostrophic equilibrium and enhancing order while preserving balance.

Contribution

It introduces a novel Godunov-type scheme with steady state detection to preserve all steady states and improve accuracy in rotating shallow water equations.

Findings

Scheme accurately preserves steady states including geostrophic equilibrium.

Second-order scheme improves accuracy while maintaining well-balanced property.

Numerical experiments validate the scheme's effectiveness.

Abstract

The present work is devoted to the derivation of a fully well-balanced and positivepreserving numerical scheme for the shallow water equations with Coriolis force. The first main issue consists in preserving all the steady states, including the geostrophic equilibrium. Our strategy relies on a Godunov-type scheme with suitable source term and steady state discretisations. The second challenge lies in improving the order of the scheme while preserving the fully well-balanced property. A modification of the classical methods is required since no conservative reconstruction can preserve all the steady states in the case of rotating shallow water equations. A steady state detector is used to overcome this matter. Some numerical experiments are presented to show the relevance and the accuracy of both first-order and second-order schemes.