Communications in Mathematical Sciences

Volume 20 (2022)

Number 7

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

Pages: 1875 – 1900

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n7.a4

Authors

Vivien Desveaux (Laboratoire Amiénois de Mathématique Fondamentale et Appliquée (LAMFA) UMR CNRS 7352, Université de Picardie Jules Verne, Amiens, France)

Alice Masset (Laboratoire Amiénois de Mathématique Fondamentale et Appliquée (LAMFA) UMR CNRS 7352, Université de Picardie Jules Verne, Amiens, France)

Abstract

The present work is devoted to the derivation of a fully well-balanced and positivity-preserving numerical scheme for the shallow water equations with Coriolis force. The first main issue consists in preserving all the steady states. Our strategy relies on a Godunov-type scheme with suitable source term and steady state discretisations. The preservation of moving steady states may lead to ill-defined intermediate states in the Riemann solver. Therefore, a proper correction is introduced in order to obtain a fully well-balanced scheme. 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 accuracy of both first-order and second-order schemes.

Keywords

shallow water equations, Coriolis force, fully well-balanced schemes, Godunov-type schemes, high-order approximation

2010 Mathematics Subject Classification

65M08, 65M12

Received 11 October 2021

Received revised 19 January 2022

Accepted 27 January 2022

Published 21 October 2022