A large time-step and well-balanced Lagrange-Projection type scheme for the shallow-water equations

TL;DR

This paper introduces a novel Lagrange-Projection scheme for the shallow-water equations that allows larger time steps, maintains accuracy in subsonic flows, and preserves the well-balanced property, with demonstrated stability and effectiveness.

Contribution

It extends implicit-explicit schemes to the shallow-water equations, enabling larger time steps and improved stability while ensuring well-balanced discretization of non-conservative terms.

Findings

The scheme is unconditionally stable for certain test cases.

It accurately captures steady states and dynamic flows.

The method demonstrates robustness across various test scenarios.

Abstract

This work focuses on the numerical approximation of the Shallow Water Equations (SWE) using a Lagrange-Projection type approach. We propose to extend to this context recent implicit-explicit schemes developed in the framework of compressibleflows, with or without stiff source terms. These methods enable the use of time steps that are no longer constrained by the sound velocity thanks to an implicit treatment of the acoustic waves, and maintain accuracy in the subsonic regime thanks to an explicit treatment of the material waves. In the present setting, a particular attention will be also given to the discretization of the non-conservative terms in SWE and more specifically to the well-known well-balanced property. We prove that the proposed numerical strategy enjoys important non linear stability properties and we illustrate its behaviour past several relevant test cases.