# An all-regime and well-balanced Lagrange-projection type scheme for the shallow water equations on unstructured meshes

## TL;DR

This paper introduces a novel numerical scheme for the 2D shallow water equations that is well-balanced, capable of handling all flow regimes including low-Froude, and maintains positive water height on unstructured meshes.

## Contribution

It proposes a Lagrange-projection based scheme that ensures well-balancedness, all-regime applicability, and positivity, improving robustness over traditional methods.

## Key findings

- Successfully preserves lake at rest equilibrium solutions.
- Effective in all flow regimes, including low-Froude.
- Maintains positive water height throughout simulations.

## Abstract

In this work, we focus on the numerical approximation of the shallow water equations in two space dimensions. Our aim is to propose a well-balanced, all-regime and positive scheme. By well-balanced, it is meant that the scheme is able to preserve the so-called lake at rest smooth equilibrium solutions. By all-regime, we mean that the scheme is able to deal with all flow regimes, including the low-Froude regime which is known to be challenging when using usual Godunov-type finite volume schemes. At last, the scheme should be positive which means that the water height stays positive for all time. Our approach is based on a Lagrange-projection decomposition which allows to naturally decouple the acoustic and transport terms. Numerical experiments on unstructured meshes illustrate the good behaviour of the scheme.