A robust and stable numerical scheme for a depth-averaged Euler system

TL;DR

This paper introduces a new stable and robust numerical scheme for a shallow water model that accurately captures non-hydrostatic effects, ensuring positivity, well-balancing, and stability even at minimal water depths.

Contribution

The paper presents a kinetic interpretation-based projection-correction scheme for a non-hydrostatic shallow water model, improving stability and accuracy over existing methods.

Findings

The scheme satisfies positivity, well-balancing, and entropy properties.

It remains stable as water depth approaches zero.

Validated against analytical solutions with promising results.

Abstract

We propose an efficient numerical scheme for the resolution of a non-hydrostatic Saint-Venant type model. The model is a shallow water type approximation of the incompressbile Euler system with free surface and slightly differs from the Green-Naghdi model. The numerical approximation relies on a kinetic interpretation of the model and a projection-correction type scheme. The hyperbolic part of the system is approximated using a kinetic based finite volume solver and the correction step implies to solve an elliptic problem involving the non-hydrostatic part of the pressure. We prove the numerical scheme satisfies properties such as positivity, well-balancing and a fully discrete entropy inequality. The numerical scheme is confronted with various time-dependent analytical solutions. Notice that the numerical procedure remains stable when the water depth tends to zero.