A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime

TL;DR

This paper develops a Godunov-type numerical scheme for the shallow water equations with discontinuous topography, effectively handling the complex resonant regime and ensuring convergence in most cases.

Contribution

It introduces a novel Riemann solver and a well-balanced Godunov-type scheme tailored for shallow water equations with discontinuous topography, addressing resonance issues.

Findings

The scheme converges in the non-resonant regime.

Numerical experiments confirm the scheme's effectiveness.

Convergence issues arise only in the limiting resonance case.

Abstract

We investigate the Riemann problem for the shallow water equations with variable and (possibly) discontinuous topography and provide a complete description of the properties of its solutions: existence; uniqueness in the non-resonant regime; multiple solutions in the resonant regime. This analysis leads us to a numerical algorithm that provides one with a Riemann solver. Next, we introduce a Godunov-type scheme based on this Riemann solver, which is well-balanced and of quasi-conservative form. Finally, we present numerical experiments which demonstrate the convergence of the proposed scheme even in the resonance regime, except in the limiting situation when Riemann data precisely belong to the resonance hypersurface.