Finite Volume Evolution Galerkin Methods for the Shallow Water Equations with Dry Beds

TL;DR

This paper introduces a new finite volume evolution Galerkin scheme for shallow water equations that effectively handles dry beds and preserves water height positivity, improving accuracy near vacuum states.

Contribution

It extends existing FVEG methods to handle dry boundaries and introduces flux limiting and entropy fixes for better physical fidelity.

Findings

Successfully handles dry beds in shallow water simulations

Preserves positivity of water height in all computational cells

Improves reproduction of sonic rarefaction waves

Abstract

We present a new Finite Volume Evolution Galerkin (FVEG) scheme for the solution of the shallow water equations (SWE) with the bottom topography as a source term. Our new scheme will be based on the FVEG methods presented in (Luk\'a\v{c}ov\'a, Noelle and Kraft, J. Comp. Phys. 221, 2007), but adds the possibility to handle dry boundaries. The most important aspect is to preserve the positivity of the water height. We present a general approach to ensure this for arbitrary finite volume schemes. The main idea is to limit the outgoing fluxes of a cell whenever they would create negative water height. Physically, this corresponds to the absence of fluxes in the presence of vacuum. Well-balancing is then re-established by splitting gravitational and gravity driven parts of the flux. Moreover, a new entropy fix is introduced that improves the reproduction of sonic rarefaction waves.