Entropy stable discontinuous Galerkin methods for the shallow water equations with subcell positivity preservation

TL;DR

This paper introduces a high order entropy stable discontinuous Galerkin method for 2D shallow water equations that maintains positivity of water heights and is stable near shocks, improving solution reliability.

Contribution

It develops a novel combination of low order positivity preservation with high order entropy stability using convex limiting for shallow water equations.

Findings

Ensures positivity of water heights in high order schemes.

Maintains entropy stability and well-balanced properties.

Effective on 2D triangular meshes with continuous bathymetry.

Abstract

High order schemes are known to be unstable in the presence of shock discontinuities or under-resolved solution features, and have traditionally required additional filtering, limiting, or artificial viscosity to avoid solution blow up. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi-discrete entropy inequality independently of discretization parameters. However, additional measures must be taken to ensure that solutions satisfy physical constraints such as positivity. In this work, we present a high order entropy stable discontinuous Galerkin (ESDG) method for the nonlinear shallow water equations (SWE) on two-dimensional (2D) triangular meshes which preserves the positivity of the water heights. The scheme combines a low order positivity preserving method with a high order entropy stable method using convex limiting. This method is entropy stable and well-balanced for fitted meshes with continuous bathymetry profiles.