An Entropy Stable Nodal Discontinuous Galerkin Method for the Two Dimensional Shallow Water Equations on Unstructured Curvilinear Meshes with Discontinuous Bathymetry

TL;DR

This paper introduces a high-order nodal discontinuous Galerkin method for 2D shallow water equations on unstructured meshes, ensuring mass, momentum, and energy conservation, and demonstrating stability and well-balanced properties.

Contribution

It develops an entropy stable, high-order accurate DG scheme that preserves physical invariants and handles discontinuous bathymetry on curved meshes.

Findings

The scheme exactly preserves mass and momentum.

It maintains total energy, ensuring entropy stability.

Numerical tests confirm theoretical properties and effectiveness.

Abstract

We design an arbitrary high-order accurate nodal discontinuous Galerkin spectral element approximation for the nonlinear two dimensional shallow water equations with non-constant, possibly discontinuous, bathymetry on unstructured, possibly curved, quadrilateral meshes. The scheme is derived from an equivalent flux differencing formulation of the split form of the equations. We prove that this discretisation exactly preserves the local mass and momentum. Furthermore, combined with a special numerical interface flux function, the method exactly preserves the mathematical entropy, which is the total energy for the shallow water equations. By adding a specific form of interface dissipation to the baseline entropy conserving scheme we create a provably entropy stable scheme. That is, the numerical scheme discretely satisfies the second law of thermodynamics. Finally, with a particular discretisation of the bathymetry source term we prove that the numerical approximation is well-balanced. We provide numerical examples that verify the theoretical findings and furthermore provide an application of the scheme for a partial break of a curved dam test problem.