Entropy analysis and entropy stable DG methods for the shallow water moment equations

TL;DR

This paper proves entropy conservation and dissipation properties of the shallow water moment equations and develops an entropy stable discontinuous Galerkin method validated by numerical examples.

Contribution

It introduces an entropy stable DG scheme for the shallow water moment equations based on entropy analysis and flux construction.

Findings

The equations satisfy an energy-based entropy conservation law.

The developed scheme is entropy stable and well-balanced.

Numerical tests confirm the theoretical entropy stability and accuracy.

Abstract

We demonstrate that the shallow water moment equations satisfy an auxiliary entropy conservation law, where the entropy function corresponds to the total energy. Additionally, we show that the classical Newtonian slip friction and Manning friction terms are entropy dissipative with respect to the developed entropy variables. The results from the continuous entropy analysis are used to construct an entropy stable and well-balanced nodal discontinuous Galerkin spectral element method for the spatial approximation. Key to ensure the entropy stability of the scheme is the derivation of entropy conservative numerical fluxes that satisfy a discrete version of the entropy flux compatibility condition. Finally, numerical examples demonstrate the performance of the scheme and validate the theoretical results.