A new class of entropy stable fluctuations for the discontinuous Galerkin method with application to the Saint-Venant-Exner model

TL;DR

This paper introduces a novel class of entropy stable fluctuations for discontinuous Galerkin methods, enabling the construction of entropy conservative schemes for nonconservative hyperbolic systems like the Saint-Venant-Exner model, with verified numerical robustness.

Contribution

It presents a new approach to design entropy stable fluctuations without system-specific derivations, including a blending procedure for dissipation terms, applied to high-order schemes for complex systems.

Findings

Successfully developed entropy stable, high-order schemes for the Saint-Venant-Exner system.

Numerical tests confirm the scheme's stability and accuracy.

The method enhances robustness for nonconservative hyperbolic systems.

Abstract

In this work we consider entropy stable discontinuous Galerkin methods applied to nonconservative hyperbolic systems. We introduce a new class of entropy conservative fluctuations that allow us to construct entropy conservative schemes without any system-specific derivations. We demonstrate that a loss of entropy symmetrization for nonconservative systems restricts the design of entropy stable fluctuations and propose a novel blending procedure to construct entropy stable dissipation terms from general numerical viscosity matrices. The resulting methodology is applied to develop a high-order, entropy stable, and well-balanced approximation for the Saint-Venant-Exner system. Numerical tests are presented to verify the theoretical findings and demonstrate the performance and robustness of the proposed scheme.