An artificial viscosity approach to high order entropy stable discontinuous Galerkin methods

TL;DR

This paper introduces an entropy correction artificial viscosity approach for high order entropy stable discontinuous Galerkin methods, enhancing robustness and computational efficiency without requiring explicit flux expressions.

Contribution

It proposes a novel, parameter-free artificial viscosity technique that maintains entropy stability and high order accuracy in DG methods, simplifying implementation for complex systems.

Findings

Method preserves semi-discrete entropy inequality.

Artificial viscosity is locally computable and parameter-free.

Maintains high order accuracy and stability.

Abstract

Entropy stable discontinuous Galerkin (DG) methods improve the robustness of high order DG simulations of nonlinear conservation laws. These methods yield a semi-discrete entropy inequality, and rely on an algebraic flux differencing formulation which involves both summation-by-parts (SBP) discretization matrices and entropy conservative two-point finite volume fluxes. However, explicit expressions for such two-point finite volume fluxes may not be available for all systems, or may be computationally expensive to compute. This paper proposes an alternative approach to constructing entropy stable DG methods using an entropy correction artificial viscosity, where the artificial viscosity coefficient is determined based on the local violation of a cell entropy inequality and the local entropy dissipation. The resulting method is a modification of the entropy correction introduced by Abgrall, Offner, and Ranocha (2022) in "Reinterpretation and Extension of Entropy Correction Terms for Residual Distribution and Discontinuous Galerkin Schemes: Application to Structure Preserving Discretization", and recovers the same global semi-discrete entropy inequality that is satisfied by entropy stable flux differencing DG methods. The entropy correction artificial viscosity coefficients are parameter-free and locally computable over each cell, and the resulting artificial viscosity preserves both high order accuracy and a hyperbolic maximum stable time-step size under explicit time-stepping.