Entropy correction artificial viscosity for high order DG methods using multiple artificial viscosities

TL;DR

This paper introduces an entropy correction artificial viscosity approach for high-order DG methods, enabling precise control over physical phenomena while maintaining robustness in complex simulations.

Contribution

It extends entropy correction artificial viscosity to multiple types, providing analytical expressions for optimal parameters and improved targeting of physical effects.

Findings

The method allows more precise targeting of physical phenomena.

It retains robustness for various problem settings.

Analytical expressions for viscosity parameters are derived.

Abstract

Entropy stable discontinuous Galerkin (DG) methods display improved robustness for problems with shocks, turbulence, and under-resolved features by enforcing an entropy inequality. Such methods have traditionally relied on entropy conservative (EC) fluxes that are computationally expensive to evaluate. An alternative approach for enforcing an entropy inequality is through a minimally dissipative ``entropy correction" artificial viscosity. We review how to construct such an artificial viscosity formulation and extend this approach to multiple types of viscosity (e.g., viscosity and thermal diffusivity). We determine simple analytical expressions for optimal viscosity parameters. We compare this to the case of a single monolithic viscosity parameter for different 1D and 2D problems, and show that the proposed method allows users to more precisely target specific physical phenomena while retaining robustness for general problem settings.