A positivity preserving strategy for entropy stable discontinuous Galerkin discretizations of the compressible Euler and Navier-Stokes equations

TL;DR

This paper introduces a positivity limiting strategy for high-order entropy-stable discontinuous Galerkin methods applied to the compressible Euler and Navier-Stokes equations, ensuring thermodynamic quantities remain positive while maintaining accuracy.

Contribution

A novel blending-based positivity limiting strategy that preserves entropy stability and positivity in high-order DG discretizations of fluid dynamics equations.

Findings

Numerical experiments demonstrate high order accuracy.

The method is robust across various test cases.

Positivity is preserved without sacrificing stability.

Abstract

High-order entropy-stable discontinuous Galerkin methods for the compressible Euler and Navier-Stokes equations require the positivity of thermodynamic quantities in order to guarantee their well-posedness. In this work, we introduce a positivity limiting strategy for entropy-stable discontinuous Galerkin discretizations constructed by blending high order solutions with a low order positivity-preserving discretization. The proposed low order discretization is semi-discretely entropy stable, and the proposed limiting strategy is positivity preserving for the compressible Euler and Navier-Stokes equations. Numerical experiments confirm the high order accuracy and robustness of the proposed strategy.