Entropy stable modal discontinuous Galerkin schemes and wall boundary conditions for the compressible Navier-Stokes equations

TL;DR

This paper develops entropy stable discontinuous Galerkin schemes for the compressible Navier-Stokes equations, enabling robust and accurate boundary condition implementation for viscous flows.

Contribution

It introduces a discretization of viscous terms that allows explicit enforcement of entropy stable wall boundary conditions in DG methods.

Findings

Numerical results demonstrate robustness and accuracy.

The schemes satisfy a semi-discrete entropy inequality.

Effective implementation of no-slip and reflective boundary conditions.

Abstract

Entropy stable schemes ensure that physically meaningful numerical solutions also satisfy a semi-discrete entropy inequality under appropriate boundary conditions. In this work, we describe a discretization of viscous terms in the compressible Navier-Stokes equations which enables a simple and explicit imposition of entropy stable no-slip (adiabatic and isothermal) and reflective (symmetry) wall boundary conditions for discontinuous Galerkin (DG) discretizations. Numerical results confirm the robustness and accuracy of the proposed approaches.