# Stability of Wall Boundary Condition Procedures for Discontinuous Galerkin Spectral Element Approximations of the Compressible Euler Equations

## TL;DR

This paper analyzes the stability of wall boundary condition procedures in discontinuous Galerkin spectral element methods for the compressible Euler equations, highlighting conditions for energy and entropy stability.

## Contribution

It provides a comprehensive stability analysis of different boundary procedures, including new insights into their dissipation properties and robustness.

## Key findings

- Exact upwind and Lax-Friedrichs solvers are energy dissipative.
- Standard Riemann solvers like Lax-Friedrichs, HLL, HLLC are entropy stable.
- Roe flux is entropy stable under certain conditions.

## Abstract

We perform a linear and entropy stability analysis for wall boundary condition procedures for discontinuous Galerkin spectral element approximations of the compressible Euler equations. Two types of boundary procedures are examined. The first defines a special wall boundary flux that incorporates the boundary condition. The other is the commonly used reflection condition where an external state is specified that has an equal and opposite normal velocity. The internal and external states are then combined through an approximate Riemann solver to weakly impose the boundary condition. We show that with the exact upwind and Lax-Friedrichs solvers the approximations are energy dissipative, with the amount of dissipation proportional to the square of the normal Mach number. Standard approximate Riemann solvers, namely Lax-Friedrichs, HLL, HLLC are entropy stable. The Roe flux is entropy stable under certain conditions. An entropy conserving flux with an entropy stable dissipation term (EC-ES) is also presented. The analysis gives insight into why these boundary conditions are robust in that they introduce large amounts of energy or entropy dissipation when the boundary condition is not accurately satisfied, e.g. due to an impulsive start or under resolution.