Abstract

AbstractWe 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.

Highlights

  • The ingredients for a reliable numerical method for the approximation of partial differential equations, e.g. one that will not blow up, include stable inter-element and physical boundary condition implementations

  • The recognition that the discontinuous Galerkin spectral element method (DGSEM) with Gauss-Lobatto quadratures satisfies a summation-by-parts (SBP) operators [4, 7] has allowed for the analysis of these schemes and to connect them with penalty collocation and SBP finite difference schemes

  • In [5], we showed that a split form approximation of the compressible Navier–Stokes equations was both linearly and entropy stable provided that the boundary conditions were properly imposed

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Summary

Introduction

The ingredients for a reliable numerical method for the approximation of partial differential equations, e.g. one that will not blow up, include stable inter-element and physical boundary condition implementations. We analyze both the linear and entropy stability of two types of commonly used wall boundary condition procedures used with the DGSEM applied to the compressible Euler equations. In both cases, wall boundary conditions are implemented through a numerical flux. The boundary condition might be implemented through a special wall numerical flux that includes the boundary condition, or a fictitious external state applied to a Riemann solver approximation. We show that the use of Riemann solvers at the boundaries introduce numerical dissipation in an amount that depends on the size of the normal Mach number at the wall

The Compressible Euler Equations and the Wall Boundary Condition
Stability Bounds for the DGSEM
Linear Stability of Wall Boundary Condition Approximations
Entropy Stability of Wall Boundary Condition Approximations
Exact Solution of the Riemann Problem
Using Approximate Riemann Solvers for the Boundary Flux
Discussion
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