The BR1 Scheme is Stable for the Compressible Navier-Stokes Equations

TL;DR

This paper demonstrates how to modify the original BR1 scheme to achieve provable stability for the compressible Navier-Stokes equations using a discontinuous Galerkin spectral element method on curvilinear meshes, ensuring energy or entropy stability.

Contribution

The authors provide a modified BR1 scheme with proven stability for 3D curvilinear meshes, incorporating specific discretizations and flux choices for the compressible Navier-Stokes equations.

Findings

BR1 scheme can be made stable with proper metric and flux discretizations

The modified scheme preserves stability without artificial dissipation at interior faces

Achieves energy or entropy stability for linear and nonlinear NSE

Abstract

We show how to modify the original Bassi and Rebay scheme (BR1) [F. Bassi and S. Rebay, A High Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations, Journal of Computational Physics, 131:267--279, 1997] to get a provably stable discontinuous Galerkin collocation spectral element method (DGSEM) with Gauss-Lobatto (GL) nodes for the compressible Navier-Stokes equations (NSE) on three dimensional curvilinear meshes. Specifically, we show that the BR1 scheme can be provably stable if the metric identities are discretely satisfied, a two-point average for the metric terms is used for the contravariant fluxes in the volume, an entropy conserving split form is used for the advective volume integrals, the auxiliary gradients for the viscous terms are computed from gradients of entropy variables, and the BR1 scheme is used for the interface fluxes. Our analysis shows that even with three dimensional curvilinear grids, the BR1 fluxes do not add artificial dissipation at the interior element faces. Thus, the BR1 interface fluxes preserve the stability of the discretization of the advection terms and we get either energy stability or entropy-stability for the linear or nonlinear compressible NSE, respectively.