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

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.

An entropy–stable discontinuous Galerkin approximation for the incompressible Navier–Stokes equations with variable density and artificial compressibility

“…only considering spatial discretization errors. The approximation uses a split-form DG [13,14], with the exact Riemann solver [15], and the Bassi-Rebay 1 (BR1) [16] to compute inter-element boundary fluxes. We complete the analysis with a stability study of solid wall boundary conditions.…”

Entropy–stable discontinuous Galerkin approximation with summation–by–parts property for the incompressible Navier–Stokes/Cahn–Hilliard system

“…For all meshes, we compute the solution radius R e , the analytical pressure jump ∆p a , which are compared to the numerical solution interior pressure, p i , exterior pressure p e , and pressure jump ∆p. Here we refer to the static pressure, and not the auxiliary pressure (16). The results show that the pressure jump converges to that given by the Poisson law (155) as we refine the grid.…”

“…The use of the standard BR1 method identically cancels the boundary integrals (see [16] for the compressible NSE, [9] for the incompressible NSE, [37] for the MHD equations, and [8] for the Cahn-Hilliard equation). Then, we prove that the extra interface stabilization terms are dissipative.…”

“…We use the volume weighted contravariant basis J a i to transform differential operators from physical ( ∇) to reference ( ∇ ξ ) space. The divergence of a vector is [16]…”

“…and then replace the test functions. Following [16,9], we insert ϕ q = W, and ↔ ϕ g = ↔ F v , and following [8], ϕ µ = C t , and ϕ c = 3 2 σε G c ,…”