[Paper Review] The BR1 Scheme is Stable for the Compressible Navier-Stokes Equations

Gregor J. Gassner, Andrew R. Winters+2 more|Kölner Universitäts PublikationsServer (Universität zu Köln)|Apr 12, 2017

Computational Fluid Dynamics and Aerodynamics28 references5 citations

TL;DR

This paper proves that the BR1 scheme for the compressible Navier-Stokes equations is provably stable when combined with discrete metric identity satisfaction, a two-point average for metric terms, entropy-conserving split form for advective fluxes, entropy-variable-based viscous gradients, and BR1 interface fluxes. The scheme achieves energy or entropy stability on 3D curvilinear meshes without introducing artificial dissipation at interior faces.

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.

Motivation & Objective

• To establish provable stability of the BR1 scheme for the compressible Navier-Stokes equations in three dimensions.
• To resolve long-standing concerns about instability in BR1 despite its widespread use in simulations.
• To identify the minimal set of modifications required to achieve energy or entropy stability.
• To demonstrate that BR1 does not introduce artificial dissipation at interior element faces when properly formulated.
• To provide a stable, parameter-free DGSEM framework using Gauss-Lobatto nodes for complex curvilinear meshes.

Proposed method

• Discretely satisfy metric identities using Gauss-Lobatto nodes and consistent metric reconstruction.
• Apply a two-point average for contravariant fluxes in the volume integrals to maintain consistency.
• Use an entropy-conserving split form for the advective fluxes to preserve conservation properties.
• Compute auxiliary gradients for viscous terms from entropy variables to ensure stability.
• Employ BR1 numerical fluxes at element interfaces to maintain simplicity and consistency.
• Ensure the discrete weak form satisfies entropy or energy stability conditions via summation-by-parts structure.

Experimental results

Research questions

• RQ1 Can the BR1 scheme be proven stable for the compressible Navier-Stokes equations on 3D curvilinear grids?
• RQ2 What modifications are necessary to ensure energy or entropy stability in the BR1 framework?
• RQ3 Does the BR1 scheme introduce artificial dissipation at interior element faces?
• RQ4 How can metric identities be discretely satisfied to preserve conservation and stability?
• RQ5 Is the BR1 scheme stable under explicit time integration for advection-dominated flows?

Key findings

• The BR1 scheme is provably stable for the compressible Navier-Stokes equations when metric identities are satisfied discretely.
• The scheme achieves entropy-stability for the nonlinear equations and energy-stability for the linearized equations.
• No artificial dissipation is introduced at interior element faces, preserving the scheme's low dissipation for resolved scales.
• The method is parameter-free and does not require penalty parameters or additional tuning.
• The stability result holds for three-dimensional curvilinear hexahedral elements using Gauss-Lobatto nodes.
• The analysis confirms that the BR1 interface fluxes preserve stability when combined with entropy-conserving split forms and consistent metric approximations.