Journal of Scientific Computing

, Volume 77, Issue 1, pp 154–200 | Cite as

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

  • Gregor J. Gassner
  • Andrew R. Winters
  • Florian J. Hindenlang
  • David A. Kopriva


In this work we prove that the original (Bassi and Rebay in J Comput Phys 131:267–279, 1997) scheme (BR1) for the discretization of second order viscous terms within the discontinuous Galerkin collocation spectral element method (DGSEM) with Gauss Lobatto nodes is stable. More precisely, we prove in the first part that the BR1 scheme preserves energy stability of the skew-symmetric advection term DGSEM discretization for the linearized compressible Navier–Stokes equations (NSE). In the second part, we prove that the BR1 scheme preserves the entropy stability of the recently developed entropy stable compressible Euler DGSEM discretization of Carpenter et al. (SIAM J Sci Comput 36:B835–B867, 2014) for the non-linear compressible NSE, provided that the auxiliary gradient equations use the entropy variables. Both parts are presented for fully three-dimensional, unstructured curvilinear hexahedral grids. Although the focus of this work is on the BR1 scheme, we show that the proof naturally includes the Local DG scheme of Cockburn and Shu.


Discontinuous Galerkin Bassi and Rebay Viscous terms Linearized Navier–Stokes equations Compressible Navier–Stokes Energy stability Skew-symmetry Entropy stability 



This work was supported by a grant from the Simons Foundation (#426393, David Kopriva). G. G has been supported by the European Research Council (ERC) under the European Union’s Eights Framework Program Horizon 2020 with the research project Extreme, ERC Grant Agreement No. 714487.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Gregor J. Gassner
    • 1
  • Andrew R. Winters
    • 1
  • Florian J. Hindenlang
    • 2
  • David A. Kopriva
    • 3
  1. 1.Mathematical InstituteUniversity of CologneCologneGermany
  2. 2.Max Planck Institute for Plasma PhysicsGarchingGermany
  3. 3.Department of MathematicsThe Florida State UniversityTallahasseeUSA

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