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The BR1 scheme is stable for the compressible Navier–Stokes equations
Mathematical Institute, University of Cologne, Cologne, Germany.
Mathematical Institute, University of Cologne, Cologne, Germany.ORCID iD: 0000-0002-5902-1522
Max Planck Institute for Plasma Physics, Garching, Germany.
Department of Mathematics, The Florida State UniversityTallahassee, USA.
2018 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 77, no 1, p. 154-200Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Springer, 2018. Vol. 77, no 1, p. 154-200
Keywords [en]
Discontinuous Galerkin, Bassi and Rebay, viscous terms, linearized Navier-Stokes equations, compressible Navier-Stokes, energy stability, skew-symmetry, entropy-stability
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-156860DOI: 10.1007/s10915-018-0702-1ISI: 000443708200007Scopus ID: 2-s2.0-85045052289OAI: oai:DiVA.org:liu-156860DiVA, id: diva2:1315787
Funder
EU, European Research Council, 714487Available from: 2019-05-14 Created: 2019-05-14 Last updated: 2019-05-24Bibliographically approved

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Winters, Andrew Ross

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