Schnücke, Gero, Krais, Nico, Bolemann, Thomas and Gassner, Gregor J. (2019). Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes with Summation-by-Parts Property for Hyperbolic Conservation Laws. Technical Report.


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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.

Item Type: Preprints, Working Papers or Reports (Technical Report)
Translated abstract:
This work is focused on the entropy analysis of a semi-discrete nodal discontinuous Galerkin spectral element method (DGSEM) on moving meshes for hyperbolic conservation laws. The DGSEM is constructed with a local tensor-product Lagrange-polynomial basis computed from Legendre-Gauss-Lobatto (LGL) points. Furthermore, the collocation of interpolation and quadrature nodes is used in the spatial discretization. This approach leads to discrete derivative approximations in space that are summation-by-parts (SBP) operators. On a static mesh, the SBP property and suitable two-point flux functions, which satisfy the entropy condition from Tadmor, allow to mimic results from the continuous entropy analysis on the discrete level. In this paper, Tadmor’s condition is extended to the moving mesh framework. Based on the moving mesh entropy condition, entropy conservative two-point flux functions for the homogeneous shallow water equations and the compressible Euler equations are constructed. Furthermore, it will be proven that the semi-discrete moving mesh DGSEM is an entropy conservative scheme when a two-point flux function, which satisfies the moving mesh entropy condition, is applied in the split form DG framework. This proof does not require any exactness of quadrature in the spatial integrals of the variational form. Nevertheless, entropy conservation is not sufficient to tame discontinuities in the numerical solution and thus the entropy conservative moving mesh DGSEM is modified by adding numerical dissipation matrices to the entropy conservative fluxes. Then, the method becomes entropy stable such that the discrete mathematical entropy is bounded at any time by its initial and boundary data when the boundary conditions are specified appropriately. Besides the entropy stability, the time discretization of the moving mesh DGSEM will be investigated and it will be proven that the moving mesh DGSEM satisfies the free stream preservation property for an arbitrary s-stage Runge-Kutta method. The theoretical properties of the moving mesh DGSEM will be validated by numerical experiments for the compressible Euler equations.English
CreatorsEmailORCIDORCID Put Code
Schnücke, Gerogschnuec@math.uni-koeln.deUNSPECIFIEDUNSPECIFIED
Krais, Nicokrais@iag.uni-stuttgart.deUNSPECIFIEDUNSPECIFIED
Bolemann, Thomasbolemann@iag.uni-stuttgart.deUNSPECIFIEDUNSPECIFIED
Gassner, Gregor J.ggassner@math.uni-koeln.deUNSPECIFIEDUNSPECIFIED
URN: urn:nbn:de:hbz:38-95189
Series Name at the University of Cologne: Technical report series. Center for Data and Simulation Science
Volume: 2019,9
Date: 25 January 2019
Language: English
Faculty: Central Institutions / Interdisciplinary Research Centers
Divisions: Weitere Institute, Arbeits- und Forschungsgruppen > Center for Data and Simulation Science (CDS)
Subjects: Data processing Computer science
Technology (Applied sciences)
Uncontrolled Keywords:
Discontinuous GalerkinEnglish
Moving MeshesEnglish
Entropy StabilityEnglish
Free Stream PreservationEnglish


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