Schnücke, Gero, Krais, Nico, Bolemann, Thomas and Gassner, Gregor J.
(2019).
Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes with SummationbyParts Property for Hyperbolic Conservation Laws.
Technical Report.
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 NavierStokes Equations, Journal of Computational Physics, 131:267–279, 1997] to get a provably stable discontinuous Galerkin collocation spectral element method (DGSEM) with GaussLobatto (GL) nodes for the compressible NavierStokes 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 twopoint 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 entropystability for the linear or nonlinear compressible NSE, respectively.
Item Type: 
Preprints, Working Papers or Reports
(Technical Report)

Translated abstract: 
Abstract  Language 

This work is focused on the entropy analysis of a semidiscrete nodal discontinuous Galerkin spectral element method (DGSEM) on moving meshes for hyperbolic conservation laws. The DGSEM is constructed with a local tensorproduct Lagrangepolynomial basis computed from LegendreGaussLobatto (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 summationbyparts (SBP) operators. On a static mesh, the SBP property and suitable twopoint 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 twopoint flux functions for the homogeneous shallow water equations and the compressible Euler equations are constructed. Furthermore, it will be proven that the semidiscrete moving mesh DGSEM is an entropy conservative scheme when a twopoint 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 sstage RungeKutta method.
The theoretical properties of the moving mesh DGSEM will be validated by numerical experiments for the compressible Euler equations.  English 

Creators: 
Creators  Email  ORCID  ORCID Put Code 

Schnücke, Gero  gschnuec@math.unikoeln.de  UNSPECIFIED  UNSPECIFIED  Krais, Nico  krais@iag.unistuttgart.de  UNSPECIFIED  UNSPECIFIED  Bolemann, Thomas  bolemann@iag.unistuttgart.de  UNSPECIFIED  UNSPECIFIED  Gassner, Gregor J.  ggassner@math.unikoeln.de  UNSPECIFIED  UNSPECIFIED 

URN: 
urn:nbn:de:hbz:3895189 
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 Mathematics Technology (Applied sciences) 
Uncontrolled Keywords: 
Keywords  Language 

Discontinuous Galerkin  English  SummationbyParts  English  Moving Meshes  English  Entropy Stability  English  Free Stream Preservation  English 

URI: 
http://kups.ub.unikoeln.de/id/eprint/9518 
Downloads per month over past year
Export
Actions (login required)

View Item 