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A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations

Hennemann, Sebastian and Rueda-Ramírez, Andrés M. and Hindenlang, Florian J. and Gassner, Gregor J. (2020) A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations. Journal of Computational Physics. Elsevier. doi: 10.1016/j.jcp.2020.109935. ISSN 0021-9991. (In Press)

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Official URL: https://www.sciencedirect.com/science/article/pii/S0021999120307099


The main result in this paper is a provably entropy stable shock capturing approach for the high order entropy stable Discontinuous Galerkin Spectral Element Method (DGSEM) based on a hybrid blending with a subcell low order variant. Since it is possible to rewrite a high order summation-by-parts (SBP) operator into an equivalent conservative finite volume form, we were able to design a low order scheme directly with the Legendre-Gauss-Lobatto (LGL) nodes that is compatible to the discrete entropy analysis used for the proof of the entropy stable DGSEM. Furthermore, we present a hybrid low order/high order discretisation where it is possible to seamlessly blend between the two approaches, while still being provably entropy stable. With tensor products and careful design of the low order scheme on curved elements, we are able to extend the approach to three spatial dimensions on unstructured curvilinear hexahedral meshes. We validate our theoretical findings and demonstrate convergence order for smooth problems, conservation of the primary quantities and discrete entropy stability for an arbitrary blending on curvilinear grids. In practical simulations, we connect the blending factor to a local troubled element indicator that provides the control of the amount of low order dissipation injected into the high order scheme. We modified an existing shock indicator, which is based on the modal polynomial representation, to our provably stable hybrid scheme. The aim is to reduce the impact of the parameters as good as possible. We describe our indicator in detail and demonstrate its robustness in combination with the hybrid scheme, as it is possible to compute all the different test cases without changing the indicator. The test cases include e.g. the double Mach reflection setup, forward and backward facing steps with shock Mach numbers up to 100. The proposed approach is relatively straight forward to implement in an existing entropy stable DGSEM code as only modifications local to an element are necessary.

Item URL in elib:https://elib.dlr.de/134230/
Document Type:Article
Title:A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations
AuthorsInstitution or Email of AuthorsAuthor's ORCID iD
Hennemann, Sebastiansebastian.hennemann (at) dlr.deUNSPECIFIED
Rueda-Ramírez, Andrés M.aruedara (at) uni-koeln.deUNSPECIFIED
Hindenlang, Florian J.florian.hindenlang (at) ipp.mpg.deUNSPECIFIED
Gassner, Gregor J.ggassner (at) uni-koeln.deUNSPECIFIED
Date:21 October 2020
Journal or Publication Title:Journal of Computational Physics
Refereed publication:Yes
Open Access:No
Gold Open Access:No
In ISI Web of Science:Yes
DOI :10.1016/j.jcp.2020.109935
Status:In Press
Keywords:Compressible Euler equations Discontinuous Galerkin spectral element method Shock capturing Entropy stability Computational robustness
HGF - Research field:Aeronautics, Space and Transport
HGF - Program:Aeronautics
HGF - Program Themes:propulsion systems
DLR - Research area:Aeronautics
DLR - Program:L ER - Engine Research
DLR - Research theme (Project):L - Virtual Engine and Validation methods (old)
Location: Köln-Porz
Institutes and Institutions:Institute of Propulsion Technology
Institute of Test and Simulation for Gas Turbines
Deposited By: Hennemann, Sebastian
Deposited On:10 Nov 2020 10:35
Last Modified:10 Nov 2020 10:35

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