Universität zu Köln

Hennemann, Sebastian, Rueda-Ramirez, Andres M., Hindenlang, Florian J. and Gassner, Gregor J. (2021). A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations. J. Comput. Phys., 426. SAN DIEGO: ACADEMIC PRESS INC ELSEVIER SCIENCE. ISSN 1090-2716

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Abstract

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. (C) 2020 Elsevier Inc. All rights reserved.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Hennemann, Sebastian UNSPECIFIED UNSPECIFIED UNSPECIFIED Rueda-Ramirez, Andres M. UNSPECIFIED UNSPECIFIED UNSPECIFIED Hindenlang, Florian J. UNSPECIFIED UNSPECIFIED UNSPECIFIED Gassner, Gregor J. UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-583291
DOI:	10.1016/j.jcp.2020.109935
Journal or Publication Title:	J. Comput. Phys.
Volume:	426
Date:	2021
Publisher:	ACADEMIC PRESS INC ELSEVIER SCIENCE
Place of Publication:	SAN DIEGO
ISSN:	1090-2716
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language DISCONTINUOUS GALERKIN METHODS; SUMMATION; PARTS; SCHEMES; GAUSS; FLOW Multiple languages Computer Science, Interdisciplinary Applications; Physics, Mathematical Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/58329