Universität zu Köln

Gassner, Gregor J., Svärd, Magnus and Hindenlang, Florian J. (2020). Stability issues of entropy-stable and/or split-form high-order schemes -- Analysis of local linear stability. Technical Report.

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Abstract

The focus of the present research is on the analysis of local linear stability of high-order (including split-form) summation-by-parts methods, with e.g. two-point entropy-conserving fluxes, approximating non-linear conservation laws. Our main finding is that local linear stability is not guaranteed even when the scheme is non-linearly stable and that this has grave implications for simulation results. We show that entropy-conserving two-point fluxes are inherently locally linearly unstable, as they can be dissipative or anti-dissipative. Unfortunately, these fluxes are at the core of many commonly used high-order entropy-stable extensions, including split-form summation-by-parts discontinuous Galerkin spectral element methods (or spectral collocation methods). For the non-linear Burgers equation, we demonstrate numerically that such schemes cause exponential growth of errors. Furthermore, we demonstrate a similar abnormal behaviour for the compressible Euler equations. Finally, we demonstrate numerically that other commonly used split-forms, such as the Kennedy and Gruber splitting, are also locally linearly unstable.

Item Type:	Preprints, Working Papers or Reports (Technical Report)
Creators:	Creators Email ORCID ORCID Put Code Gassner, Gregor J. ggassner@math.uni-koeln.de UNSPECIFIED UNSPECIFIED Svärd, Magnus Magnus.Svard@uib.no UNSPECIFIED UNSPECIFIED Hindenlang, Florian J. florian.hindenlang@ipp.mpg.de UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-115311
Series Name at the University of Cologne:	Technical report series. Center for Data and Simulation Science
Volume:	2020,3
Date:	20 July 2020
Language:	English
Faculty:	Central Institutions / Interdisciplinary Research Centers
Divisions:	Weitere Institute, Arbeits- und Forschungsgruppen > Center for Data and Simulation Science (CDS)
Subjects:	Natural sciences and mathematics Mathematics Technology (Applied sciences)
Uncontrolled Keywords:	Keywords Language entropy-conserving two-point flux English entropy-stable high order scheme English summation-by-parts English split-form English discontinuous Galerkin English non-linear stability English linear stability English
URI:	http://kups.ub.uni-koeln.de/id/eprint/11531