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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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)
CreatorsEmailORCIDORCID Put Code
Gassner, Gregor J.ggassner@math.uni-koeln.deUNSPECIFIEDUNSPECIFIED
Svärd, MagnusMagnus.Svard@uib.noUNSPECIFIEDUNSPECIFIED
Hindenlang, Florian J.florian.hindenlang@ipp.mpg.deUNSPECIFIEDUNSPECIFIED
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
Technology (Applied sciences)
Uncontrolled Keywords:
entropy-conserving two-point fluxEnglish
entropy-stable high order schemeEnglish
discontinuous GalerkinEnglish
non-linear stabilityEnglish
linear stabilityEnglish


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