Preventing Pressure Oscillations Does Not Fix Local Linear Stability Issues of Entropy-Based Split-Form High-Order Schemes

Hendrik Ranocha, Gregor J. Gassner

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

Recently, it was discovered that the entropy-conserving/dissipative high-order split-form discontinuous Galerkin discretizations have robustness issues when trying to solve the simple density wave propagation example for the compressible Euler equations. The issue is related to missing local linear stability, i.e., the stability of the discretization towards perturbations added to a stable base flow. This is strongly related to an anti-diffusion mechanism, that is inherent in entropy-conserving two-point fluxes, which are a key ingredient for the high-order discontinuous Galerkin extension. In this paper, we investigate if pressure equilibrium preservation is a remedy to these recently found local linear stability issues of entropy-conservative/dissipative high-order split-form discontinuous Galerkin methods for the compressible Euler equations. Pressure equilibrium preservation describes the property of a discretization to keep pressure and velocity constant for pure density wave propagation. We present the full theoretical derivation, analysis, and show corresponding numerical results to underline our findings. In addition, we characterize numerical fluxes for the Euler equations that are entropy-conservative, kinetic-energy-preserving, pressure-equilibrium-preserving, and have a density flux that does not depend on the pressure. The source code to reproduce all numerical experiments presented in this article is available online (10.5281/zenodo.4054366).
Original languageEnglish (US)
JournalCommunications on Applied Mathematics and Computation
DOIs
StatePublished - Jul 31 2021

Bibliographical note

KAUST Repository Item: Exported on 2021-10-18
Acknowledgements: Research reported in this publication was supported by the King Abdullah University of Science and Technology (KAUST). Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure. Gregor Gassner is supported by the European Research Council (ERC) under the European Union’s Eights Framework Program Horizon 2020 with the research project Extreme, ERC Grant Agreement No. 714487.

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