Recently,it was discovered that the entropy-conserving/dissipative high-order split-form discontinuous Galerkin discretizations have robustness issues when trying to solve the sim-ple 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 per-turbations added to a stable base flow.This is strongly related to an anti-diffusion mech-anism,that is inherent in entropy-conserving two-point fluxes,which are a key ingredi-ent 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().

Hendrik Ranocha;Gregor J.Gassner

Applied Mathematics Munster,University of Münster,48149 Münster,Germany;Computer Electrical and Mathematical Science and Engineering(CEMSE)Division,King Abdullah University of Science and Technology(KAUST),Thuwal 23955-6900,Saudi Arabia;Department of Mathematics and Computer Science,Center for Data and Simulation Science,University of Cologne,Cologne,Germany

应用数学与计算数学学报

[1]Hendrik Ranocha;Gregor J.Gassner-.Preventing Pressure Oscillations Does Not Fix Local Linear Stability Issues of Entropy-Based Split-Form High-Order Schemes)[J].应用数学与计算数学学报,2022(03):880-903

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