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 (https://doi.org/10.5281/zenodo.4054366).
Highlights
Discontinuous Galerkin (DG) spectral collocation methods with summation-by-parts (SBP) property have gained a lot of traction in the high-order community [4, 5, 11, 30, 31, 45,46,47], due to the possibility to construct entropy-conservative/dissipative [7, 10, 27, 32, 50, 51] and/or kinetic-energy-preserving (KEP) [22, 26, 33, 34] discretizations
A key building block in these novel high-order collocation discretizations is a special twopoint flux formulation of the volume terms introduced by LeFloch et al for central finite differences in periodic domains [27], Fisher et al for SBP finite differences in bounded domains [10], and by Carpenter et al for discontinuous spectral collocation schemes [2, 3]
The same is for instance true for kinetic energy preservation [14, 33]; as we will show in this paper, it holds for pressure equilibrium preservation
Summary
Discontinuous Galerkin (DG) spectral collocation methods with summation-by-parts (SBP) property have gained a lot of traction in the high-order community [4, 5, 11, 30, 31, 45,46,47], due to the possibility to construct entropy-conservative/dissipative [7, 10, 27, 32, 50, 51] and/or kinetic-energy-preserving (KEP) [22, 26, 33, 34] discretizations. It turns out that the linearized spectrum shows spurious modes with exponential growth, that may cause fatal crashing of the simulation In another recent paper, Shima et al [44] investigated the capability of their KEP twopoint flux to retain what they call pressure equilibrium. Shima et al [44] found exponential spurious growth for a similar density-wave test case when using their KEP two-point flux [26] When they modified the two-point flux to discretely preserve pressure equilibrium, they could demonstrate numerically that the novel scheme robustly solves the density-wave, even for very long simulation times.
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