  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 (DOI: 10.5281/zenodo.4054366).
