论文标题
防止压力振荡不能解决基于熵的分裂高阶方案的本地线性稳定性问题
Preventing pressure oscillations does not fix local linear stability issues of entropy-based split-form high-order schemes
论文作者
论文摘要
最近，发现在试图解决可压缩欧拉方程的简单密度传播示例时，熵的/耗散/耗散的高阶拆分格式不连续的盖尔金离散是有鲁棒的问题。该问题与缺少局部线性稳定性有关，即离散化对稳定基础流动的扰动的稳定性。这与一种抗扩散机制密切相关，该机制是熵的两点通量固有的，这是高阶不连续的盖尔金扩展的关键成分。在本文中，我们研究了这些最近发现的熵保守性/耗散性高阶拆分格式不连续的盖尔金方法的局部线性稳定性问题是否是对这些最近发现的局部线性稳定性问题的补救措施。压力平衡保存描述了离散化的特性，以保持压力和速度恒定以使纯密度波传播。我们介绍了完整的理论推导，分析，并显示了相应的数值结果，以强调我们的发现。此外，我们表征了熵保守，动力学能量，具有压力平衡的欧拉尔方程的数值通量，并具有不取决于压力的密度通量。重现本文中介绍的所有数值实验的源代码可在线获得（doi：10.5281/Zenodo.4054366）。
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).
