The shock instability problem commonly arises in flow simulations involving strong shocks, particularly when employing high-order schemes, limiting their applications in hypersonic flow simulations. This study focuses on exploring the numerical characteristics and underlying mechanisms of shock instabilities in fifth-order finite-volume WENO schemes. To this end, for the first time, we have established the matrix stability analysis method for the fifth-order scheme. By predicting the evolution of perturbation errors in the exponential growth stage, this method provides quantitative insights into the behavior of shock-capturing and helps elucidate the mechanisms that cause shock instabilities. Results reveal that even dissipative solvers also suffer from shock instabilities when the spatial accuracy is increased to fifth-order. Further investigation indicates that this is due to the excessively high spatial accuracy of the WENO scheme near the numerical shock structure. Moreover, the shock instability problem of fifth-order schemes is demonstrated to be a multidimensional coupling problem. To stably capture strong shocks, it is crucial to have sufficient dissipation on transverse faces and ensure at least two points within the numerical shock structure in the direction perpendicular to the shock. The source location of instability is also clarified by the matrix stability analysis method, revealing that the instability arises from the numerical shock structure. Additionally, stability analysis demonstrates that local characteristic decomposition helps mitigate shock instabilities in high-order schemes, although the instability still persists. These conclusions pave the way for a better understanding of the shock instability in fifth-order schemes and provide guidance for the development of more reliable high-order shock-capturing methods for compressible flows with high Mach numbers.

翻译：强激波模拟中常见的激波不稳定性问题，尤其在高阶格式应用时更为突出，这限制了其在高超声速流动模拟中的应用。本研究聚焦于五阶有限体积WENO格式中激波不稳定性的数值特征及潜在机理。为此，我们首次建立了五阶格式的矩阵稳定性分析方法。通过预测误差扰动在指数增长阶段的演化规律，该方法可定量揭示激波捕获行为特性，并阐明导致激波不稳定性的机制。结果表明：当空间精度提升至五阶时，即使具有耗散特性的求解器也会出现激波不稳定性。进一步研究发现，这是由于数值激波结构附近WENO格式的空间精度过高所致。此外，五阶格式的激波不稳定性被证实是多维耦合问题。要稳定捕获强激波，关键需在横向界面提供足够耗散，并确保激波垂直方向数值结构内至少包含两个网格点。矩阵稳定性分析方法还明确了不稳定性源位置，证实不稳定性源于数值激波结构。稳定性分析进一步表明，局部特征分解虽有助于缓解高阶格式中的激波不稳定性，但该不稳定性问题依然存在。这些结论为深入理解五阶格式激波不稳定性机理奠定了基础，并为发展适用于高马赫数可压缩流的高可靠高阶激波捕获方法提供了指导。