Modern shock-capturing schemes often suffer from numerical shock anomalies if the flow field contains strong shocks, which may limit their further application in hypersonic flow computations. In the current study, we devote our efforts to exploring the primary numerical characteristics and the underlying mechanism of shock instability for second-order finite-volume schemes. To this end, we, for the first time, develop the matrix stability analysis method for the finite-volume MUSCL approach. Such a linearized analysis method allows to investigate the shock instability problem of the finite-volume shock-capturing schemes in a quantitative and efficient manner. Results of the stability analysis demonstrate that the shock stability of second-order scheme is strongly related to the Riemann solver, Mach number, limiter function, numerical shock structure, and computational grid. Unique stability characteristics associated with these factors for second-order methods are revealed quantitatively with the established method. Source location of instability is also clarified by the matrix stability analysis method. Results show that the shock instability originates from the numerical shock structure. Such conclusions pave the way to better understand the shock instability problem and may shed new light on developing more reliable shock-capturing methods for compressible flows with high Mach number.

翻译：现代激波捕捉格式在处理包含强激波的流场时，常会出现数值激波异常现象，这限制了其在高超声速流动计算中的进一步应用。本研究致力于探索二阶有限体积格式激波不稳定性的主要数值特征及其内在机理。为此，我们首次针对有限体积MUSCL方法建立了矩阵稳定性分析方法。这种线性化分析方法能够以定量且高效的方式研究有限体积激波捕捉格式的激波不稳定性问题。稳定性分析结果表明，二阶格式的激波稳定性与黎曼求解器、马赫数、限制器函数、数值激波结构及计算网格密切相关。通过所建立的方法，定量揭示了这些因素对二阶格式特有的稳定性特征。矩阵稳定性分析方法还明确了不稳定性的来源位置。结果表明，激波不稳定性源于数值激波结构。这些结论为更好地理解激波不稳定性问题奠定了坚实基础，并可能为开发更可靠的高马赫数可压缩流激波捕捉方法提供新思路。