Numerical stability analysis of shock-capturing methods for strong shocks II: high-order finite-volume schemes

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.