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The shock instability problem commonly arises in flow simulations involvingstrong shocks, particularly when employing high-order schemes, limiting theirapplications in hypersonic flow simulations. This study focuses on exploringthe numerical characteristics and underlying mechanisms of shock instabilitiesin fifth-order finite-volume WENO schemes. To this end, for the first time, wehave established the matrix stability analysis method for the fifth-orderscheme. By predicting the evolution of perturbation errors in the exponentialgrowth stage, this method provides quantitative insights into the behavior ofshock-capturing and helps elucidate the mechanisms that cause shockinstabilities. Results reveal that even dissipative solvers also suffer fromshock instabilities when the spatial accuracy is increased to fifth-order.Further investigation indicates that this is due to the excessively highspatial accuracy of the WENO scheme near the numerical shock structure.Moreover, the shock instability problem of fifth-order schemes is demonstratedto be a multidimensional coupling problem. To stably capture strong shocks, itis crucial to have sufficient dissipation on transverse faces and ensure atleast two points within the numerical shock structure in the directionperpendicular to the shock. The source location of instability is alsoclarified by the matrix stability analysis method, revealing that theinstability arises from the numerical shock structure. Additionally, stabilityanalysis demonstrates that local characteristic decomposition helps mitigateshock instabilities in high-order schemes, although the instability stillpersists. These conclusions pave the way for a better understanding of theshock instability in fifth-order schemes and provide guidance for thedevelopment of more reliable high-order shock-capturing methods forcompressible flows with high Mach numbers.

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