  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.
