  The numerical simulation of supersonic complex flow problems demands
capabilities in identifying multiscale structures and capturing shocks,
imposing stringent requirements on the numerical scheme. The capability to
identify multiscale structures is closely related to the spectral properties of
the numerical scheme. Currently, existing methods to improve the spectral
properties of finite difference schemes face shortcomings such as parallel
difficulties (compact methods) or introducing unnecessary dispersion errors at
low wavenumbers due to accuracy loss (spectral-like optimization methods). In
this paper, we proposed an order-preserving spectral properties optimization
method based on the group velocity control theory: the weighted group velocity
control (WGVC) scheme. This method, centered around the concept of group
velocity, achieves low-wavenumber accuracy control and mid-wavenumber group
velocity control by designing smoothness indicators and nonlinear weighting
approach for wave packets. Furthermore, by embedding the WGVC scheme into
shock-capturing schemes such as WENO/TENO scheme, we not only preserve the
spectral properties of the WGVC scheme at medium to low wavenumbers but also
enhance the shock-capturing capability of the scheme. Theoretical and numerical
experiments verify that the new method has advantages such as order-preserving,
small dispersion and dissipation errors, and is very suitable for numerical
simulation of complex flow problems such as turbulence-shock boundary layer
interactions.
