Revisit to the WGVC schemes: a nonlinear order-preserving and spectral-property-optimized methodology and its enhancement

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.