Fourier property of nonlinear weights and limiters in wave-space are analyzed. Firstly, a statistical method is designed to analyze Fourier property of nonlinear weights. Secondly, averaged values and deviations of nonlinear weights are shown, and values with change of wave-number are analyzed. It is helpful to comprehension of nonlinear schemes. The method is a guidance for designing shock-capture schemes.

To improve finite difference scheme, Fourier analysis is used to optimize dispersion and dissipation of WENO scheme. And optimal linear weights are given. A class of hybrid schemes is designed by combing optimized WENO schemes with monotonicity-preserving scheme. A weighted hybrid WENO scheme(H-WENO) is obtained. The scheme is tested with one-dimensional shock tube problem, Shu-Osher problem, two-dimensional Mach reflection problem and R-T instability problem. It shows that the scheme has strong ability to capture shock wave and high resolution for small scale wave structure, which is improved obviously compared with original WENO scheme.

Based on a high-order compact upwind scheme,a high-order shock-fitting finite difference scheme is studied for numerical simulation of the generation of boundary layer disturbance waves due to free-stream waves.Both steady and unsteady flow solutions of the receptivity problem are obtained by computing full Navier-Stokes equations.Numerical simulation of receptivity to free-stream disturbances for hypersonic boundary layers is performed.