A new way for constructing high accuracy shock-capturing generalized compact difference schemes

The generalized compact (GC) schemes and some of their important properties are presented. And a new way for constructing high order accuracy and high-resolution GC schemes is presented. The schemes constructed by using this way could satisfy some principles and demands prescribed in advance to ensure some desired properties to the schemes, such as the principle about suppression of the oscillations, the principle of stability, the order of accuracy and number of scheme points, etc. As two examples, a three-point third-order compact scheme and a three-point fifth-order GC scheme satisfying the principle about suppression of the oscillations and the principle of stability are described in this paper. Numerical results show that these schemes are shock-capturing. The time-dependent boundary conditions proposed by Thompson are well employed when the algorithm is applied to the Euler equations of gas dynamics. Fourier analysis shows that the resolution characteristics are spectral-like.