Compact high-order accurate nonlinear schemes

We develop here compact high-order accurate nonlinear schemes for discontinuities capturing. Such schemes achieve high-order spatial accuracy by the cell-centered compact schemes. Compact adaptive interpolations of variables at cell edges are designed which automatically ''jump'' to local ones as discontinuities being encountered. This is the key to make the overall compact schemes capture discontinuities in a nonoscillatory manner. The analysis shows that the basic principle to design a compact interpolation of variables at the cell edges is to prevent it from crossing the discontinuous data, such that the accuracy analysis based on Taylor series expanding is valid over all grid points. A high-order Runge-Kutta method is employed for the time integration. The conservative property, as well as the boundary schemes, is discussed. We also extend the schemes to a system of conservation laws. The extensions to multidimensional problems are straightforward. Some typical one-dimensional numerical examples, including the shock tube problem, strong shock waves with complex wave interactions, and ''shock/turbulence'' interaction, are presented. (C) 1997 Academic Press