matically ''jump'' to local ones as discontinuities are encountered. Hence the schemes are nonlinear and Gibbs phe-
We develop here compact high-order accurate nonlinear schemes for discontinuities capturing. Such schemes achieve high-order spa-nomenon is avoided. Two propositions have been proved, tial accuracy by the cell-centered compact schemes. Compact adapwhich show that in order to obtain uniformly high-order active interpolations of variables at cell edges are designed which curate compact interpolants, the interpolations of variables automatically ''jump'' to local ones as discontinuities being encounat cell edges should be prevented from crossing discontinutered. This is the key to make the overall compact schemes capture ous data, such that the accuracy analysis based on the Taylor discontinuities in a nonoscillatory manner. The analysis shows that the basic principle to design a compact interpolation of variables series expanding is valid over all grid points.
at the cell edges is to prevent it from crossing the discontinuous The organization of this paper is as follows. Section 2 data, such that the accuracy analysis based on Taylor series expresents the basic compact schemes for scalar hyperbolic panding is valid over all grid points. A high-order Runge-Kutta conservation law, including a fourth-order compact apmethod is employed for the time integration. The conservative propproximation to the first derivative, the compact adaptive erty, as well as the boundary schemes, is discussed. We also extend the schemes to a system of conservation laws. The extensions to interpolations of variables at cell edges, the analysis of the multidimensional problems are straightforward. Some typical oneinterpolations, and the time integration. In Section 3, we dimensional numerical examples, including the shock tube probdiscuss the boundary and near boundary algorithms and lem, strong shock waves with complex wave interactions, and the conservative property of the schemes. The extension ''shock/turbulence'' interaction, are presented. แฎ 1997 Academic Press to Euler equations is given in Section 4. Section 5 contains numerical experiments, including solutions to a linear scalar equation with different initial conditions, the solution 77