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Compact high-order accurate nonlinear schemes. (English) Zbl 0870.65075
A new methodology to construct compact high-order accurate nonlinear schemes for capturing discontinuities in solutions of hyperbolic conservation laws is developed. Compact adaptive interpolations of variables at cell edges are used which automatically switch to local ones as discontinuities appear.
In this way new schemes, which capture discontinuities in a nonoscillatory way, are derived. The goal is to prevent a compact interpolation of variables from crossing the discontinuous data, such that the accuracy analysis based on Taylor expansion is valid over all grid points. For the time integration a high-order Runge-Kutta method is employed. The authors derive third- and fourth-order schemes for one-dimensional nonlinear conservation laws. Some typical one-dimensional numerical examples, such as shock tube problem, strong shock waves with complex wave interactions, and shock/turbulence interaction, are presented.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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