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Nonlinearly stable compact schemes for shock calculations. (English) Zbl 0805.65085
Applications of high-order compact finite difference methods for shock calculations are discussed. The main idea is the definition of a local mean that serves as a reference for introducing a local nonlinear limiting to control spurious oscillations while keeping the accuracy of the scheme. For scalar conservation laws the resulting scheme can be proven total variation stable in one-space dimension, and maximum norm stable in multispace dimensions. The idea in this paper can also be applied to other implicit schemes.

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
35L67 Shocks and singularities for hyperbolic equations
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