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Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points. (English) Zbl 1072.65114

Summary: A new fifth-order weighted essentially non-oscillatory scheme is developed. Necessary and sufficient conditions on the weights for fifth-order convergence are derived; one more condition than previously published is found. A detailed analysis reveals that the version of this scheme implemented by G.-S. Jiang and C.-W. Shu [ibid. 126, No. 1, 202–228 (1996; Zbl 0877.65065)] is, in general, only third-order accurate at critical points. This result is verified in a simple example. The magnitude of \(\epsilon\), a parameter which keeps the weights bounded, and the level of grid resolution are shown to determine the order of the scheme in a nontrivial way. A simple modification of the original scheme is found to be sufficient to give optimal order convergence even near critical points. This is demonstrated using the one-dimensional linear advection equation. Also, four examples utilizing the compressible Euler equations are used to demonstrate the scheme’s improved behavior for practical shock capturing problems.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L45 Initial value problems for first-order hyperbolic systems
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws

Citations:

Zbl 0877.65065
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Full Text: DOI

References:

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