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Henrick, Andrew K.; Aslam, Tariq D.; Powers, Joseph M.

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

J. Comput. Phys. 207, No. 2, 542-567 (2005).
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.

Cited in 14 Reviews
Cited in 330 Documents

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

Keywords:

WENO; Hyperbolic equations; fifth-order weighted essentially non-oscillatory scheme; convergence; linear advection equation; compressible Euler equations; shock capturing problems; numerical examples; conservation laws

Citations:

Zbl 0877.65065
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\textit{A. K. Henrick} et al., J. Comput. Phys. 207, No. 2, 542--567 (2005; Zbl 1072.65114)
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