Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points

In this paper, 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 Jiang and Shu [G.-S. Jiang, C.-W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996) 202-228] is, in general, only third-order accurate at critical points. This result is verified in a simple example. The magnitude of ϵ, a parameter which keeps the weights bounded, and the level of grid resolution are shown to determine the order of the scheme in a non-trivial 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.