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 e, 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. (c) 2005 Published by Elsevier Inc.