A novel scheme, based on third-order Weighted Essentially Non-Oscillatory (WENO) reconstructions, is presented. It attains unconditionally optimal accuracy when the data is smooth enough, even in presence of critical points, and second-order accuracy if a discontinuity crosses the data. The key to attribute these properties to this scheme is the inclusion of an additional node in the data stencil, which is only used in the computation of the weights measuring the smoothness. The accuracy properties of this scheme are proven in detail and several numerical experiments are presented, which show that this scheme is more efficient in terms of the error reduction versus CPU time than its traditional third-order counterparts as well as several higher-order WENO schemes that are found in the literature.

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