  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.
