  In this article we present a modification of the algorithm for data
discretized in the point values introduced in [S. Amat, J. Ruiz, C.-W. Shu, On
a new WENO algorithm of order 2r with improved accuracy close to
discontinuities, App. Math. Lett. 105 (2020), 106-298]. In the aforementioned
work, we managed to obtain an algorithm that reaches a progressive and optimal
order of accuracy close to discontinuities for WENO-6. For higher orders, i.e.
WENO-8, WENO-10, etc. We have found that the previous algorithm presents some
shadows in the detection of discontinuities, meaning that the order of accuracy
is better than the one attained by WENO of the same order, but not optimal. In
this article we present a modification of the smoothness indicators used in the
original algorithm, oriented to solve this problem and to attain a WENO-2r
algorithm with progressive order of accuracy close to the discontinuities. We
also present proofs for the accuracy and explicit formulas for all the weights
used for any order 2r of the algorithm.
