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.

中文翻译：

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一种新的 WENO-2$r$ 算法，具有接近不连续性的渐进顺序

在本文中，我们介绍了对 [S. Amat, J. Ruiz, C.-W. Shu，关于一种新的 2r 阶 WENO 算法，其精度提高了接近不连续性，App。数学。莱特。105 (2020), 106-298]。在上述工作中，我们设法获得了一种算法，该算法达到了 WENO-6 接近不连续性的渐进和最佳精度顺序。对于更高的阶，即WENO-8、WENO-10等，我们发现之前的算法在检测不连续性时会出现一些阴影，这意味着准确度比同阶WENO获得的要好，但不是最优的。在本文中，我们对原始算法中使用的平滑度指标进行了修改，旨在解决这个问题并获得具有接近不连续性的渐进精度顺序的 WENO-2r 算法。我们还提供了用于算法的任何阶 2r 的所有权重的准确性和显式公式的证明。