The weighted essentially non-oscillatory {technique} using a stencil of $2r$ points (WENO-$2r$) is an interpolatory method that consists in obtaining a higher approximation order from the non-linear combination of interpolants of $r+1$ nodes. The result is an interpolant of order $2r$ at the smooth parts and order $r+1$ when an isolated discontinuity falls at any grid interval of the large stencil except at the central one. Recently, a new WENO method based on Aitken-Neville's algorithm has been designed for interpolation of equally spaced data at the mid-points and presents progressive order of accuracy close to discontinuities. This paper is devoted to constructing a general progressive WENO method for non-necessarily uniformly spaced data and several variables interpolating in any point of the central interval. Also, we provide explicit formulas for linear and non-linear weights and prove the order obtained. Finally, some numerical experiments are presented to check the theoretical results.

翻译：加权本质无振荡技术采用$2r$个节点模板（WENO-$2r$）是一种插值方法，通过对$r+1$个节点的插值函数进行非线性组合以获得更高逼近阶。该方法在光滑区域可获得$2r$阶插值精度，当孤立间断点落在大模板的网格区间（除中心区间外）时，仍能保持$r+1$阶精度。近期，基于Aitken-Neville算法设计的新型WENO方法针对中点等距数据插值，在间断附近表现出渐进精度阶数。本文致力于构建一种通用的渐进式WENO方法，适用于非均匀间距数据及多个变量在中心区间任意点处的插值。同时，我们给出了线性和非线性权重的显式表达式，并证明了所获得的精度阶数。最后通过数值实验验证理论结果。