In this work we present a new WENO b-spline based quasi-interpolation algorithm. The novelty of this construction resides in the application of the WENO weights to the b-spline functions, that are a partition of unity, instead to the coefficients that multiply the b-spline functions of the spline. The result obtained conserves the smoothness of the original spline and presents adaption to discontinuities in the function. Another new idea that we introduce in this work is the use of different base weight functions from those proposed in classical WENO algorithms. Apart from introducing the construction of the new algorithms, we present theoretical results regarding the order of accuracy obtained at smooth zones and close to the discontinuity, as well as theoretical considerations about how to design the new weight functions. Through a tensor product strategy, we extend our results to several dimensions. In order to check the theoretical results obtained, we present an extended battery of numerical experiments in one, two and tree dimensions that support our conclussions.

翻译：本文提出一种基于WENO B样条的新准插值算法。该构造的创新性在于将WENO权重应用于构成单位分割的B样条函数本身，而非作用于样条中B样条函数的乘性系数。所得结果在保持原始样条光滑性的同时，实现了对函数间断点的自适应处理。本文引入的另一新思想是采用不同于经典WENO算法中提出的基权重函数。除介绍新算法的构造外，我们还给出了关于光滑区域及间断附近精度阶的理论结果，以及设计新权重函数的理论考量。通过张量积策略，我们将结果推广至多维情形。为验证理论结果，我们展示了一维、二维及三维空间中的大量数值实验，这些实验结果支持了我们的结论。