Non-linear WENO B-spline based approximation method

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 form a partition of unity, instead of 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 extensive battery of numerical experiments in one, two, and three dimensions that support our conclusions.