Non-linear WENO B-spline based approximation method

TL;DR

This paper introduces a novel WENO B-spline quasi-interpolation method that applies WENO weights to B-spline functions, enhancing smoothness preservation and discontinuity adaptation, supported by theoretical analysis and extensive numerical experiments.

Contribution

It proposes a new WENO B-spline quasi-interpolation algorithm with modified weight functions, extending the approach to multiple dimensions and providing theoretical and numerical validation.

Findings

Conserves spline smoothness while adapting to discontinuities

Achieves high order accuracy in smooth regions

Effective in multi-dimensional settings

Abstract

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.