This paper introduces the Non-linear Partition of Unity Method (NL-PUM), a novel technique integrating Radial Basis Function interpolation and Weighted Essentially Non-Oscillatory algorithms. As far as we know, this is the first time that a non-linear PUM method is introduced in the literature. The main advance of this proposal is providing an algorithm that keeps the properties of the PUM at smooth zones, while introducing a non-linear modification close to the discontinuities to avoid oscillations. This is done automatically computing estimations of the smoothness of the data and replacing the PUM by Sheppard method when all the data is affected by a discontinuity. Thus, the computation of smoothness indicators and the use of compactly supported base functions ensure precision in regions affected by the presence of discontinuities. Error bounds are calculated and validate the effectiveness of the new method, showing improved interpolation capabilities at discontinuity regions as well as at smooth zones. A battery of experiments is presented to check the theoretical results provided.
