ENO and WENO cubic quasi-interpolating splines in Bernstein–Bézier form

Abstract

In this paper we propose the use of C1-continuous cubic quasi-interpolation schemes expressed in Bernstein–Bézier form to approximate functions with jumps. The construction of these schemes is explicit and consists of directly attaching the Bernstein–Bézier coefficients to appropriate combinations of the given data values. This construction can lead to quasi-interpolation schemes with free parameters. This allows to write these schemes of optimal convergence order as a non-negative convex combination of certain quasi-interpolation schemes of lower convergence order. The idea behind that, is to divide the data set used to define a quasi-interpolant of optimal order into subsets, and then define the associated quasi-interpolants. The free parameters facilitate the choice of the convex combination weights. We then apply the WENO approach to the weights to eliminate the Gibbs phenomenon that occurs when we approximate in a non-smooth region. The proposed schemes are of optimal order in the smooth regions and near optimal order is achieved in the neighboring region of discontinuity.