On a regularization-correction approach for the approximation of piecewise smooth functions

TL;DR

This paper introduces a three-stage regularization-correction method for approximating piecewise smooth functions, achieving high accuracy and regularity without oscillations or diffusion near singularities.

Contribution

A novel three-stage algorithm combining smoothing, high-order linear approximation, and correction to preserve regularity and accuracy in piecewise smooth function approximation.

Findings

Achieves high precision and regularity in approximations

Prevents oscillations and diffusion near discontinuities

Applicable to point-value and cell-average data

Abstract

Linear approximation approaches suffer from Gibbs oscillations when approximating functions with singularities. ENO-SR resolution is a local approach avoiding oscillations and with a full order of accuracy, but a loss of regularity of the approximant appears. The goal of this paper is to introduce a new approach having both properties of full accuracy and regularity. In order to obtain it, we propose a three-stage algorithm: first, the data is smoothed by subtracting an appropriate non-smooth data sequence; then a chosen high order linear approximation operator is applied to the smoothed data and finally, an approximation with the proper singularity structure is reinstated by correcting the smooth approximation with the non-smooth element used in the first stage. We apply this approach to both cases of point-value data and of cell-average data, using the 4-point subdivision algorithm in the second stage. Using the proposed approach we are able to construct approximations with high precision, with high piecewise regularity, and without diffusion nor oscillations in the presence of discontinuities.