On an algorithm to adapt spline approximations to the presence of discontinuities

Abstract

Interpolatory splines are usually useful to reconstruct data that present certain regularity. This paper is devoted to the construction and analysis of a new technique that allows to improve the accuracy of splines near corner singularities in the point values and jump discontinuities in the cell averages. The detection of discontinuities is easily done through the processing of the right hand side of the system of equations of the spline, that contains divided differences. The process of adaption will require some knowledge about the position of the discontinuity and the values of the function and its derivatives at the discontinuity. Using Harten’s multiresolution we can adapt splines to the presence of corner singularities and jump discontinuities. Thanks to the adaption, the smearing of corner singularities does not appear in the reconstruction obtained in the point values and Gibbs phenomenon and diffusion of jump discontinuities is also eliminated. These numerical effects are a consequence of assuming that the data used in the reconstruction comes from the discretization of a continuous function with certain regularity and this might not be true.