Adapting Cubic Hermite Splines to the Presence of Singularities Through Correction Terms

Abstract

Hermite interpolation is classically used to reconstruct smooth data when the function and its first order derivatives are available at certain nodes. If first order derivatives are not available, it is easy to set a system of equations imposing some regularity conditions at the data nodes in order to obtain them. This process leads to the construction of a Hermite spline. The problem of the described Hermite splines is that the accuracy is lost if the data contains singularities. The consequence is the appearance of oscillations, if there is a jump discontinuity in the function, that globally affects the accuracy of the spline, or the smearing of singularities, if the discontinuities are in the derivatives of the function. This paper is devoted to the construction and analysis of a new technique that allows for the computation of accurate first order derivatives of a function close to singularities using a Hermite spline. The idea is to correct the system of equations of the spline in order to attain the desired accuracy even close to the singularities. Once we have computed the first order derivatives with enough accuracy, a correction term is added to the Hermite spline in the intervals that contain a singularity. The aim is to reconstruct piecewise smooth functions with O(h^4) accuracy even close to the singularities. The process of adaption will require some knowledge about the position of the singularity and the jumps of the function and some of its derivatives at the singularity. The whole process can be used as a post-processing, where a correction term is added to the classical cubic Hermite spline. Proofs for the accuracy and regularity of the corrected spline and its derivatives are given. We also analyse the mechanism that eliminates the Gibbs phenomenon close to jump discontinuities in the function. The numerical experiments presented confirm the theoretical results obtained.