Monotone cubic spline interpolation for functions with a strong gradient

TL;DR

This paper develops methods for constructing monotone cubic spline interpolants that handle steep gradients and discontinuities, ensuring smoothness, accuracy, and monotonicity in data interpolation.

Contribution

It introduces new non-linear formulas for monotone cubic splines based on Hermite interpolators, with theoretical analysis and numerical validation.

Findings

Proposed methods preserve data monotonicity.

Achieved high approximation order in numerical experiments.

Validated effectiveness in handling steep gradients.

Abstract

Spline interpolation has been used in several applications due to its favorable properties regarding smoothness and accuracy of the interpolant. However, when there exists a discontinuity or a steep gradient in the data, some artifacts can appear due to the Gibbs phenomenon. Also, preservation of data monotonicity is a requirement in some applications, and that property is not automatically verified by the interpolator. In this paper, we study sufficient conditions to obtain monotone cubic splines based on Hermite cubic interpolators and propose different ways to construct them using non-linear formulas. The order of approximation, in each case, is calculated and several numerical experiments are performed to contrast the theoretical results.