Shape preserving interpolation with arbitrary smoothness based on mollification

A common problem in Physics and Engineering is to interpolate discrete samples while preserving the overall shape of the data: for example, maintaining positivity for a temperature curve, or monotonicity for a cumulative probability. The well-known methods do not have this property, and while several solutions can be found in the literature, they may not be smooth enough for some applications, too computationally expensive or difficult to implement. In this paper we present a shape preserving interpolation scheme that yields interpolants with an arbitrary differentiability class. It is suitable for modelling data in differential equations, fast and easy to implement. Our method is based on a convolutional operation called mollification, which we apply to a simpler shape preserving interpolant to achieve the desired degree of smoothness. With an appropriate polynomial mollification, the convolution has a closed form, so it can be computed quickly, giving a Cn interpolant. We define the general scheme for an arbitrary mollifier, provide an error analysis and an explicit C2 implementation. Finally, we show how the latter was applied successfully to model the plasma density in the electron cyclotron waves raytracer GRAY.