Spherical Essentially Non-Oscillatory (SENO) Interpolation

TL;DR

This paper introduces SIDER-n, a new spherical interpolation method, and SENO, an oscillation-reducing interpolation technique for curves on the sphere, improving accuracy near discontinuities.

Contribution

The paper presents the first spherical interpolation method with arbitrary order and a non-oscillatory interpolation approach inspired by ENO for $ ext{S}^2$ curves.

Findings

SIDER-n provides $C^{n}$ smooth interpolants on $ ext{S}^2$.

SENO reduces oscillations near sharp features in spherical curves.

Numerical examples demonstrate improved accuracy and stability.

Abstract

We develop two new ideas for interpolation on $\mathbb{S}^2$. In this first part, we will introduce a simple interpolation method named \textit{Spherical Interpolation of orDER} $n$ (SIDER-$n$) that gives a $C^{n}$ interpolant given $n \geq 2$. The idea generalizes the construction of the B\'{e}zier curves developed for $\mathbb{R}$. The second part incorporates the ENO philosophy and develops a new \textit{Spherical Essentially Non-Oscillatory} (SENO) interpolation method. When the underlying curve on $\mathbb{S}^2$ has kinks or sharp discontinuity in the higher derivatives, our proposed approach can reduce spurious oscillations in the high-order reconstruction. We will give multiple examples to demonstrate the accuracy and effectiveness of the proposed approaches.