Spherical Essentially Non-Oscillatory (SENO) Interpolation

We develop two new ideas for interpolation on ${\mathbb{S}}^2$. In this first part, we will introduce a simple interpolation method named $\mathit{\text{Spherical}}\mathit{\text{Interpolation}}\mathit{\text{of}}\mathit{\text{orDER}}$ $n$ (SIDER-$n$) that gives a $C^n$ interpolant given $n\ge 2$. The idea generalizes the construction of the Bézier curves developed for $\mathbb{R}$. The second part incorporates the ENO philosophy and develops a new $\mathit{\text{Spherical}}\mathit{\text{Essentially}}\mathit{\text{Non-Oscillatory}}$ (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.