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.
Categories: math.NA cs.NA