Spherical Essentially Non-oscillatory (SENO) Interpolation

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We develop two new ideas for interpolation on \(\mathbb {S}^2\). In this first part, we will introduce a simple interpolation method named Spherical Interpolation of 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 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.