A Semi-Lagrangian Spherical Essentially Non-Oscillatory (SENO) Scheme for Advection Equations of S2-valued Functions

We develop a numerical scheme for solving the advection equation of ${\mathbb{S}}^2$-valued functions of real variables, which models the time-evolution of a ${\mathbb{S}}^2$-valued mapping on the real line by a known velocity field. The idea is to extend the semi-Lagrangian method for the linear scalar advection equation. We first construct the backward flow map between two adjacent time levels and then interpolate the discrete ordered data of ${\mathbb{S}}^2$. To handle ${\mathbb{S}}^2$-functions which have kinks or sharp discontinuity in their components, we incorporate the $\mathit{\text{Spherical}}\mathit{\text{Essentially}}\mathit{\text{Non-Oscillatory}}$ (SENO) interpolation method, which effectively reduces the spurious oscillations in high-order reconstructions. We will show multiple examples to demonstrate the accuracy and effectiveness of the proposed algorithm for the partial differential equation of ${\mathbb{S}}^2$-functions.