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 Spherical Essentially 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.