Semi-Lagrangian advection on a spherical geodesic grid

A simple and efficient numerical method for solving the advection equation on the spherical surface is presented. To overcome the well-known ‘pole problem’ related to the polar singularity of spherical coordinates, the space discretization is performed on a geodesic grid derived by a uniform triangulation of the sphere; the time discretization uses a semi-Lagrangian approach. These two choices, efficiently combined in a substepping procedure, allow us to easily determine the departure points of the characteristic lines, avoiding any computationally expensive tree-search. Moreover, suitable interpolation procedures on such geodesic grid are presented and compared. The performance of the method in terms of accuracy and efficiency is assessed on two standard test cases: solid-body rotation and a deformation flow.