An extension of Draghicescu’s fast tree-code algorithm to the vortex method on a sphere

A fast and accurate algorithm to compute interactions between N point vortices and between N vortex blobs on a sphere is proposed. It is an extension of the fast tree-code algorithm developed by Draghicescu for the vortex method in the plane. When we choose numerical parameters in the fast algorithm suitably, the computational cost of O ( N 2 ) is reduced to O ( N ( log N ) 4 ) and the approximation error decreases like O ( 1 / N ) when N → ∞ , as demonstrated in the present article. We also apply the fast method to long-time evolution of two vortex sheets on the sphere to see the efficiency. A key point is to describe the equation of motion for the N points in the three-dimensional Cartesian coordinates.