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 NN point vortices and between NN 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(N2)O(N2) is reduced to O(N(logN)4)O(N(logN)4) and the approximation error decreases like O(1/N)O(1/N) when N→∞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 NN points in the three-dimensional Cartesian coordinates.