A fast tree-code algorithm for the vortex method on a sphere
Preprint Series # 847
SAKAJO, Takashi A fast tree-code algorithm for the vortex method on a sphere. (2007);
A fast and accurate algorithm to compute interaction between $N$ point vortices on a sphere is proposed. It is an extension of the fast tree-code algorithm based on the Taylor expansion developed by Draghicescu for the point vortices 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 $O(1/N)$ as $N \to \infty$, which are clearly confirmed in the present article. We also apply the fast method to a long-time evolution of two vortex sheets on the sphere. A key device is to embed the sphere into the three-dimensional space, in which we reformulate the equation of motion for the $N$ point vortices.