Fast Summation Algorithms for the Particle Simulation of Far-Field Potential and Rapidly-Decaying Potential

Fast summation algorithms for the particle simulation are presented. The diffusion velocity, which appears in the vortex methods, is treated as an example of rapidly-decaying potential and the convection velocity as far-field potential. Computational count can be reduced from O(N^2) to O(N), where N is the particle number, for the diffusion velocity and to O(N log N) for the convection velocity by the use of blocks with various widths which are arranged in a computational domain based on simple principles. Our method is applied to heat-vortex interaction and it is shown that the theoretical efficiency is achieved, and that our code is sufficiently parallelised.