Pseudoparticle multipole method: A simple method to implement a high-accuracy tree code

In this Letter, we describe the pseudoparticle multipole method ((PM2)-M-2) , which is a new method to express multipole expansion by a distribution of pseudoparticles. We can use this distribution of particles to calculate hi gh-order terms in both the Barnes-Hut tree code and the fast multipole meth od. The primary advantage of (PM2)-M-2 is that it works on GRAPE (GRAvity P ipE). GRAPE is special-purpose hardware for the calculation of the gravitat ional force between particles. Although the tree code has been implemented on GRAPE, we could only handle terms up to a dipole since GRAPE can calcula te forces from point-mass particles only. Thus, the calculation cost grows quickly when high accuracy is required. With (PM2)-M-2, the multipole expan sion is expressed by particles, and thus GRAPE can calculate high-order ter ms. Using (PM2)-M-2, we implemented an arbitrary-order tree code on GRAPE-4 . The timing result shows that GRAPE-4 accelerates the calculation by a fac tor of between 10 (for low accuracy) and 100 (for high accuracy). Even on g eneral-purpose programmable computers, our method offers the advantage that the mathematical formulae, and therefore the actual program, are much simp ler than that of the direct implementation of multipole expansion, although the calculation cost becomes somewhat higher.