Pseudoparticle Multipole Method: A Simple Method to Implement a High-Accuracy Tree Code

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Abstract
In this Letter, we describe the pseudoparticle multipole method (P2M2), which is a new method to express multipole expansion by a distribution of pseudoparticles. We can use this distribution of particles to calculate high-order terms in both the Barnes-Hut tree code and the fast multipole method. The primary advantage of P2M2 is that it works on GRAPE (GRAvity PipE). GRAPE is special-purpose hardware for the calculation of the gravitational force between particles. Although the tree code has been implemented on GRAPE, we could only handle terms up to a dipole since GRAPE can calculate forces from point-mass particles only. Thus, the calculation cost grows quickly when high accuracy is required. With P2M2, the multipole expansion is expressed by particles, and thus GRAPE can calculate high-order terms. Using P2M2, we implemented an arbitrary-order tree code on GRAPE-4. The timing result shows that GRAPE-4 accelerates the calculation by a factor of between 10 (for low accuracy) and 100 (for high accuracy). Even on general-purpose programmable computers, our method offers the advantage that the mathematical formulae, and therefore the actual program, are much simpler than that of the direct implementation of multipole expansion, although the calculation cost becomes somewhat higher.
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