Performance analysis of high-accuracy tree code based on the pseudoparticle multipole method

In this paper, we describe the performance and accuracy of the (PM2)-M-2 tree code. The (PM2)-M-2 tree code is a high-accuracy tree code based on the pseudoparticle multipole method ((PM2)-M-2). (PM2)-M-2 is a method to express multipole expansion using a small number of pseudoparticles. The potential field of physical particles is approximated by the field generated by the pseudoparticles. The primary advantage of the (PM2)-M-2 tree code is that it can use Gravity Pipe (GRAPE) special-purpose computers efficiently for high-accuracy calculations. Although the tree code has been implemented on GRAPE, it could not handle terms of the multipole expansion higher than dipole, since GRAPE can calculate forces from point mass particles only. Thus, the calculation cost grows quickly when high accuracy is required. In the (PM2)-M-2 tree code, the multipole expansion is expressed by particles, and thus we can evaluate high-order terms on GRAPE. We implemented the (PM2)-M-2 tree code on both MDGRAPE-2 and a conventional workstation and measured the performance. The results show that MDGRAPE-2 accelerates the calculation by a factor between 20 (for low accuracy) and 200 (for high accuracy). Even on general-purpose programmable computers, the (PM2)-M-2 tree code 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.