PPPM and TreePM Methods on GRAPE Systems for Cosmological N-body   Simulations
  We present Particle-Particle-Particle-Mesh (PPPM) and Tree Particle-Mesh (TreePM) implementations on GRAPE-5 and GRAPE-6A systems, special-purpose hardware accelerators for gravitational many-body simulations. In our PPPM and TreePM implementations on GRAPE, the computational time is significantly reduced compared with the conventional implementations without GRAPE, especially under the strong particle clustering, and almost constant irrespective of the degree of particle clustering. We carry out the survey of two simulation parameters, the PM grid spacing and the opening parameter for the most optimal combination of force accuracy and computational speed. We also describe the parallelization of these implementations on a PC-GRAPE cluster, in which each node has one GRAPE board, and present the optimal configuration of simulation parameters for good parallel scalability. 
Kohji Yoshikawa and Toshiyuki Fukushige