Particle-Particle Particle-Tree: A Direct-Tree Hybrid Scheme for   Collisional N-Body Simulations
  In this paper, we present a new hybrid algorithm for the time integration of collisional N-body systems. In this algorithm, gravitational force between two particles is divided into short-range and long-range terms, using a distance-dependent cutoff function. The long-range interaction is calculated using the tree algorithm and integrated with the constant-timestep leapfrog integrator. The short-range term is calculated directly and integrated with the high-order Hermite scheme. We can reduce the calculation cost per orbital period from O(N^2) to O(N log N), without significantly increasing the long-term integration error. The results of our test simulations show that close encounters are integrated accurately. Long-term errors of the total energy shows random-walk behaviour, because it is dominated by the error caused by tree approximation. 
Shoichi Oshino, Yoko Funato, Junichiro Makino
astro-ph.EP