I Scientific Highlights 047

For the time integration of collisional-body systems, such as star clusters and systems of planetesimals, the combination of direct summation for force calculation and the individual timestep algorithm has been the standard method for nearly a half century. It is not impossible to combine an individual timestep algorithm and a fast and approximate force calculation. However, it was difficult to achieve good performance on distributedmemory parallel computers for such a scheme. Figure 1 shows the long-term variation of the relative energy error. In this case, the energy error reaches about 7.5 × 10−8 after the time integration for 10 yr (10 orbital periods), while it is 9.8 × 10−9 for 10 yr (10 periods). In other words, the energy error grows to be 10 times larger as the integration time becomes 1000 times longer. Thi