Parallel implementation of an adaptive and parameter-free \(N\)-body integrator (Q538625)

This paper is concerned with numerical integration of \(N\)-body problems in a parallel implementation, where \(N\) is assumed to be large. The authors consider only \(N\)-body systems under gravitational interactions, and the original \(6N\)-dimensional system in positions \({\mathbf{x}}_i\), \(i=1, \dots ,N\), and velocities \({\mathbf{v}}_i\), \(i=1, \dots ,N\), is extended to an autonomous polynomial system of dimension \( 6N +N(N-1)/2 \) that includes the inverse distances \( S_{ij}= | {\mathbf{x}}_i -{\mathbf{x}}_j |^{-1}\), \( 1 \leq i < j \leq N\). The time evolution of such a system is carried out by a Taylor series expansion of all variables up to order \(m \leq 28\), and variable step sizes are selected dynamically according to a local error estimate algorithm. Parallelization is achieved by a suitable distribution of the bodies in the processors. In particular, a detailed study of the parallelization with 4 processors is given estimating the operation count per processor. In the second part, some numerical experiments are presented to show the advantages and shortcomings of the proposed method. After some introductory numerical experiments with a two-body problem, the authors examine as a test problem long time evolution of the solar system under gravitational forces and some ``swarms'' of particles with \(N\) between 18 and 480. The results are compared with other standard Runge-Kutta methods.