Fast Force Algorithms and Solar System Integrations

Abstract

Symplectic integrators have been the backbone of much theoretical solar system research over the past decade. As implemented, they involve the direct computation of the distances between each pair of N particles, a process whose effort grows as O(N2). A variety of fast [that is, with effort growing more slowly than O(N2)] but approximate force calculation methods have been developed in other areas of research. Several of these algorithms are examined here, and their speed and accuracy are compared with traditional methods, with an eye toward their suitability for solar system research in particular. We find that approximate force algorithms can provide, in some situations, a suitable alternative to traditional ones, with break-even in terms of computation time at particle numbers as low as a few hundred, and often with only modest increases in the short-term error. Though empirically stable on the systems tested here, the effect of approximate methods on the phase-space manifold of the problem remains a concern.