Multi-point Hermite methods for the N-body problem

Abstract

Numerical integration methods are central to the study of self-gravitating systems, especially those comprised of many bodies or otherwise beyond the reach of analytical methods. Predictor–corrector schemes, both multi-step methods and those based on 2-point Hermite interpolation, have found great success in the simulation of star clusters and other collisional systems. Higher-order methods, such as those based on Gaussian quadratures and Richardson extrapolation, have also proven popular for high-accuracy integrations of few-body systems, particularly those that may undergo close encounters. This work presents a family of high-order schemes based on multi-point Hermite interpolation. When applied as multi-step multi-derivative schemes, these can be seen as generalizing both Adams–Bashforth–Moulton methods and 2-point Hermite methods; I present results for the 6th-, 9th-, and 12th-order 3-point schemes applied in this manner using variable timesteps. In a star cluster-like test problem, the 3-point 6th-order predictor–corrector scheme matches or outperforms the standard 2-point 4th-order Hermite scheme at negligible $O\left(N\right)$ additional cost, potentially reducing the necessary number of force evaluations in simulations of large-$N$ collisional systems by factors of $\sim 3$ or more. I also present a number of high-order time-symmetric schemes up to 18th order, which have the potential to improve the accuracy and efficiency of long-duration simulations.