6th and 8th Order Hermite Integrator for N-body Simulations

TL;DR

This paper introduces sixth- and eighth-order Hermite integrators for astrophysical N-body simulations, improving accuracy and efficiency by using higher derivatives of acceleration, with minimal additional computational cost.

Contribution

The paper develops and analyzes high-order Hermite integrators that are easier to implement and more efficient than traditional methods, especially for high-accuracy simulations.

Findings

Eighth-order scheme requires about twice the FLOPs of fourth-order.

Sixth-order scheme outperforms traditional fourth-order in most cases.

High-order schemes enable larger parallel particle integrations.

Abstract

We present sixth- and eighth-order Hermite integrators for astrophysical $N$-body simulations, which use the derivatives of accelerations up to second order ({\it snap}) and third order ({\it crackle}). These schemes do not require previous values for the corrector, and require only one previous value to construct the predictor. Thus, they are fairly easy to implemente. The additional cost of the calculation of the higher order derivatives is not very high. Even for the eighth-order scheme, the number of floating-point operations for force calculation is only about two times larger than that for traditional fourth-order Hermite scheme. The sixth order scheme is better than the traditional fourth order scheme for most cases. When the required accuracy is very high, the eighth-order one is the best. These high-order schemes have several practical advantages. For example, they allow a larger number of particles to be integrated in parallel than the fourth-order scheme does, resulting in higher execution efficiency in both general-purpose parallel computers and GRAPE systems.