OPTIMAL ORDER AND TIME-STEP CRITERION FOR AARSETH-TYPE N-BODY INTEGRATORS

We discuss the optimal order and time-step criterion for Aarseth-type N-body integrators. We compare two different classes of integrators. The first one is the standard Aarseth-type predictor-corrector scheme that uses the Newton interpolation to construct the predictor and the corrector. The second one is a similar predictor-corrector scheme that calculates the force derivative df/dt directly and applies the Hermite interpolation to construct the predictor and the corrector. We measured the dependence of the integration error on the step-size and the number of particles. The Hermite scheme allows a time step twice as large as that for the standard Aarseth scheme for the same accuracy. The calculation cost of the Hermite scheme per time step is roughly twice as much as that of the standard Aarseth scheme. Thus the efficiency depends strongly on the characteristics of the hardware. On vector machines the Hermite scheme would perform better because of the reduction in the amount of scalar calculations. The optimal order of the integrator depends on both the particle number and the accuracy required. However, the calculation cost is rather insensitive to the selection of the order and the four-step integrator adopted by Aarseth gives the calculation cost within 30% of the minimum, for practical range of the accuracy. The time-step criterion of the standard Aarseth scheme was found to be not applicable to higher order integrators and we propose a more uniformly reliable criterion.