Variational Integrators for the Gravitational N-Body Problem

Abstract

This paper describes a fourth-order integration algorithm for the gravitational N-body problem based on discrete Lagrangian mechanics. When used with shared timesteps, the algorithm is momentum conserving and symplectic. We generalize the algorithm to handle individual time steps; this introduces fifth-order errors in angular momentum conservation and symplecticity. We show that using adaptive block power of two timesteps does not increase the error in symplecticity. In contrast to other high-order, symplectic, individual timestep, momentum-preserving algorithms, the algorithm takes only forward timesteps. We compare a code integrating an N-body system using the algorithm with a direct-summation force calculation to standard stellar cluster simulation codes. We find that our algorithm has about 1.5 orders of magnitude better symplecticity and momentum conservation errors than standard algorithms for equivalent numbers of force evaluations and equivalent energy conservation errors.

31 pages, 8 figures. v2: Revised individual-timestepping description, expanded comparison with other methods, corrected error in predictor equation. ApJ, in press

References in corpus (3)

• The cosmological simulation code GADGET-2
• A Time-Symmetric Block Time-Step Algorithm for N-Body Simulations
• Forward Symplectic Integrators and the Long Time Phase Error in Periodic Motions

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