Should Nbody integrators be symplectic everywhere in phase space?
Abstract
Symplectic integrators are the preferred method of solving conservative Nbody problems in cosmological, stellar cluster, and planetary system simulations because of their superior error properties and ability to compute orbital stability. Newtonian gravity is scale free, and there is no preferred time or lengthscale: this is at odds with construction of traditional symplectic integrators, in which there is an explicit timescale in the timestep. Additional timescales have been incorporated into symplectic integration using various techniques, such as hybrid methods and potential decompositions in planetary astrophysics, integrator subcycling in cosmology, and block timestepping in stellar astrophysics, at the cost of breaking or potentially breaking symplecticity at a few points in phase space. The justification provided, if any, for this procedure is that these trouble points where the symplectic structure is broken should be rarely or never encountered in practice. We consider the case of hybrid integrators, which are used ubiquitously in astrophysics and other fields, to show that symplecticity breaks at a few points are sufficient to destroy beneficial properties of symplectic integrators, which is at odds with some statements in the literature. We show how to solve this problem in the case of hybrid integrators by requiring Lipschitz continuity of the equations of motion. For other techniques, such as timestep subdivision, consequences to this problem are not explored here, and the fact that symplectic structure is broken should be taken into account by Nbody simulators, who may find an alternative nonsymplectic integrator performs similarly.
 Publication:

Monthly Notices of the Royal Astronomical Society
 Pub Date:
 July 2019
 DOI:
 10.1093/mnras/stz884
 arXiv:
 arXiv:1902.03684
 Bibcode:
 2019MNRAS.486.5231H
 Keywords:

 methods: numerical;
 celestial mechanics;
 planets and satellites: dynamical evolution and stability;
 globular clusters: general;
 Galaxy: kinematics and dynamics;
 galaxies: evolution;
 Astrophysics  Earth and Planetary Astrophysics;
 Astrophysics  Cosmology and Nongalactic Astrophysics;
 Astrophysics  Astrophysics of Galaxies;
 Astrophysics  Instrumentation and Methods for Astrophysics
 EPrint:
 8 pages, 9 figures. Matches accepted MNRAS version