A Class of Symplectic Integrators with Adaptive Time Step for Separable Hamiltonian Systems

Abstract

Symplectic integration algorithms are well suited for long-term integrations of Hamiltonian systems, because they preserve the geometric structure of the Hamiltonian Ñow. However, this desirable property is generally lost when adaptive time step control is added to a symplectic integrator. We describe an adaptive time step, symplectic integrator that can be used if the Hamiltonian is the sum of kinetic and potential energy components and the required time step depends only on the potential energy (e.g., testparticle integrations in Ðxed potentials). In particular, we describe an explicit, reversible, symplectic, leapfrog integrator for a test particle in a near-Keplerian potential ; this integrator has a time step proportional to distance from the attracting mass and has the remarkable property of integrating orbits in an inverse-square force Ðeld with only "" along-track ÏÏ errors ; i.e., the phase-space shape of a Keplerian orbit is reproduced exactly, but the orbital period is in error by O(N~2), where N is the number of steps per period.

N-body integrators with individual time steps from Hierarchical splitting

“…For general N-body simulation codes, where particles experience very different dynamical timescales during their evolution, we need to relax this condition and let the particles move between timestep sets. Using adaptive timesteps dependent on the phase space coordinates will in general destroy the symplecticity (and hence the conservation properties) of the integrator (Skeel and Gear, 1992;Preto and Tremaine, 1999; Figure 1: Schematic overview of the different integrators. Panel A shows the interactions that are calculated between two particles (1 and 2) by the SHARED integrator for a fiducial trajectory of these two particles, where particle 1 has a time step 4× larger than particle 2, which is also globally in the smallest time step bin (with the rest of the system is represented by the greyed out trajectory).…”

“…The difference with panel B is the fact that each kick of the velocity of particle 2 is matched by the corresponding kick on particle 1. Mikkola and Tanikawa, 1999), although with some restrictions it is possible to construct adaptive timestep symplectic integrators by considering the Hamiltonian in an extended phase space (Preto and Tremaine, 1999;Mikkola and Tanikawa, 1999) or using a variational approach (Farr and Bertschinger, 2007). For the integrators we present here it is however possible to recover long term energy conservation by ensuring time reversibility of the integrator.…”

“…For the integrators we present here it is however possible to recover long term energy conservation by ensuring time reversibility of the integrator. This can be done by using a time-symmetrized time step criterion for the particles (Hut et al, 1995;Preto and Tremaine, 1999).…”

N-Body Integrators for Planets in Binary Star Systems

“…Close encounters between planets can be dealt with in the same way as for systems with a single star by partitioning the planet interaction terms between H Kep and H Int as in (13).…”

“…As with the hybrid integrator described by (13), H Large has to be integrated numerically whenever Λ = 0. In addition, for this choice of Λ, H Small must also be integrated numerically.…”