N-body Integrators with Individual Time Steps from Hierarchical   Splitting
  We review the implementation of individual particle time-stepping for N-body dynamics. We present a class of integrators derived from second order Hamiltonian splitting. In contrast to the usual implementation of individual time-stepping, these integrators are momentum conserving and show excellent energy conservation in conjunction with a symmetrized time step criterion. We use an explicit but approximate formula for the time symmetrization that is compatible with the use of individual time steps. No iterative scheme is necessary. We implement these ideas in the HUAYNO (available online at www.amusecode.org) code and present tests of the integrators and show that the presented integration schemes shows good energy conservation, with little or no systematic drift, while conserving momentum and angular momentum to machine precision for long term integrations. 
Federico I. Pelupessy, J\"urgen J\"anes, Simon F. Portegies Zwart
astro-ph.IM physics.comp-ph