Time Symmetrization Meta-Algorithms

We present two types of meta-algorithm that can greatly improve the accuracy of existing algorithms for integrating the equations of motion of dynamical systems. The first meta-algorithm takes an integrator that is time-symmetric only for constant time steps, and ensures time-symmetry even in the case of varying time steps. The second meta-algorithm can be applied to any self-starting integration scheme to create time-symmetry, both for constant and for variable time steps, even if the original scheme was not time-symmetric. Our meta-algorithms are most effective for Hamilton systems or systems with periodic solutions. If the system is not Hamiltonian (for example, if some dissipative force exists), our methods are still useful so long as the dissipation is small.