Explicit, time reversible, adaptive step size control

Adaptive step size control is difficult to combine with geometric numerical integration. As classical step size control is based on "past" information only, time symmetry is destroyed and with it the qualitative properties of the method. In this paper we develop completely explicit, reversible, symmetry-preserving, adaptive step size selection algorithms for geometric numerical integrators such as the Stormer-Verlet method. A new step density controller is proposed and analyzed using backward error analysis and reversible perturbation theory. For integrable reversible systems we show that the resulting adaptive method nearly preserves all action variables and, in particular, the total energy for Hamiltonian systems. It has the same excellent long-term behavior as that obtained when constant steps are used. With variable steps, however, both accuracy and efficiency are greatly improved.