On the nonlinear stability of symplectic integrators

Abstract

The modified Hamiltonian is used to study the nonlinear stability of symplectic integrators, especially for nonlinear oscillators. We give conditions under which an initial condition on a compact energy surface will remain bounded for exponentially long times for sufficiently small time steps. For example, the implicit midpoint rule achieves this for the critical energy surface of the H´enon- Heiles system, while the leapfrog method does not. We construct explicit methods which are nonlinearly stable for all simple mechanical systems for exponentially long times. We also address questions of topological stability, finding conditions under which the original and modified energy surfaces are topologically equivalent.