On the Qualitative Behaviour of Symplectic Integrators. Part II. Integrable Systems

In this paper the numerical integration of integrable Hamiltonian systems is considered. Symplectic one-step methods are used. The discrete system is shown to be integrable up to a remainder which is exponentially small with respect to the step size of the one-step method. As a consequence it is shown that the global error grows linearly for exponentially long times.