On the qualitative behaviour of symplectic integrators. II: Integrable systems
[For part I see Numer. Math. 77, No. 4, 535-547 (1997, Zbl 0885.65088).] This paper deals with the qualitative error behaviour of symplectic one step integrators on completely integrable Hamiltonian systems. As it is well konwn, \textit{N. N. Nekhoroshev} [Russ. Math. Surveys, 32, No. 6, 1-65 (1977; Zbl 0389.70028)] derived stability bounds which hold on exponentially long time intervals for perturbed Hamiltonian systems. Since the solution of a symplectic method can be considered as the exact solution of a perturbed Hamiltonian it is conceivable that under suitable assumptions some Nekhoroshev type results could be established for symplectic integrators. In this paper, the author proves that for completely integrable Hamiltonian systems all first integrals are approximately preserved (in the sense that the errors grow linearly) for exponentially long times. This implies that the global error of symplectic integrators grows linearly for such intervals in contrast with nonsymplectic integrators in which global errors grow in general quadratically.
completely integrable Hamiltonian systems
scientific article; zbMATH DE number 1115483