On the KAM and Nekhoroshev theorems for symplectic integrators and implications for error growth.

In this study an alternative to backward error analysis (BEA)for geometric numerical integration schemes is introduced. It allows us to prove KAM and Nekhoroshev stability theorems for symplectic discretizations of close to integrable Hamiltonian systems, without the restrictions of traditional BEA. The results are qualitative in nature and explain in further detail why symplectic integration of Hamiltonian systems is so successful.