Backward Error Analysis of Symplectic Integrators for Linear Separable Hamiltonian Systems

Abstract

Symplecticness, stability, and asymptotic properties of Runge-Kutta, partitioned Runge-Kutta, and Runge-Kutta-Nyström methods applied to the simple Hamiltonian system $\dot{p}= -vq, \dot{q}= kp$ are studied. Some new results in connection with P-stability are presented. The main part is focused on backward error analysis. The numerical solution produced by a symplectic method with an appropriate stepsize is the exact solution of a perturbed Hamiltonian system at discrete points. This system is studied in detail and new results are derived. Numerical examples are presented.