Error Growth in the Numerical Integration of Periodic Orbits, with Application to Hamiltonian and Reversible Systems

Abstract

We analyze in detail the growth with time (of the coefficients of the asymptotic expansion) of the error in the numerical integration with one-step methods of periodic solutions of systems of ordinary differential equations. Variable stepsizes are allowed. We successively consider "general," Hamiltonian, and reversible problems. For Hamiltonian and reversible systems and under fairly general hypotheses on the orbit being integrated, numerical methods with relevant geometric properties (symplecticness, energy-conservation, reversibility) are proved to have better error growth than "general" methods.