Error growth in the numerical integration of periodic orbits

This paper is concerned with the long term behaviour of the error generated by one step methods in the numerical integration of periodic flows. Assuming numerical methods where the global error possesses an asymptotic expansion and a periodic flow with the period depending smoothly on the starting point, some conditions that ensure an asymptotically linear growth of the error with the number of periods are given. A study of the error growth of first integrals is also carried out. The error behaviour of Runge–Kutta methods implemented with fixed or variable step size with a smooth step size function, with a projection technique on the invariants of the problem is considered.

► The behavior of the global error of one step methods in the numerical integration of periodic flows has been studied.
► We proved linear growth of the global error if orthogonal or directional projection techniques are used to preserve the period.
► It has been shown that the growth of the error in the first integrals is asymptotically linear for any numerical one step method.