A Priori Estimates for the Global Error Committed by Runge-Kutta Methods for a Nonlinear Oscillator

The Alekseev-Gr{\"o}bner lemma is combined with the theory of modified equations to obtain an \emph{a priori} estimate for the global error of numerical integrators. This estimate is correct up to a remainder term of order $h^{2p}$, where $h$ denotes the step size and $p$ the order of the method. It is applied to a class of nonautonomous linear oscillatory equations, which includes the Airy equation, thereby improving prior work which only gave the $h^p$ term. Next, nonlinear oscillators whose behaviour is described by the Emden-Fowler equation $y'' + t^\nu y^n = 0$ are considered, and global errors committed by Runge-Kutta methods are calculated. Numerical experiments show that the resulting estimates are generally accurate. The main conclusion is that we need to do a full calculation to obtain good estimates: the behaviour is different from the linear case, it is not sufficient to look only at the leading term, and merely considering the local error does not provide an accurate picture either.