A Priori Estimates for the Global Error Committed by Runge-Kutta Methods for a Nonlinear Oscillator
Jitse Niesen
TL;DR
This work develops a priori global-error estimates for Runge–Kutta methods solving nonautonomous oscillatory ODEs by merging the Alekseev–Gröbner lemma with modified equations and a B-series framework. It extends known results for linear oscillators (e.g., Airy) to higher-order terms and, for the nonlinear Emden–Fowler equation, derives a detailed asymptotic expansion that reveals substantial differences from the linear case. Numerical experiments with y''+t y^3=0 validate the theory and show that long-time behavior can be dominated by terms beyond the leading $O(h^p)$ term, with the possibility to design tuned methods that cancel certain higher-order contributions. The findings highlight the necessity of a complete, problem-aware calculation to accurately predict global errors in nonlinear oscillatory systems and point to avenues for developing specialized integrators for such regimes.
Abstract
The Alekseev-Gr{ö}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 $y'' + t^νy^n = 0$ 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.