A priori estimates for the global error committed by Runge–Kutta methods for a nonlinear oscillator

Abstract: The Alekseev–Groebner lemma is combined with the theory of modified equations to obtain an 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 nonlinear oscillators whose behaviour is described by the Emden–Fowler equation y'' + t^νyⁿ = 0. The result shows explicitly that later terms sometimes blow up faster than the leading term of order h^p, necessitating the whole computation. This is supported by numerical experiments.