On the global error of discretization methods for highly-oscillatory ordinary differential equations

Commencing from a global-error formula, originally due to Henrici, we investigate the accumulation of global error in the numerical solution of linear highly-oscillating systems of the form $y''+g(t)y=0$, where $g(t)\stackrel{t\rightarrow\infty}{\longrightarrow} \infty$. Using WKB analysis we derive an explicit form of the global-error envelope for Runge--Kutta and Magnus methods. Our results are closely matched by numerical experiments. Motivated by the superior performance of Lie-group methods, we present a modification of the Magnus expansion which displays even better long-term behaviour in the presence of oscillations.