Think globally, act locally: solving highly-oscillatory ordinary differential equations

In this paper we explore the solution of highly-oscillatory differential equations, with a special reference to the linear oscillator $y''+g(t)y=0$, where $g(t)\stackrel{t\rightarrow\infty}{\longrightarrow} +\infty$. Commencing from a global-error formula, we explore the accumulation of the error by Runge--Kutta and Magnus methods. Motivated by our analysis, we present a modification of the Magnus method which results in substantially better performance.