On the global error of discretization methods for highly-oscillatory ordinary differential equations (Q1864778)

The author considers the numerical integration of linear highly-oscillating systems of the form \(y''+g(t)y=0\), where \(g(t) \to \infty\) for \(t \to \infty\), e.g. \(g(t)= t^2, g(t)=t\) or \(g(t)=\log(1+t)\) leading to Riccati's and Airy's equation and the log-oscillator, respectively. Using WKB analysis an explicit form of the global-error envelope is derived for Runge-Kutta and Magnus methods. The starting point for the analysis is a global-error expansion of an arbitrary time-stepping method dating back to P. Henrici. The error formula provides a thorough explanation for the superior performance of the Magnus methods compared to classical solvers for the type of ordinary differential equations considered. The author also proposes a modification of the Magnus method based on local linearisation. The theoretical results are illustrated by nicely chosen numerical examples.