  We consider initial value problems for
$\varepsilon^2\,\varphi''+a(x)\,\varphi=0$ in the highly oscillatory regime,
i.e., with $a(x)>0$ and $0<\varepsilon\ll 1$. We discuss their efficient
numerical integration on coarse grids, but still yielding accurate solutions.
The $\mathcal{O}(h^2)$ one-step method from [2] is based on an analytic
WKB-preprocessing of the equation. Here we extend this method to
$\mathcal{O}(h^3)$ accuracy.
