  We discuss the numerical solution of 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 analyze and implement an
approximate solution based on the well-known WKB-ansatz. The resulting
approximation error is of magnitude $\mathcal{O}(\varepsilon^{N})$ where $N$
refers to the truncation order of the underlying asymptotic series. When the
optimal truncation order $N_{opt}$ is chosen, the error behaves like
$\mathcal{O}(\varepsilon^{-2}\exp(-c\varepsilon^{-1}))$ with some $c>0$.
