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 $c>0$.
Categories: math.NA cs.NA