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 $c>0$.

翻译：我们讨论在高度振荡区域中初值问题$\varepsilon^2\,\varphi''+a(x)\,\varphi=0$的数值求解，其中$a(x)>0$且$0<\varepsilon\ll 1$。基于著名的WKB拟设，我们分析并实现了一种近似解。由此产生的近似误差量级为$\mathcal{O}(\varepsilon^{N})$，其中$N$为渐近级数的截断阶数。当选择最优截断阶数$N_{opt}$时，误差行为表现为$\mathcal{O}(\varepsilon^{-2}\exp(-c\varepsilon^{-1}))$，其中$c>0$。