This paper is dedicated to the efficient numerical computation of solutions to the 1D stationary Schr\"odinger equation in the highly oscillatory regime. We compute an approximate solution based on the well-known WKB-ansatz, which relies on an asymptotic expansion w.r.t. the small parameter $\varepsilon$. Assuming that the coefficient in the equation is analytic, we derive an explicit error estimate for the truncated WKB series, in terms of $\varepsilon$ and the truncation order $N$. For any fixed $\varepsilon$, this allows to determine the optimal truncation order $N_{opt}$ which turns out to be proportional to $\varepsilon^{-1}$. When chosen this way, the resulting error of the optimally truncated WKB series behaves like $\mathcal{O}(\varepsilon^{-2}\exp(-r/\varepsilon))$, with some parameter $r>0$. The theoretical results established in this paper are confirmed by several numerical examples.

翻译：本文致力于高振荡区域下一维稳态薛定谔方程解的高效数值计算。我们基于著名的WKB渐近展开方法，构造了一个与小参数$\varepsilon$相关的渐近逼近解。假设方程中的系数为解析函数，我们推导了截断WKB级数关于$\varepsilon$和截断阶数$N$的显式误差估计。对于任意固定的$\varepsilon$，可由此确定最优截断阶数$N_{opt}$，其与$\varepsilon^{-1}$成正比。当采用该最优截断阶数时，最优截断WKB级数的误差行为表现为$\mathcal{O}(\varepsilon^{-2}\exp(-r/\varepsilon))$，其中参数$r>0$。本文建立的理论结果通过多个数值算例得到了验证。