This paper is concerned with an efficient numerical method for solving the 1D stationary Schr\"odinger equation in the highly oscillatory regime. Being a hybrid, analytical-numerical approach it does not have to resolve each oscillation, in contrast to standard schemes for ODEs. We build upon the WKB-based marching method from [1] and extend it in two ways: By comparing the $\mathcal{O}(h)$ and $\mathcal{O}(h^{2})$ methods from [1] we design an adaptive step size controller for the WKB method. While this WKB method is very efficient in the highly oscillatory regime, it cannot be used close to turning points. Hence, we introduce for such regions an automated method coupling, choosing between the WKB method for the oscillatory region and a standard Runge-Kutta-Fehlberg 4(5) method in smooth regions. A similar approach was proposed recently in [2, 3], however, only for a $\mathcal{O}(h)$-method. Hence, we compare our new strategy to their method on two examples (Airy function on the spatial interval $[0,\,10^{8}]$ with one turning point at $x=0$ and a parabolic cylinder function having two turning points), and illustrate the advantages of the new approach w.r.t.\ accuracy and efficiency.
翻译:本文所关注的是一种高效的数字方法, 用于解决高度悬浮系统中的 1D 固定 Schr\\\\" odinger 方程式 。 这是一种混合、 分析- 数字方法, 它并不需要解决每个振动, 与 ODE 的标准方案相比。 因此, 我们从 [ 1 建立基于 WKB 的行进方法, 并以两种方式扩展它: 在平滑区域比较 $\ mathcal{ O} (h) 美元 和$\ mathcal{ O} (h) 2 (h) 2} 我们为 WKB 设计了一个适应性步步控控制器。 在高度振动性制度中, 这个WKB 方法非常高效, 但它不能使用非常接近于转折点。 因此, 我们为这些区域引入一个自动的行进法, 选择星系区域的WKB 方法, 和平滑动区域中的Rungge- Kutta- Fehlberg 4(5) 方法。 最近在 [ 2 3] 中提出了类似方法, 然而, 仅为 $\\\\\\\\ x cal_ a ex 和 ex ex ycreal ex ex 和 ex ex ex ex ex ex ex ex ex ycrecuring ex ycol yclection yclection 。