High-order WKB-based Method For The 1D Stationary Schr\"odinger Equation In The Semi-classical Limit

TL;DR

This paper develops an advanced numerical method based on WKB approximation to efficiently solve the highly oscillatory 1D Schrödinger equation in the semi-classical limit, achieving higher accuracy on coarse grids.

Contribution

The paper extends a WKB-based numerical method from second to third order accuracy for solving the semi-classical Schrödinger equation.

Findings

Achieves third-order accuracy in WKB-based numerical solutions.

Enables efficient computation on coarse grids.

Improves upon previous second-order methods.

Abstract

We consider 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 discuss their efficient numerical integration on coarse grids, but still yielding accurate solutions. The $\mathcal{O}(h^2)$ one-step method from [2] is based on an analytic WKB-preprocessing of the equation. Here we extend this method to $\mathcal{O}(h^3)$ accuracy.