High-order WKB-based Method For The 1D Stationary Schrödinger Equation In The Semi-classical Limit

We consider initial value problems for ${\epsilon }^2{\phi }^{\mathrm{\prime }\mathrm{\prime }}+a(x)\phi =0$ in the highly oscillatory regime, i.e., with $a(x)>0$ and $0<\epsilon \ll 1$. 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.