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

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