WKB-Based Schemes for the Oscillatory 1D Schrödinger Equation in the Semiclassical Limit

An efficient and accurate numerical method is presented for the solution of highly oscillatory differential equations. While standard methods would require a very fine grid to resolve the oscillations, the presented approach uses first an analytic (second order) WKB-type transformation, which filters out the dominant oscillations. The resulting ODE is much smoother and can hence be discretized on a much coarser grid, with significantly reduced numerical costs. In many practically relevant examples, the method is even asymptotically correct w.r.t. the small parameter ε that identifies the oscillation wavelength. Indeed, in these cases, the error then vanishes for ε→0, even on a fixed spatial mesh. Applications to the stationary Schrödinger equation are presented.