We are concerned with the numerical integration of ODE‐initial value problems of the form ε2φxx+a(x)φ = 0 with given a(x)≥a0>0 in the highly oscillatory regime 0<ε⋘1 (appearing as a stationary Schrödinger equation, e.g.). In two steps we derive an accurate finite difference scheme that does not need to resolve each oscillation: With a WKB‐ansatz the dominant oscillations are “transformed out”, yielding a much smoother ODE. For the resulting oscillatory integrals we devise an asymptotic expansion both in ε and h. The resulting scheme typically has a step size restriction of h = o(ε). If the phase of the WKB‐transformation can be computed explicitly, then the scheme is asymptotically correct with an error bound of the order o3h2). As an application we present simulations of a 1D‐model for ballistic quantum transport in a MOSFET (metal oxide semiconductor fieldeffect transistor).

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