Asymptotically Correct Finite Difference Schemes for Highly Oscillatory ODEs

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(appearing as a stationary Schroedinger 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 o(ε3h2). As an application we present simulations of a 1D-model for ballistic quantum transport in a MOSFET (metal oxide semiconductor field-effect transistor).