We are concerned with the numerical integration of ODE‐initial value problems of the form with given in the highly oscillatory regime (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 If the phase of the WKB‐transformation can be computed explicitly, then the scheme is asymptotically correct with an error bound of the order As an application we present simulations of a 1D‐model for ballistic quantum transport in a MOSFET (metal oxide semiconductor field‐effect transistor).
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Research Article| September 30 2010
Asymptotically Correct Finite Difference Schemes for Highly Oscillatory ODEs
AIP Conf. Proc. 1281, 206–209 (2010)
Anton Arnold, Jens Geier; Asymptotically Correct Finite Difference Schemes for Highly Oscillatory ODEs. AIP Conf. Proc. 30 September 2010; 1281 (1): 206–209. https://doi.org/10.1063/1.3498358
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