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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30 September 2010
ICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics 2010
19–25 September 2010
Rhodes (Greece)
Research Article|
September 30 2010
Asymptotically Correct Finite Difference Schemes for Highly Oscillatory ODEs Available to Purchase
Anton Arnold;
Anton Arnold
Inst. f. Analysis u. Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8, A‐1040 Wien, Austria
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Jens Geier
Jens Geier
Inst. f. Analysis u. Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8, A‐1040 Wien, Austria
Search for other works by this author on:
Anton Arnold
Inst. f. Analysis u. Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8, A‐1040 Wien, Austria
Jens Geier
Inst. f. Analysis u. Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8, A‐1040 Wien, Austria
AIP Conf. Proc. 1281, 206–209 (2010)
Citation
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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