Asymptotically Correct Finite Difference Schemes for Highly Oscillatory ODEs

Journal Article · Thu Sep 30 00:00:00 EDT 2010 · AIP Conference Proceedings

DOI:https://doi.org/10.1063/1.3498358· OSTI ID:21428589

Arnold, Anton; Geier, Jens ^[1]

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1. Inst. f. Analysis u. Scientific Computing, Technische Universitaet Wien, Wiedner Hauptstr. 8, A-1040 Wien (Austria)

We are concerned with the numerical integration of ODE-initial value problems of the form {epsilon}{sup 2{phi}}{sub xx}+a(x){phi} = 0 with given a(x){>=}a{sub 0}>0 in the highly oscillatory regime 0<{epsilon}(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 {epsilon} and h. The resulting scheme typically has a step size restriction of h = o({radical}({epsilon})). 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({epsilon}{sup 3}h{sup 2}). As an application we present simulations of a 1D-model for ballistic quantum transport in a MOSFET (metal oxide semiconductor field-effect transistor).

• OSTI ID: 21428589
• Journal Information: AIP Conference Proceedings, Vol. 1281, Issue 1; Conference: ICNAAM 2010: International conference of numerical analysis and applied mathematics 2010, Rhodes (Greece), 19-25 Sep 2009; Other Information: DOI: 10.1063/1.3498358; (c) 2010 American Institute of Physics; ISSN 0094-243X
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGEBRA
ASYMPTOTIC SOLUTIONS
COMPUTERIZED SIMULATION
INTEGRAL EQUATIONS
MATRICES
MOSFET
OSCILLATIONS
SCHROEDINGER EQUATION
WKB APPROXIMATION
APPROXIMATIONS
CALCULATION METHODS
DIFFERENTIAL EQUATIONS
EQUATIONS
FIELD EFFECT TRANSISTORS
MATHEMATICAL SOLUTIONS
MATHEMATICS
MOS TRANSISTORS
PARTIAL DIFFERENTIAL EQUATIONS
SEMICONDUCTOR DEVICES
SIMULATION
TRANSISTORS
WAVE EQUATIONS