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Asymptotically correct finite difference schemes for highly oscillatory ODEs
 

Summary: Asymptotically correct finite difference schemes
for highly oscillatory ODEs
Anton Arnold and Jens Geier
Inst. f. Analysis u. Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8, A-1040 Wien, Austria
Abstract. We are concerned with the numerical integration of ODE-initial value problems of the form 2xx + a(x) = 0
with given a(x) a0 > 0 in the highly oscillatory regime 0 < 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 and h. The resulting scheme typically has a step size restriction of h = O(

). 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(3h2). As an application we present simulations of a 1D-model for ballistic quantum transport in a MOSFET
(metal oxide semiconductor field-effect transistor).
Keywords: Schrödinger equation; highly oscillating wave functions; higher order WKB-approximation
PACS: MSC2010 classification: 65L11, 34E20, 65P10, 82D37, 82D77
INTRODUCTION
This paper deals with an efficient scheme for the numerical solution of highly oscillating differential equations of the
type
2

  

Source: Arnold, Anton - Institut für Analysis und Scientific Computing, Technische Universität Wien

 

Collections: Mathematics