 
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, A1040 Wien, Austria
Abstract. We are concerned with the numerical integration of ODEinitial 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 WKBansatz
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 WKBtransformation 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 1Dmodel for ballistic quantum transport in a MOSFET
(metal oxide semiconductor fieldeffect transistor).
Keywords: Schrödinger equation; highly oscillating wave functions; higher order WKBapproximation
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
