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Asymptotics of a Time-Splitting Scheme for the Random Schrödinger Equation with Long-Range Correlations

Published online by Cambridge University Press:  20 February 2014

Christophe Gomez  and
Olivier Pinaud
Christophe Gomez
Affiliation:
Laboratoire d’Analyse, Topologie, Probabilités, UMR 7353, Aix-Marseille Université, Marseille, France.. christophe.gomez@latp.univ-mrs.fr
Olivier Pinaud
Affiliation:
Department of Mathematics, Colorado State University, Fort Collins, CO, USA.; pinaud@math.colostate.edu
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Abstract

This work is concerned with the asymptotic analysis of a time-splitting scheme for the Schrödinger equation with a random potential having weak amplitude, fast oscillations in time and space, and long-range correlations. Such a problem arises for instance in the simulation of waves propagating in random media in the paraxial approximation. The high-frequency limit of the Schrödinger equation leads to different regimes depending on the distance of propagation, the oscillation pattern of the initial condition, and the statistical properties of the random medium. We show that the splitting scheme captures these regimes in a statistical sense for a time stepsize independent of the frequency.

Keywords

Random Schrödinger equationlong-range correlationshigh frequency asymptoticssplitting scheme
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 2: Multiscale problems and techniques , March 2014 , pp. 411 - 431
Copyright
© EDP Sciences, SMAI, 2014

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