Commun. Comput. Phys.,
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Volume 4.


A Review of Transparent and Artificial Boundary Conditions Techniques for Linearand Nonlinear Schrodinger Equations

Xavier Antoine 1, Anton Arnold 2, Christophe Besse 3, Matthias Ehrhardt 4*, Achim Schadle 5

1 Institut Elie Cartan Nancy (IECN), Universite Henri Poincare Nancy 1, UMR 7502, INRIA Corida, Nancy-Universite, France.
2 Institut fur Analysis und Scientific Computing, Technische Universitat Wien, Wiedner Hauptstr. 8, 1040 Wien, Austria.
3 Project-Team SIMPAF INRIA Lille Nord Europe Research Centre, Laboratoire Paul Painleve U.M.R CNRS 8524 Universite des Sciences et Technologies de Lille, France.
4 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany.
5 Zuse-Institut Berlin, Takustrasse 7, 14195 Berlin, Germany.

Received 28 February 2008; Accepted (in revised version) 3 April 2008
Available online 16 May 2008

Abstract

In this review article we discuss different techniques to solve numerically the time-dependent Schrodinger equation on unbounded domains. We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works. We conclude with several numerical examples from different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case.

AMS subject classifications: 65M12, 35Q40, 45K05
PACS: 02.70.Bf, 31.15.Fx
Key words: Schrodinger equation, transparent boundary conditions, discrete convolution, unbounded domain, finite difference schemes, finite elements.

*Corresponding author.
Email: Xavier.Antoine@iecn.u-nancy.fr (X. Antoine), anton.arnold@tuwien.ac.at (A. Arnold), Christophe.Besse@math.univ-lille1.fr (C. Besse), ehrhardt@wias-berlin.de (M. Ehrhardt), schaedle@zib.de (A. Schadle)
 

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