Preprint 18-2007

A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations

Author(s) : Xavier Antoine , Anton Arnold , Christophe Besse , Matthias Ehrhardt , Achim Schädle

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 18-2007

PACS : 02.70.Bf, 31.15.Fx

Abstract :
In this review article we discuss different techniques to solve numerically the time-dependent Schrödinger equation on unbounded domains. We present and compare several approaches to implement the classical transparent boundary condition into finite difference and finite element discretizations. We present in detail the approaches of the authors 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.

Keywords : Schrödinger equation, transparent boundary conditions, discrete convolution, unbounded domain, finite difference schemes, finite elements

Notes :
submitted to: Applied Numerical Mathematics, Elsevier.
supplementary software is located at http://www.tbc-review.de.vu/