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Preprint 11-2007

Discrete artificial boundary conditions for nonlinear Schrödinger equations

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Author(s) : Andrea Zisowsky , Matthias Ehrhardt

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 11-2007

MSC 2000

65M06 Finite difference methods
35Q40 Equations from quantum mechanics

Abstract :
In this work we construct and analyze discrete artificial boundary conditions (ABCs) for different finite difference schemes to solve nonlinear Schrödinger equations. These new discrete boundary conditions are motivated by the continuous ABCs recently obtained by the potential strategy of Szeftel. Since these new nonlinear ABCs are based on the discrete ABCs for the linear problem we first review the well-known results for the linear Schrödinger equation. We present our approach for a couple of finite difference schemes, including the Crank-Nicholson scheme, the Duran-Sanz-Serna scheme, the DuFort-Frankel method and several split-step (fractional step) methods such as the Lie splitting, the Strang splitting and the relaxation scheme of Besse. Finally, several numerical tests illustrate the accuracy and stability of our new discrete approach for the considered finite difference schemes.

Keywords : Nonlinear Schrödinger equation, unbounded domains, discrete artificial boundary conditions, finite difference scheme, split-step method

Notes :
Submitted to: Mathematical and Computer Modelling.
The first author was supported by the Berliner Programm zur Förderung der Chancengleichheit fuer Frauen in Forschung und Lehre.

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