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.