Author(s) :
Anton Arnold
,
Matthias Ehrhardt
,
Maike Schulte
,
Ivan Sofronov
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 41-2007
MSC 2000
- 65M12 Stability and convergence of numerical methods
-
35Q40 Equations from quantum mechanics
-
45K05 Integro-partial differential equations
Abstract :
We propose transparent boundary conditions for the
time-dependent Schrödinger equation on a circular computational domain.
First we derive the two-dimensional discrete TBCs in
conjunction with a
conservative Crank-Nicolson-type finite difference scheme.
The presented discrete boundary-valued problem is unconditionally stable
and completely reflection-free at the boundary.
Then, since the discrete TBCs for the
Schrödinger equation with a spatially dependent potential
include a convolution w.r.t. time with a weakly
decaying kernel, we construct approximate discrete TBCs with a kernel
having the form of a finite sum of exponentials, which can be efficiently
evaluated by recursion. Finally, we describe several numerical examples
illustrating the accuracy, stability and efficiency of the proposed method.
Keywords :
two-dimensional Schrödinger equation, transparent boundary conditions, discrete convolution, sum of exponentials, Padé approximations, finite difference schemes
Notes :
submitted to: Communications in Mathematical Sciences