Author(s) :
Anton Arnold
,
Matthias Ehrhardt
,
Maike Schulte
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 6-2008
MSC 2000
- 65M12 Stability and convergence of numerical methods
-
35Q40 Equations from quantum mechanics
-
45K05 Integro-partial differential equations
Abstract :
In this chapter we propose transparent boundary conditions (TBCs)
for the time-dependent Schrödinger
equation on a two-dimensional domain.
First we derive the two-dimensional discrete TBCs
in conjunction with a
conservative Crank-Nicolson-type finite difference scheme
and a novel compact nine-point scheme.
For this difference equations we derive discrete transparent boundary conditions (DTBCs) in order to get highly accurate solutions for open boundary problems.
The presented discrete boundary-valued problem is unconditionally stable
and completely reflection-free at the boundary.
Several numerical experiments illustrate the perfect absorption
of outgoing waves independent of their impact angle at the boundary.
Finally, we apply inhomogeneous DTBCs to the transient simulation of quantum waveguides with a prescribed electron inflow.
Keywords :
Quantum waveguide, Schrödinger equation, transparent boundary condition, finite difference scheme
Notes :
Chapter in: F.Columbus (ed.), Very-Large-Scale Integration (VLSI): Architecture, Performance and Nano Applications,
to appear fall 2008.