Author(s) :
Anton Arnold
,
Matthias Ehrhardt
,
Ivan Sofronov
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 753-2002
MSC 2000
- 65M12 Stability and convergence of numerical methods
-
35Q40 Equations from quantum mechanics
-
45K05 Integro-partial differential equations
Abstract :
This paper is concerned with transparent boundary conditions (TBCs)
for the time-dependent Schrödinger equation in one and two dimensions.
Discrete TBCs are introduced in the numerical simulations of whole space
problems in order to reduce the computational domain to a finite region.
Since the discrete TBC for the Schrödinger equation includes a
convolution w.r.t. time with a weakly decaying kernel, its numerical
evaluation becomes very costly for large-time simulations.
As a remedy we construct approximate TBCs with a kernel having the form of a
finite sum-of-exponentials, which can be evaluated in a very efficient recursion.
We prove stability of the resulting initial-boundary value scheme,
give error estimates for the considered approximation of the boundary condition,
and illustrate the efficiency of the proposed method on several examples.
Keywords :
Schrödinger equation, transparent boundary conditions, discrete convolution, sum of exponentials, Padé approximations, finite difference schemes
Notes :
(submitted to: "Communications in Mathematical Sciences")