We propose a way to efficiently treat the well-known transparent boundary conditions
for the Schrödinger equation. Our approach is based on two ideas: to write out a discrete transparent
boundary condition (DTBC) using the Crank-Nicolson finite difference scheme for the governing
equation, and to approximate the discrete convolution kernel of DTBC by sum-of-exponentials for a
rapid recursive calculation of the convolution.
¶ 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.
Publié le : 2003-09-15
Classification:
Schrödinger equation,
transparent boundary conditions,
discrete convolution,
sum of exponentials,
Padé approximations,
finite difference schemes,
65M12,
35Q40,
45K05
@article{1250880098,
author = {Arnold, Anton and Ehrhardt, Matthias and Sofronov, Ivan},
title = {Discrete transparent boundary conditions for the Schr\"odinger equation: fast calculation, approximation, and stability},
journal = {Commun. Math. Sci.},
volume = {1},
number = {1},
year = {2003},
pages = { 501-556},
language = {en},
url = {http://dml.mathdoc.fr/item/1250880098}
}
Arnold, Anton; Ehrhardt, Matthias; Sofronov, Ivan. Discrete transparent boundary conditions for the Schrödinger equation: fast calculation, approximation, and stability. Commun. Math. Sci., Tome 1 (2003) no. 1, pp. 501-556. http://gdmltest.u-ga.fr/item/1250880098/