The solutions of the nonlinear Schrödinger equation are of great
importance for ab initio calculations. It can be shown that such solutions
conserve a countable number of quantities, the simplest being the local norm
square conservation law. Numerical solutions of high quality, especially for
long time intervals, must necessarily obey these conservation laws. In this
work we first give the conservation laws that can be calculated by means of
Lie theory and then critically compare the quality of different finite
difference methods that have been proposed in geometric integration with
respect to conservation laws. We find that finite difference schemes derived
by writing the Schrödinger equation as an (artificial)
Hamiltonian system do not necessarily conserve important physical quantities
better than other methods.
@article{1199377550,
author = {Heitzinger, Clemens and Ringhofer, Christian and Selberherr, Siegfried},
title = {Finite difference solutions of the nonlinear Schr\"odinger equation
and their conservation of physical quantities},
journal = {Commun. Math. Sci.},
volume = {5},
number = {1},
year = {2007},
pages = { 779-788},
language = {en},
url = {http://dml.mathdoc.fr/item/1199377550}
}
Heitzinger, Clemens; Ringhofer, Christian; Selberherr, Siegfried. Finite difference solutions of the nonlinear Schrödinger equation
and their conservation of physical quantities. Commun. Math. Sci., Tome 5 (2007) no. 1, pp. 779-788. http://gdmltest.u-ga.fr/item/1199377550/