Modified energy for split-step methods applied to the linear Schr\"odinger equation
Numerical Analysis
2009-01-12 v1
Abstract
We consider the linear Schr\"odinger equation and its discretization by split-step methods where the part corresponding to the Laplace operator is approximated by the midpoint rule. We show that the numerical solution coincides with the exact solution of a modified partial differential equation at each time step. This shows the existence of a modified energy preserved by the numerical scheme. This energy is close to the exact energy if the numerical solution is smooth. As a consequence, we give uniform regularity estimates for the numerical solution over arbitrary long time
Cite
@article{arxiv.0901.1190,
title = {Modified energy for split-step methods applied to the linear Schr\"odinger equation},
author = {Arnaud Debussche and Erwan Faou},
journal= {arXiv preprint arXiv:0901.1190},
year = {2009}
}