Energy preserving methods for nonlinear Schr\"odinger equations
Abstract
This paper is concerned with the numerical integration in time of nonlinear Schr\"odinger equations using different methods preserving the energy or a discrete analog of it. The Crank-Nicolson method is a well known method of order 2 but is fully implicit and one may prefer a linearly implicit method like the relaxation method introduced in [10] for the cubic nonlinear Schr{\"o}dinger equation. This method is also an energy preserving method and numerical simulations have shown that its order is 2. In this paper we give a rigorous proof of the order of this relaxation method and propose a generalized version that allows to deal with general power law nonlinearites. Numerical simulations for different physical models show the efficiency of these methods.
Keywords
Cite
@article{arxiv.1812.04890,
title = {Energy preserving methods for nonlinear Schr\"odinger equations},
author = {Christophe Besse and Stephane Descombes and Guillaume Dujardin and Ingrid Lacroix-Violet},
journal= {arXiv preprint arXiv:1812.04890},
year = {2018}
}