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arXiv · 1812.04890

Energy preserving methods for nonlinear Schr\"odinger equations

Christophe Besse·Stephane Descombes·Guillaume Dujardin·Ingrid Lacroix-Violet
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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.

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Christophe Besse, Stephane Descombes, Guillaume Dujardin, Ingrid Lacroix-Violet. 2018-12-12. Energy preserving methods for nonlinear Schr\"odinger equations. https://arxiv.org/abs/1812.04890

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