Energy-preserving methods for nonlinear Schrodinger equations

This paper is concerned with the numerical integration in time of nonlinear Schrodinger equations using different methods preserving the energy or a discrete analogue 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 Besse (1998, Analyse numerique des systemes de Davey-Stewartson. Ph.D. Thesis, Universite Bordeaux) for the cubic nonlinear Schrodinger 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 one to deal with general power law nonlinearites. Numerical simulations for different physical models show the efficiency of these methods.

Automated summary

This paper explores numerical integration methods for nonlinear Schrodinger equations that can preserve energy. The relaxation method, introduced in Besse (1998), is shown to be effective for cubic nonlinear Schrodinger equations, with rigorous proof of its second-order accuracy provided. A generalized version of the method is proposed for handling general power law nonlinearities, with numerical simulations demonstrating efficiency across different physical models.