Energy-preserving methods for nonlinear Schrödinger equations

Abstract

This paper is concerned with the numerical integration in time of nonlinear Schrödinger 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 numérique des systèmes de Davey-Stewartson. Ph.D. Thesis, Université Bordeaux) for the cubic nonlinear Schrödinger 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.

Error estimation of the Besse Relaxation Scheme for a semilinear heat equation

“…Independently of the present work [16], C. Besse et al [6] focusing on the cubic NLS equation, completed the convergence analysis of [4] by a proper consistency argument and arrived at a error bound consisting of the error approximating ( ( 1 2 , •)) along with a second order, with respect to the time step, term. However, the latter error estimate fails to explain the second order, experimental convergence of the (RS) under the choice ( 0 (•)) as an initial approximation of ( ( 12 , •)) (see Rem.…”

“…1.1). In addition, the technique used in [5,6] for the convergence analysis of the time-discrete (RS) is not suitable for the error estimation of a fully-discrete version of the (RS), because it is based on the derivation of a priori bounds of the (RS) time-discrete approximations in higher order Sobolev norms.…”

Finite difference methods for the one-dimensional Chern-Simons gauged models

“…In general, it is very challenging develop the energy-preserving numerical schemes, even for the (nonlinear) Schrödinger equation. To the extent of the authors' knowledge, there exist limited number of literature about energy-preserving schemes for the nonlinear Schrödinger equations, for example [4,11,32]. When the Chern-Simons gauge is coupled with the Schrödinger equation, it becomes much harder to devise the energy-preserving numerical scheme, mainly due to the interaction between the gauge field and the matter field.…”

A note on optimal $$H^1$$-error estimates for Crank-Nicolson approximations to the nonlinear Schrödinger equation

“…For the subclass of power law nonlinearities of the form γ (|u| 2 ) = K k=1 κ k |u| 2σ k for σ k ≥ 0 and α k ∈ R, a mass and energy conserving relaxation scheme was proposed and analyzed by Besse [23,24]. Thanks to its properties, the scheme shows a very good performance in realistic physical setups [21].…”