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.

中文翻译：

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非线性Schrödinger方程的能量守恒方法

本文关注的是非线性Schrödinger方程在时间上的数值积分，采用了不同的方法来保留能量或能量的离散模拟。Crank–Nicolson方法是一种众所周知的订单方法| $ 2 $ | 但是它是完全隐式的，因此可能更喜欢线性隐式方法，例如Besse（1998年，戴维-史陶森大学的Analyzenumériquedessystèmesde Davey-Stewartson博士，UniversitéBordeaux）引入的松弛方法，用于三次非线性Schrödinger方程。此方法还是一种节能方法，数值模拟表明其顺序为| $ 2 $ |⁠。在本文中，我们给出了这种松弛方法阶数的严格证明，并提出了一种通用版本，该版本可以处理一般幂律非线性。不同物理模型的数值模拟表明了这些方法的有效性。