分数阶非线性薛定谔方程的数值格式研究  

NUMERICAL SCHEMES FOR FRACTIONAL NONLINEAR SCHRÖDINGER EQUATION

作　　者：王俊杰 Wang Junjie(Mathematics Department of Pu'er University,Pu'er 665000,China)

机构地区：[1]普洱学院数学与统计学院,普洱665000

出　　处：《数值计算与计算机应用》2024年第2期136-153,共18页Journal on Numerical Methods and Computer Applications

基　　金：国家自然科学基金(12161070);云南省兴滇青年人才(XDYC-QNRC-2022-0038)资助。

摘　　要：本文研究一类分数阶非线性薛定谔方程的数值格式,该格式满足分数阶系统的一个或多个守恒性质.首先,我们基于\BDF格式、Crank-Nicolson格式和松弛格式来离散时间导数,并分析半离散格式的守恒性和色散误差.其次,在周期边界条件下,利用中心差分格式和紧致差分格式对分数阶非线性薛定谔方程的空间分数阶导数进行离散,并证明这些数值格式保持质量和能量守恒律.最后,对一些分数阶非线性薛定谔方程进行了数值实验,验证了理论结果的正确性.In this paper,we develop some numerical schemes to solve fractional nonlinear Schröd\-inger equation,which preserve one or more analytical properties of the fractional system.First,we apply BDF scheme,Crank-Nicolson scheme and relaxation scheme to discrete time derivative,and analytic conservation and dispersion error of the discrete schemes.Second,we use central difference scheme and compact difference scheme to discrete space fractional derivative of the fractional nonlinear Schrödinger equation with periodic boundary condition.We find that central difference scheme and compact difference scheme preserve mass and energy conservation laws very well for periodic boundary condition.Finally,the numerical experiments of some fractional nonlinear Schrödinger equations are given to verify the correctness of theoretical results.

关 键 词：分数阶薛定谔方程 守恒律 CRANK-NICOLSON格式 紧差分格式 松弛格式 

分 类 号：O241.82[理学—计算数学]