二维非线性Schr?dinger方程的两类局部守恒算法  

Numerical analysis of two local conservative methods for two-dimensional nonlinear Schr?dinger equation

作　　者：钱旭 宋松和[1] Xu Qian;Songhe Song

机构地区：[1]国防科技大学理学院,长沙410072 [2]国防科技大学海洋科学与工程研究院,长沙410073

出　　处：《中国科学：数学》2018年第2期345-360,共16页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号：11501570和11571366);国防科技大学科研计划(批准号：JC15-02-02);中国博士后科学基金(批准号：2017M613362)资助项目

摘　　要：本文针对二维非线性Schr?dinger方程,提出两类局部守恒算法.不需要考虑边界条件,即可保持任意时空区域上相应的局部能量守恒律和局部动量守恒律.在合适的边界条件下,它们能自然地保持电荷、全局能量或全局动量守恒律.本文同时对算法进行了守恒分析和误差分析.在数值实验部分,本文构造了类似的多辛Preissman算法进行比较,数值结果验证了其长时间计算的优势.In this paper, we propose two local conservative methods for solving two-dimensional nonlinear Schr？dinger equation. Without consideration of the boundary conditions, they can preserve corresponding local energy and momentum conservation laws exactly at arbitrary spatial-temporal regions. Meanwhile, the charge,global energy and global momentum conservation laws can be conserved under suitable boundary conditions.Numerical analysis, including conservative properties and error estimation analysis, are investigated. Furthermore, a similar multi-symplectic Preissman scheme is constructed as comparison. Numerical experiments show the advantages of the proposed methods during long-time numerical simulations and validate the analysis.

关 键 词：SCHRODINGER方程 局部守恒特征 能量守恒律 动量守恒律 保结构算法 

分 类 号：O175[理学—数学]