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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2025, Vol. 42 ›› Issue (2): 311-328.doi: 10.3969/j.issn.1005-3085.2025.02.007

• • 上一篇    下一篇

一类耦合非线性薛定谔方程的线性隐式差分格式

李胜平,  王俊杰   

  1. 普洱学院数学与统计学院，普洱 665000
  • 收稿日期:2022-07-11 接受日期:2023-03-29 出版日期:2025-04-15 发布日期:2025-06-15
  • 基金资助:
    国家自然科学基金(12161070).

A Linearly Implicit Conservative Scheme for a Coupled Nonlinear Schr\"odinger Equations

LI Shengping,  WANG Junjie   

  1. School of Mathematics and Statistics, Pu'er University, Pu'er 665000
  • Received:2022-07-11 Accepted:2023-03-29 Online:2025-04-15 Published:2025-06-15
  • Supported by:
    The National Natural Science Foundation of China (12161070).

摘要：

薛定谔方程一类重要的数学物理方程，在工程领域具有重要的应用。基于高阶有限差分法、Crank-Nicolson方法和Leap-frog方法，对耦合非线性薛定谔方程的守恒差分格式进行了研究。所提出的数值格式是解耦的、线性的，并满足离散质量和能量守恒律。同时，也讨论了数值格式的存在性、唯一性、稳定性和收敛性，证明该格式的精度为$O(\tau^2+h^4)$。最后，给出了数值实验结果，验证了该格式的有效性。

关键词: 薛定谔方程, 守恒格式, 稳定性, 收敛性

Abstract:

The Schr\"odinger equation is an important class of mathematical physics equations, and it has significant applications in engineering. The conservative difference scheme for the coupled nonlinear Schr\"odinger equation is studied based on the high-order finite difference method, the Crank-Nicolson, and the Leap-frog method. Moreover, the proposed numerical formulation is decoupled, linear, and it satisfies the discrete mass and energy conservation laws. The existence, uniqueness, stability and convergence of the numerical formulation are discussed, and it is shown that the numerical formulation is of the accuracy $O(\tau^2+h^4)$. The numerical experiments are given, and their results verify the efficiency of the scheme.

Key words: Schr\"odinger equation, conservative scheme, stability, convergence

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