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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 ›› 2018, Vol. 35 ›› Issue (6): 693-706.doi: 10.3969/j.issn.1005-3085.2018.06.009

• • 上一篇    下一篇

非线性耗散Schrödinger方程的紧致差分格式

王廷春,   张   雯,   王国栋   

  1. 南京信息工程大学数学与统计学院，南京  210044
  • 收稿日期:2016-10-09 接受日期:2017-03-01 出版日期:2018-12-15 发布日期:2019-02-15
  • 基金资助:
    国家自然科学基金(11571181)；国家级大学生实践创新训练计划项目(201510300026Z)；江苏省青蓝工程.

Compact Finite Difference Schemes for the Dissipative Nonlinear Schrödinger Equation

WANG Ting-chun,   ZHANG Wen,   WANG Guo-dong   

  1. College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044
  • Received:2016-10-09 Accepted:2017-03-01 Online:2018-12-15 Published:2019-02-15
  • Supported by:
    The National Natural Science Foundation of China (11571181); the National College Students Innovation and Entrepreneurship Training Program (201510300026Z); Jiangsu Qianlan Project.

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摘要： 本文对非线性耗散Schrödinger方程提出并分析了两个紧致有限差分格式．由于数值解的先验估计很难得到，这给格式的收敛性分析带来本质困难．为此，本文将非线性项的系数函数光滑截断为一个全局Lipschitz连续函数，并结合标准的能量方法，在对网格比没有任何要求的前提下建立了格式在最大模意义下的最优误差估计，证明数值解在空间和时间方向的收敛阶在最大模意义下分别为4阶和2阶．数值结果验证了理论分析的正确性，并展示了新格式较已有格式的优越性．

关键词: 非线性耗散Schr?dinger方程, 紧致差分格式, 最优逐点误差估计

Abstract: In this paper, two compact finite difference schemes are proposed and analyzed for solving a dissipative nonlinear Schrödinger equation. Due to the difficulty in obtaining the a priori estimate of numerical solutions, it is hard to prove the convergence of the proposed schemes. In order to overcome the difficulty, we truncate the coefficient function of the nonlinear term into a global Lipschitz continuous function and use the standard energy method to establish the optimal error estimates of numerical solutions in terms of the maximum norm without any restriction on the grid ratio. The convergence rates are proved to be of the fourth-order in space and the second-order in time, respectively. Numerical results support the theoretical analysis and show the advantages of the proposed schemes compared with the existing ones.

Key words: dissipative nonlinear Schr?dinger equation, compact finite difference scheme, optimal point-wise error estimate

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