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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 ›› 2023, Vol. 40 ›› Issue (6): 929-940.doi: 10.3969/j.issn.1005-3085.2023.06.006

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Banach空间中复合刚性Volterra泛函微分方程隐显Euler方法的稳定性分析

龙  滔,  余越昕   

  1. 湘潭大学数学与计算科学学院，湘潭 411105
  • 收稿日期:2021-04-27 接受日期:2021-08-23 出版日期:2023-12-15 发布日期:2024-02-15
  • 基金资助:
    国家自然科学基金 (12271367)；湖南省教育厅重点项目 (21A0115).

Stability Analysis of Implicit-explicit Euler Method for Composite Stiff Volterra Functional Differential Equations in Banach Space

LONG Tao,  YU Yuexin   

  1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105
  • Received:2021-04-27 Accepted:2021-08-23 Online:2023-12-15 Published:2024-02-15
  • Supported by:
    The National Natural Science Foundation of China (12271367); the Scientific Research Fund of Hunan Provincial Education Department (21A0115).

摘要：

刚性泛函微分方程数值方法的研究大多是在内积空间中基于单边Lipschitz常数具有适度大小的条件下进行；然而对于某些刚性问题，其单边Lipschitz常数却不可避免地取非常巨大的正值。因此有必要突破内积空间和单边Lipschitz常数的限制，直接在Banach空间中探讨相应的数值方法。针对Banach空间中的非线性复合刚性Volterra泛函微分方程，对其非刚性部分采用显式Euler方法求解，刚性部分采用隐式Euler方法求解，得到了求解该问题的隐显Euler方法，论证了方法的稳定性和渐近稳定性。数值试验结果验证了所获理论的正确性。

关键词: 复合刚性微分方程, 稳定性, 渐近稳定性, 隐显Euler方法, Banach空间

Abstract:

The study of numerical methods for stiff functional differential equations is mostly carried out in the inner product space based on the assumption that the one-sided Lipschitz constant has a moderate size, whereas for some stiff problems, the one-sided Lipschitz constant inevitably takes very large positive values. Therefore, it is necessary to break through the limitations of inner product space and one-sided Lipschitz constant and study the related numerical methods directly in the Banach space. For the nonlinear composite stiff Volterra functional differential equation in Banach space, the non-stiff part is solved by the explicit Euler method and the stiff part is solved by the implicit Euler method, obtained from which is the implicit-explicit Euler method for solving the problem. The stability and asymptotic stability about the proposed method are established, and the numerical results verify the correctness of the obtained theory.

Key words: composite stiff differential equations, stability, asymptotic stability, implicit-explicit Euler methods, Banach space

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