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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 ›› 2026, Vol. 43 ›› Issue (1): 38-62.doi: 10.3969/j.issn.1005-3085.2026.01.003cstr: 32411.14.cjem.CN61-1269/O1.2026.01.003

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求解一类非线性抛物方程的单调有限差分方法

梁雨欣,  邓定文   

  1. 南昌航空大学数学与信息科学学院，南昌 330063
  • 收稿日期:2023-06-08 接受日期:2024-06-27 出版日期:2026-02-15 发布日期:2026-04-15
  • 通讯作者: 邓定文 E-mail: dengdingwen2010@163.com
  • 基金资助:
    国家自然科学基金 (12461070)；江西省自然科学基金重点项目 (20242BAB26005)；江西省杰出青年基金 (20212ACB211006).

A Class of Monotone Difference Schemes for Solving a Type of Nonlinear Parabolic Equations

LIANG Yuxin, DENG Dingwen   

  1. School of Mathematics and Information Science, Nanchang Hangkong University, Nanchang 330063
  • Received:2023-06-08 Accepted:2024-06-27 Online:2026-02-15 Published:2026-04-15
  • Contact: D. Deng. E-mail address: dengdingwen2010@163.com
  • Supported by:
    The National Natural Science Foundation of China (12461070); the Key Projects of Natural Science Foundation of Jiangxi Province (20242BAB26005); the Outstanding Youth Foundation  of Jiangxi Province (20212ACB211006).

摘要：

对一维非线性Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP) 方程构造一类单调的差分格式。运用能量分析法证明了当网格比和方程参数满足一定条件时，该格式能够保持原问题的一些性质，包括保正性、保界性和保单调性。此外，运用离散的极值原理证明了数值解在最大范数下的收敛阶。当它用于求解 Allen-Cahn 方程时，所得数值解满足最大值原理，并保持能量耗散定律。这类算法及其理论结果可以直接推广到任意高维问题。最后，数值实验表明数值结果与理论结果相吻合。

关键词: Fisher-KPP方程, Allen-Cahn方程, 单调差分格式, 收敛性分析

Abstract:

In this article, a class of monotone difference schemes (FDMs) for solving the one-dimensional nonlinear Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP) equation are derived. By using the discrete maximum principle method, it is shown that as the ratios between spatial and temporal meshsizes, and parameters satisfy certain conditions, this type of FDMs can inherit some properties of the original problem, such as  positivity, boundedness and monotonicity, and are convergent in maximum norm. They also can be used to solve the Allen-Cahn equation, whose numerical solutions satisfy the maximum principle and inherit the energy dissipation law. The current methods and corresponding analyses can be directly extended to high-dimensional problems. Finally, numerical experiments show that the numerical results are consistent with the theoretical results.

Key words: Fisher-KPP equation, Allen-Cahn equation, monotone FDMs, convergence analyses

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