二维Fisher-KPP方程的一组显式单调的有限差分法

张佳豪; 邓定文

doi:10.21656/1000-0887.450288

二维Fisher-KPP方程的一组显式单调的有限差分法

doi: 10.21656/1000-0887.450288

张佳豪,
邓定文^,

南昌航空大学数学与信息科学学院, 南昌 330063

基金项目:

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)

详细信息

作者简介:
张佳豪(2000—), 男, 硕士生(E-mail: 910342047@qq.com);邓定文(1981—), 男, 教授, 博士(通信作者. E-mail: dengdingwen2010@163.com).

通讯作者:
邓定文(1981—), 男, 教授, 博士(通信作者. E-mail: dengdingwen2010@163.com).

中图分类号: O357.41
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- 文章访问数: 16
- HTML全文浏览量: 4
- PDF下载量: 2
- 被引次数: 0
出版历程
- 收稿日期: 2024-10-28
- 修回日期: 2025-02-26
- 网络出版日期: 2026-04-30

A Class of Explicit and Monotonic Finite Difference Methods for 2D Fisher-KPP Equations

ZHANG Jiahao,
DENG Dingwen^,

College of Mathematics and Information Science, Nanchang Hangkong University, Nanchang 330063, P.R.China

Funds:

The National Science Foundation of China(12461070)

摘要

摘要: 运用一类加权的差分公式和显式Euler法离散扩散项及一阶时间导数项,从而对二维Fisher-Kolmogorov-Petrovsky-Piscounov(Fisher-KPP)方程构造一组两层、显式、单调的差分格式.经分析,证明了当网格步长和参数α，p，θ满足一定约束条件时,该格式能够保持原问题解的保正性、有界性和单调性等数学性质,并且获得了数值解在无穷范数下的误差估计.数值实验验证了数值结果与理论结果相吻合.
- Fisher-KPP方程 /
- 保正性 /
- 有界性 /
- 单调性 /
- 有限差分法
Abstract: With a class of weighted difference formulas and explicit Euler methods to discretize diffusion terms and the 1st-order temporal derivative, respectively, a new type of 2-level, explicit and monotonic finite difference methods is established for 2D Fisher-KPP equations. As α，p，θ and the grid step size satisfy some constraining conditions, their numerical solutions can inherit the properties of the exact solutions, such as positivity preservation, boundedness and monotonicity. Furthermore, the maximum norm error estimate is obtained, rigorously. Numerical experiments illustrate that the numerical results agree well with the theoretical findings.
- Fisher-KPP equation /
- positivity preservation /
- boundedness /
- monotonicity /
- finite difference method

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参考文献(13)

FISHER R A. The wave of advance of advantageous genes[J].Annals of Eugenics,1937,7(4): 355-369.

[2]KOLMOGOROV A, PETROVSKY I, PISCOUNOV N. Tudes de l’quation aved croissance de la quantité de matière et son application àun problème biologique[J].Moscow University Mathematics Bulletin,1937,1: 1-25.

[3]ARONSON D G, WEINBERGER H F. Multidimensional nonlinear diffusion arising in population genetics[J].Advances in Mathematics,1978,30(1): 33-76.

[4]WANG X Y. Exact and explicit solitary wave solutions for the generalised fisher equation[J].Physics Letters A,1988,131(4/5): 277-279.

[5]TYSON J J, BRAZHNIK P K. On traveling wave solutions of Fisher’s equation in two spatial dimensions[J].SIAM Journal on Applied Mathematics,2000,60(2): 371-391.

[6]PAO C V.Nonlinear Parabolic and Elliptic Equations[M]. New York: Plenum Press, 1992.

[7]满淑敏, 高强, 钟万勰. 非完整约束Hamilton动力系统保结构算法[J]. 应用数学和力学, 2020,41(6): 581-590.(MAN Shumin, GAO Qiang, ZHONG Wanxie. A structure-preserving algorithm for Hamiltonian systems with nonholonomic constraints[J].Applied Mathematics and Mechanics,2020,41(6): 581-590. (in Chinese))

[8]秦于越, 邓子辰, 胡伟鹏. 无限维Hamilton系统稳态解的保结构算法[J]. 应用数学和力学, 2014,35(1): 22-28.(QIN Yuyue, DENG Zichen, HU Weipeng. Structure-preserving algorithm for steady-state solution to the infinite dimensional Hamilton system[J].Applied Mathematics and Mechanics,2014,35(1): 22-28. (in Chinese))

[9]刘雪梅, 邓子辰, 胡伟鹏. 饱和多孔弹性杆流固耦合动力响应的保结构算法[J]. 应用数学和力学, 2016,37(10): 1050-1059.(LIU Xuemei, DENG Zichen, HU Weipeng. Structure-preserving algorithm for fluid-solid coupling dynamic responses of saturated poroelastic rods[J].Applied Mathematics and Mechanics,2016,37(10): 1050-1059. (in Chinese))

[10]QIN W, DING D, DING X. Two boundedness and monotonicity preserving methods for a generalized Fisher-KPP equation[J].Applied Mathematics and Computation,2015,252: 552-567.

[11]MACAS-DAZ J E, PURI A. An explicit positivity-preserving finite-difference scheme for the classical Fisher-Kolmogorov-Petrovsky-Piscounov equation[J].Applied Mathematics and Computation,2012,218(9): 5829-5837.

[12]MACAS-DAZ J E, REJNIAK K A. On a conditionally stable nonlinear method to approximate some monotone and bounded solutions of a generalized population model[J].Applied Mathematics and Computation,2014,229: 273-282.

[13]孙志忠. 偏微分方程数值解法[M]. 2版. 北京: 科学出版社, 2012.(SUN Zhizhong.Numerical Solution to Partial Differential Equations[M]. 2nd ed. Beijing: Science Press, 2012. (in Chinese))

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二维Fisher-KPP方程的一组显式单调的有限差分法

doi: 10.21656/1000-0887.450288

作者简介:
张佳豪(2000—), 男, 硕士生(E-mail: 910342047@qq.com);邓定文(1981—), 男, 教授, 博士(通信作者. E-mail: dengdingwen2010@163.com).

通讯作者:
邓定文(1981—), 男, 教授, 博士(通信作者. E-mail: dengdingwen2010@163.com).

计量

A Class of Explicit and Monotonic Finite Difference Methods for 2D Fisher-KPP Equations

计量

目录

留言板

二维Fisher-KPP方程的一组显式单调的有限差分法

doi: 10.21656/1000-0887.450288

作者简介: 张佳豪(2000—), 男, 硕士生(E-mail: 910342047@qq.com);邓定文(1981—), 男, 教授, 博士(通信作者. E-mail: dengdingwen2010@163.com).

通讯作者: 邓定文(1981—), 男, 教授, 博士(通信作者. E-mail: dengdingwen2010@163.com).

计量

出版历程

A Class of Explicit and Monotonic Finite Difference Methods for 2D Fisher-KPP Equations

计量

出版历程

目录

作者简介:
张佳豪(2000—), 男, 硕士生(E-mail: 910342047@qq.com);邓定文(1981—), 男, 教授, 博士(通信作者. E-mail: dengdingwen2010@163.com).

通讯作者:
邓定文(1981—), 男, 教授, 博士(通信作者. E-mail: dengdingwen2010@163.com).