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

ZHANG Jiahao; DENG Dingwen

doi:10.21656/1000-0887.450288

Volume 47 Issue 4

Apr. 2026

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ZHANG Jiahao, DENG Dingwen. A Class of Explicit and Monotonic Finite Difference Methods for 2D Fisher-KPP Equations[J]. Applied Mathematics and Mechanics, 2026, 47(4): 505-515. doi: 10.21656/1000-0887.450288

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

doi: 10.21656/1000-0887.450288

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)

Received Date: 2024-10-28
Rev Recd Date: 2025-02-26

Available Online: 2026-04-30

Abstract

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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References(13)

References

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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