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)

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
计量
- 文章访问数: 163
- HTML全文浏览量: 47
- PDF下载量: 24
- 被引次数: 0
出版历程
- 收稿日期: 2024-10-28
- 修回日期: 2025-02-26
- 刊出日期: 2026-04-01

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

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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图 1 当p=8，16，32，α=1，b=1，h=1/2，τ=1/100和θ取不同值时, 在x=0处的误差曲面

Figure 1. Error surfaces with p=8, 16, 32, α=1, b=1, h=1/2, τ=1/100 and different θ values for x=0

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图 2 当p=8，32，300，α=1, b=2，h=1/2，τ=1/100，θ取不同值时, 在x=2.5处的数值解曲面

Figure 2. Numerical solution surfaces with p=8, 32, 300, α=1, b=2, h=1/2, τ=1/100 and different θ values for x=2.5

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图 3 当p=8，32，300，α=1，b=2，h=1/2，τ=1/100和θ取不同值时, 在t=0.5处的数值解曲面

Figure 3. Numerical solution surfaces with p=8, 32, 300, α=1, b=2, h=1/2, τ=1/100 and different θ values for t=0.5

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图 4 当p=8，α=1，b=0.1，10，100，h=1/2，τ=1/100和θ取不同值时, 在x=2.5处的数值解曲面

Figure 4. Numerical solution surfaces with p=8, α=1, b=0.1, 10, 100, h=1/2, τ=1/100 and different θ values for x=2.5

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表 1 当p=8, 16, 32时差分格式(8a)—(8d)的数值结果

Table 1. The numerical results of difference schemes (8a)—(8d) with p=8, 16, 32

	h_x=h_y	τ	p=8		p=16		p=32
	h_x=h_y	τ	E_∞(h_x, h_y, τ)	O_order	E_∞(h_x, h_y, τ)	O_order	E_∞(h_x, h_y, τ)	O_order	T_CPU/s
θ=0, τ=h_x²/8	1/2	1/32	2.251 3E-3	-	3.866 5E-3	-	5.148 3E-3	-	0.014 4
	1/4	1/128	5.794 7E-4	1.958 0	9.177 2E-4	2.074 9	1.185 5E-3	2.118 6	0.018 9
	1/8	1/512	1.442 2E-4	2.006 5	2.293 9E-4	2.000 3	2.890 6E-4	2.036 1	0.087 2
	1/16	1/2 048	3.601 6E-5	2.001 5	5.719 4E-5	2.003 9	7.186 1E-4	2.008 1	1.164 5
θ=0.5, τ=h_x⁴/4	1/2	1/64	1.371 1E-2	-	1.155 7E-2	-	9.215 1E-3	-	0.014 9
	1/4	1/1 024	3.923 3E-3	1.805 2	3.114 3E-3	1.891 8	2.530 6E-3	1.864 5	0.054 4
	1/8	1/16 384	1.000 2E-3	1.971 9	8.046 0E-4	1.952 6	6.534 2E-4	1.953 4	2.004 0
	1/16	1/262 144	2.513 2E-4	1.992 6	2.035 0E-4	1.983 2	1.648 7E-4	1.986 7	161.700
θ=1, τ=h_x⁴/4	1/2	1/64	2.648 6E-2	-	2.237 0E-2	-	1.778 9E-2	-	0.013 4
	1/4	1/1 024	7.800 8E-3	1.763 5	6.153 0E-3	1.862 2	4.961 8E-3	1.842 0	0.063 7
	1/8	1/16 384	1.998 7E-3	1.964 5	1.608 5E-3	1.935 5	1.284 0E-3	1.950 2	2.026 1
	1/16	1/262 144	5.028 7E-4	1.990 8	4.054 0E-4	1.988 3	3.239 5E-4	1.986 8	121.119

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参考文献(13)

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