三维非线性神经传播方程的四阶和六阶Richardson外推法

张佳豪; 邓定文

doi:10.21656/1000-0887.450021

三维非线性神经传播方程的四阶和六阶Richardson外推法

doi: 10.21656/1000-0887.450021

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:

江西省杰出青年基金 20212ACB211006

国家自然科学基金 12461070

江西省自然科学基金重点项目 20242BAB26005

详细信息

作者简介:
张佳豪(2000—), 男, 硕士生

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

中图分类号: O357.41
计量
- 文章访问数: 40
- HTML全文浏览量: 10
- PDF下载量: 17
- 被引次数: 0
出版历程
- 收稿日期: 2024-01-27
- 修回日期: 2024-04-04
- 刊出日期: 2025-06-01

The 4th- and 6th-Order Richardson Extrapolation Methods for Solving 3D Nonlinear Nerve Conduction Equations

ZHANG Jiahao,
DENG Dingwen^,

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

摘要

摘要: 该文对一类非线性神经传播方程建立了一类交替方向隐式(ADI)紧致差分方法. 其在时间上有二阶精度, 在空间上有四阶精度. 运用Fourier分析法和能量法可证该方法是无条件线性稳定的. 此外, 对这类方法, 该文提出了两类Richardson外推方法, 以便分别获得时、空方向均有四阶或者六阶精度的外推解, 节省了计算成本. 数值结果验证了该方法的精度和有效性.
- ADI法 /
- 紧致差分法 /
- Richardson外推法 /
- 稳定性
Abstract: An alternating direction implicit (ADI) compact finite difference method (CFDM) was proposed for the numerical solution of the nonlinear nerve conduction equations. The method has 2nd-order accuracy in time and 4th-order accuracy in space, respectively. With the Fourier method and the discrete energy method, the proposed method was proved to be unconditionally linearly stable. In addition, 2 kinds of Richardson extrapolation methods used along with this ADI CFDM, are also developed to get time-space numerical extrapolation solutions with 4th-order or 6th-order accuracy, respectively, and improve the computational efficiency. Numerical results verify the accuracy and efficiency of the proposed method.
- ADI method /
- compact finite difference method /
- Richardson extrapolation method /
- stability

HTML全文

表 1 在t=1时刻，运用ADI紧致法获得的数值结果(Δt=h²)

Table 1. Case 1: numerical results at t=1 with the compact ADI method(Δt=h²)

h	E_u	R_∞	L_u	R_L²	E_v	R_∞	L_v	R_L²	t_CPU/s
1/2²	1.062 7E-3	-	3.745 6E-4	-	1.033 8E-4	-	3.605 9E-5	-	0.091 9
1/2³	6.808 5E-5	3.962 8	2.402 2E-5	3.962 8	6.783 4E-6	3.929 8	2.378 2E-6	3.922 4	0.538
1/2⁴	4.262 0E-6	3.997 7	1.503 8E-6	3.997 6	4.252 6E-7	3.995 6	1.491 5E-7	3.995 0	15.339
1/2⁵	2.664 0E-7	3.999 8	9.400 0E-8	3.999 8	2.658 4E-8	3.999 7	9.323 8E-9	3.999 7	539.05

下载: 导出CSV

表 2 例1: 在t=1时刻，运用Richardson-Ⅰ获得的数值结果(Δt=h)

Table 2. Case 1: numerical results at t=1 with the Richardson-Ⅰ method(Δt=h)

h	E_u	R_∞	L_u	R_L²	E_v	R_∞	L_v	R_L²	t_CPU/s
1/2²	1.115 9E-3	-	3.972 7E-4	-	1.763 4E-4	-	6.240 0E-5	-	0.018
1/2³	9.796 7E-5	3.509 7	3.553 8E-5	3.482 7	1.955 4E-5	3.172 8	7.294 2E-6	3.096 7	0.218
1/2⁴	6.691 8E-6	3.871 8	2.460 2E-6	3.852 5	1.394 4E-6	3.809 8	5.377 3E-7	3.761 8	2.836
1/2⁵	4.269 1E-7	3.970 4	1.579 1E-7	3.961 6	8.958 8E-8	3.960 2	3.506 1E-8	3.938 9	47.01

下载: 导出CSV

表 3 例1: 在t=1时刻，运用Richardson-Ⅱ获得的数值解(Δt=h)

Table 3. Case 1: numerical results at t=1 with the Richardson-Ⅱ method(Δt=h)

h	E_u	R_∞	L_u	R_L²	E_v	R_∞	L_v	R_L²	t_CPU/s
1/2²	3.010 6E-5	-	1.139 7E-5	-	9.101 4E-6	-	3.610 3E-6	-	0.238
1/2³	6.067 8E-7	5.632 7	2.559 2E-7	5.476 8	1.879 4E-7	5.597 8	8.791 1E-8	5.359 9	2.989
1/2⁴	9.788 9E-9	5.953 9	4.506 4E-9	5.827 6	3.577 3E-9	5.715 2	1.602 9E-9	5.777 3	49.77
1/2⁵	1.672 1E-10	5.871 4	7.503 7E-11	5.908 2	6.524 8E-11	5.776 8	2.683 9E-11	5.900 2	810.65

下载: 导出CSV

表 4 例2: 在t=1时刻，运用紧ADI法获得的数值结果(Δt=h²)

Table 4. Case 2: numerical results at t=1 with the compact ADI method(Δt=h²)

h	E_u	R_∞	L_u	R_L²	E_v	R_∞	L_v	R_L²	t_CPU/s
1/3	7.564 1E-2	-	3.445 5E-2	-	8.157 9E-2	-	3.737 3E-2	-	0.084 8
1/6	7.487 1E-3	3.336 7	2.632 5E-3	3.710 2	7.806 1E-3	3.385 5	2.748 5E-3	3.765 3	0.143
1/12	4.733 5E-4	3.983 4	1.668 9E-4	3.979 5	4.922 8E-4	3.987 1	1.737 3E-4	3.983 7	2.646
1/24	2.960 5E-5	3.999 0	1.044 0E-5	3.998 7	3.078 4E-5	3.999 2	1.086 5E-5	3.999 0	74.23

下载: 导出CSV

表 5 例2: 在t=1时刻，运用Richardson-Ⅰ获得的数值结果(Δt=h)

Table 5. Case 2: numerical results at t=1 with the Richardson-Ⅰ method(Δt=h)

h	E_u	R_∞	L_u	R_L²	E_v	R_∞	L_v	R_L²	t_CPU/s
1/3	9.214 4E-2	-	4.175 2E-2	-	9.342 8E-2	-	4.240 8E-2	-	0.037
1/6	1.946 7E-2	2.242 9	6.799 2E-3	2.618 4	1.727 5E-2	2.435 2	6.023 3E-3	2.815 7	0.085
1/12	1.750 2E-3	3.475 4	6.003 7E-4	3.501 4	1.459 8E-3	3.564 9	4.993 5E-4	3.592 4	0.766
1/24	1.213 8E-4	3.849 9	4.142 2E-5	3.857 4	9.910 5E-5	3.880 7	3.373 8E-5	3.887 6	9.97
1/48	7.792 5E-6	3.961 3	2.656 2E-6	3.962 9	6.333 9E-6	3.967 8	2.148 9E-6	3.972 7	164.89

下载: 导出CSV

表 6 例2: 在t=1时刻，运用Richardson-Ⅱ获得的数值解(Δt=h)

Table 6. Case 2: numerical results at t=1 with the Richardson-Ⅱ method(Δt=h)

h	E_u	R_∞	L_u	R_L²	E_v	R_∞	L_v	R_L²	t_CPU/s
1/3	9.434 3E-3	-	4.147 1E-3	-	7.605 2E-3	-	3.316 6E-3	-	0.085
1/6	5.453 2E-4	4.112 7	1.845 6E-4	4.489 9	3.928 9E-4	4.274 8	1.293 9E-4	4.679 9	0.825
1/12	1.279 1E-5	5.413 9	4.174 9E-6	5.466 2	8.392 8E-6	5.548 8	2.714 0E-6	5.575 2	10.44
1/24	2.251 1E-7	5.828 4	7.248 6E-8	5.847 9	1.725 3E-7	5.604 2	4.957 5E-8	5.774 6	186.29

下载: 导出CSV

参考文献(26)

[1]	PAO C V. A mixed initial boundary-value problem arising in neurophysiology[J]. Journal of Mathematical Analysis and Applications, 1975, 52 (1): 105-119.
[2]	PONCE G. Global existence of small solutions to a class of nonlinear evolution equations[J]. Nonlinear Analysis: Theory, Methods & Applications, 1985, 9 (5): 399-418.
[3]	刘亚成, 于涛. 神经传播型方程解的blow-up[J]. 应用数学学报, 1995, 18 (2): 264-272. LIU Yacheng, YU Tao. Blow-up of solutions for equation of nerve conduction type[J]. Acta Mathematicae Applicatae Sinica, 1995, 18 (2): 264-272. (in Chinese)
[4]	万维明, 刘亚成. 神经传播型方程初边值问题解的长时间行为[J]. 应用数学学报, 1999, 22 (2): 311-314. WAN Weiming, LIU Yacheng. Long-time behavier of initial boundary problem for equation of nerve conduction[J]. Acta Mathematicae Applicatae Sinica, 1999, 22 (2): 311-314. (in Chinese)
[5]	MOLL V, ROSENCRANS S I. Calculation of the threshold surface for nerve equations[J]. SIAM Journal on Applied Mathematics, 1990, 50 (5): 1419-1441.
[6]	DENG S J, WANG W K, ZHAO H L. Existence theory and L^p estimates for the solution of nonlinear viscous wave equation[J]. Nonlinear Analysis: Real World Applications, 2010, 11 (5): 4404-4414.
[7]	BUCKINGHAM M J. Causality, Stokes' wave equation, and acoustic pulse propagation in a viscous fluid[J]. Physical Review E, 2005, 72 (2): 026610.
[8]	DAI W, NASSAR R. A finite difference scheme for solving a three-dimensional heat transport equation in a thin film with microscale thickness[J]. International Journal for Numerical Methods in Engineering, 2001, 50 (7): 1665-1680.
[9]	DAI W, NASSAR R. A compact finite difference scheme for solving a three-dimensional heat transport equation in a thin film[J]. Numerical Methods for Partial Differential Equations, 2000, 16 (5): 441-458.
[10]	高兴宝, 万桂华, 陈开周. 方程u_tt=u_xxt +f(u_x)_x初边值问题的差分法[J]. 计算数学, 2000, 22 (2): 167-176. GAO Xingbao, WAN Guihua, CHEN Kaizhou. The finite difference method for the initial-boundary-value problem of the equation u_tt=u_xxt +f(u_x)_x[J]. Mathematica Numerical Sinica, 2000, 22 (2): 166-176. (in chinese)
[11]	DENG D W, ZHANG C J. A new fourth-order numerical algorithm for a class of three-dimensional nonlinear evolution equations[J]. Numerical Methods for Partial Differential Equations, 2013, 29 (1): 102-130.
[12]	DENG D W, ZHANG C J. Analysis and application of a compact multistep ADI solver for a class of nonlinear viscous wave equations[J]. Applied Mathematical Modelling, 2015, 39 (3/4): 1033-1049.
[13]	张志跃. 一类非线性发展方程的交替分段显隐并行数值方法[J]. 计算力学学报, 2022, 19 (2): 154-158. ZHANG Zhiyue. ASE-I parallel numerical method for a class of nonlinear evolution equation[J]. Chinese Journal of Computational Mechanics, 2022, 19 (2): 154-158. (in Chinese)
[14]	ZHANG Z Y. The finite element numerical analysis for a class nonlinear evolution equations[J]. Applied Mathematics and Computation, 2005, 166 (3): 489-500.
[15]	LI H R. Analysis and application of finite volume element methods to a class of partial differential equations[J]. Journal of Mathematical Analysis and Applications, 2009, 358 (1): 47-55.
[16]	DENG D W, ZHANG Z Y. A new high-order algorithm for a class of nonlinear evolution equation[J]. Journal of Physics A: Mathematical and Theoretical, 2008, 41 (1): 015202.
[17]	LELE S K. Compact finite difference schemes with spectral-like resolution[J]. Journal of Cmputational Pysics, 1992, 103 (1): 16-42.
[18]	KARAA S, ZHANG J. High order ADI method for solving unsteady convection-diffusion problems[J]. Journal of Computational Physics, 2004, 198 (1): 1-9.
[19]	DENG D W, LIANG D. The time fourth-order compact ADI methods for solving two-dimensional nonlinear wave equations[J]. Applied Mathematics and Computation, 2018, 329: 188-209.
[20]	WU F Y, CHENG X J, LI D F, et al. A two-level linearized compact ADI scheme for two-dimensional nonlinear reaction-diffusion equations[J]. Computers & Mathematics With Applications, 2018, 75 (8): 2835-2850.
[21]	GU Y X, LIAO W Y, ZHU J P. An efficient high-order algorithm for solving systems of 3-D reaction-diffusion equations[J]. Journal of Computational and Applied Mathematics, 2003, 155 (1): 1-17.
[22]	DENG D W, ZHANG C J. A family of new fourth-order solvers for a nonlinear damped wave equation[J]. Computer Physics Communications, 2013, 184 (1): 86-101.
[23]	魏剑英, 葛永斌. 一种求解三维非稳态对流扩散反应方程的高精度有限差分格式[J]. 应用数学和力学, 2022, 43 (2): 187-197. doi: 10.21656/1000-0887.420151 WEI Jianying, GE Yongbin. A high-order finite difference scheme for 3D unsteady convection diffusion reaction equations[J]. Applied Mathematics and Mechanics, 2022, 43 (2): 187-197. (in Chinese) doi: 10.21656/1000-0887.420151
[24]	胡健伟, 汤怀民. 微分方程数值方法[M]. 北京: 科学出版社, 2011. HU Jianwei, TANG Huaiming. Numerical Methods for Differential Equations[M]. Beijing: Science Press, 2011. (in Chinese)
[25]	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 Solutions for Partial Differential Equations[M]. 2nd ed. Beijing: Science Press, 2012. (in Chinese)
[26]	程爱杰. 交替方向隐格式稳定性和收敛性的改进[J]. 应用数学和力学, 1999, 20 (1): 71-78. doi: 10.21656/1000-0887.450301 CHENG Aijie. Improvement of stability and convergence of alternating direction implicit scheme[J]. Applied Mathematics and Mechanics, 1999, 20 (1): 71-78. (in Chinese) doi: 10.21656/1000-0887.450301