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

ZHANG Jiahao; DENG Dingwen

doi:10.21656/1000-0887.450021

Volume 46 Issue 6

Jun. 2025

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Article Navigation > Applied Mathematics and Mechanics > 2025 > 46(6): 800-808

ZHANG Jiahao, DENG Dingwen. The 4th- and 6th-Order Richardson Extrapolation Methods for Solving 3D Nonlinear Nerve Conduction Equations[J]. Applied Mathematics and Mechanics, 2025, 46(6): 800-808. doi: 10.21656/1000-0887.450021

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

doi: 10.21656/1000-0887.450021

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: 2024-01-27
Rev Recd Date: 2024-04-04

Available Online: 2025-06-30

Abstract

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

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References(26)

References

[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.

PAO C V. A mixed initial boundary-value problem arising in neurophysiology[J].Journal of Mathematical Analysis and Applications,1975,52(1): 105-119.

[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utt =uxxt +f(ux)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 utt =uxxt +f(ux)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. (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))

[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. (CHENG Aijie. Improvement of stability and convergence of alternating direction implicit scheme[J].Applied Mathematics and Mechanics,1999,20(1): 71-78. (in Chinese))

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