基于格子玻尔兹曼方法的一类FitzHugh-Nagumo系统仿真研究

何郁波; 唐先华; 林晓艳

doi:10.7498/aps.65.154701

摘要

格子玻尔兹曼方法在复杂的流体系统中得到了广泛的应用. 本文针对在高于阈值常电流刺激下神经元动作电位周期性振荡的FitzHugh-Nagumo系统，构造了一类带源项和修正项的仿真格子玻尔兹曼模型（LBM）. 通过合理选择适当的局部平衡态分布函数和修正函数，再应用Chapman-Enskog多尺度分析，可以正确恢复出一类宏观非线性方程. 通过积分法得到了修正函数的构造方法，并分析了格子玻尔兹曼模型L稳定的充分条件. 利用网格相关性分析，本文所构造的模型具有二阶空间精度. 应用本文所提出的模型，仿真模拟了几个具有解析解的初边值系统，并与传统的改进有限差分格式（MFDM）进行了对比，结果表明本文模型所得的数值解与解析解吻合，其模拟误差小于MFDM. 此外，还针对不具有解析解的初边值系统进行了数值仿真，并与MFDM进行了对比. 数值结果表明，两种计算格式的数值解比较吻合，进一步证明了本文所构造模型的有效性和稳定性.

关键词:

Abstract

The lattice Boltzmann method (LBM) was proposed as a novel mesoscopic numerical method, and is widely used to simulate complex nonlinear fluid systems. In this paper, we develop a lattice Boltzmann model with amending function and source term to solve a class of initial value problems of the FitzHugh Nagumo systems, which arises in the periodic oscillations of neuronal action potential under constant current stimulation higher than the threshold value. Firstly, we construct a non-standard lattice Boltzmann model with the proper amending function and source term. For different evolution equations, local equilibrium distribution functions and amending function are selected, and the nonlinear FitzHugh Nagumo systems can be recovered correctly by using the Chapman Enskog multi-scale analysis. Secondly, through the integral technique, we obtain a new method on how to construct the amending function. In order to guarantee the stability of the present model, the L stability of the lattice Boltzmann model is analyzed by using the extremum principle, and we get a sufficient condition for the stability that is the initial value u0(x) must satisfy |u0(x)|1 and the parameters must satisfy i-(1+)(t)/(x), (i=1-4). Thirdly, based on the results of the grid independent analysis and numerical simulation, it can be concluded that the present model is convergent with two order space accuracy. Finally, some initial boundary value problems with analytical solutions are simulated to verify the effectiveness of the present model. The results are compared with the analytical solutions and numerical solutions obtained by the modified finite difference method (MFDM). It is shown that the numerical solutions agree well with the analytical solutions and the global relative errors obtained by the present model are smaller than the MFDM. Furthermore, some test problems without analytical solutions are numerically studied by the present model and the MFDM. The results show that the numerical solutions obtained by the present model are in good agreement with those obtained by the MFDM, which can validate the effectiveness and stability of the LBM. In conclusion, our model not only can enrich the applications of the lattice Boltzmann model in simulating nonlinear partial difference equations, but also help to provide valuable references for solving more complicated nonlinear partial difference systems. Therefore, this research has important theoretical significance and application value.

Keywords:

作者及机构信息

1.
中南大学, 数学与统计学院, 长沙 410083;

2.
怀化学院数学与应用数学系, 长沙 418008

通信作者: 何郁波, heyinprc@csu.edu.cn;tangxh@csu.edu.cn ; 唐先华, heyinprc@csu.edu.cn;tangxh@csu.edu.cn ;

基金项目: 国家自然科学基金（批准号：11471137，11571370，11501232）资助的课题.

Authors and contacts

1.
School of Mathematics and Statistics, Central South University, Changsha 410083, China;

2.
Department of Mathematics and Applied Mathematics, Huaihua University, Huaihua 418008, China

Corresponding author: He Yu-Bo, heyinprc@csu.edu.cn;tangxh@csu.edu.cn ; Tang Xian-Hua, heyinprc@csu.edu.cn;tangxh@csu.edu.cn ;

Funds: Project supported by National Natural Science Foundation of China (Grant Nos. 11471137, 11571370, 11501232).

参考文献

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[9]	Koido T, Furusawa T, Moriyama K 2008 J. Power. Sour. 175 127
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[14]	Shu C W, Osher S {1998 J. Comput. Phys. 77 439
[15]	Duan Y L, Liu R X 2007 J. Comp. Appl. Math. 206 432
[16]	Zhang J Y, Yan G W 2008 Physica A 387 4771
[17]	Ma C F, Tang J, Chen X H {2007 Chin. J. Appl. Mech. 24 519 (in Chinese) [马昌凤, 唐嘉, 陈小红 2007 应用力学学报 24 519]
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[19]	He Y B, Lin X Y, Dong X L {2013 Acta Phys. Sin. 62 194701 (in Chinese) [何郁波, 林晓艳, 董晓亮 2013 62 194701]
[20]	Zhou Z Q, He Y B {2012 Pure. Appl. Math. 28 29 (in Chinese) [周志强, 何郁波 2012 纯粹数学与应用数学 28 29]
[21]	Yung K L, Lei Y M, Xu Y 2010 Chin. Phys. B 19 010503
[22]	FitzHugh R {1961 Biophys. J. 6 445
[23]	Nagumo J S, Arimoto S, Yoshizawa S 1962 Proc. IRE 50 2061
[24]	Gan C B, Matja P, Wang Q Y 2010 Chin. Phys. B 19 040508
[25]	Song Y L 2014 Chin. Phys. B 23 080504
[26]	Prager T, Neiman A B, Schimansky G L 2009 Euro. Phys. J. B 69 119
[27]	Llibre J, Valls C 2010 J. Geom. Phys. 60 1974
[28]	Lv Y, Wang W 2010 Nonlinear Anal. Real. 11 3091
[29]	Hsu C H, Yang T H, Yang C R 2009 J. Differ. Equations 247 1185
[30]	Gaiko V A 2011 Nonlinear Anal. Theor. 74 7532
[31]	Olmos D, Shizgal B {2008 Math. Comput. Simulat. 79 2258
[32]	Browne P, Nomoniat E, Mahomed F M 2008 Nonlinear. Anal. Theor. 68 1006
[33]	Kawahara T, Tanaka M 1983 Phys. Lett. A 97 311
[34]	Nucci M C, Clarkson P A 1992 Phys. Lett. A 164 49
[35]	Li H Y, Guo Y C {2006 Appl. Math. Comput. 180 524
[36]	Guo Z L, Zheng C G, Shi B C 2002 Chin. Phys. 11 366

施引文献

[1]	Kaya D 2001 Int. J. Math. Math. Sci. 27 675
[2]	Abdou M A, Soliman A A 2005 J. Comput. Appl. Math. 181 245
[3]	Ram J, Gupta R K, Vikas K 2014 Ain. Sha. Eng. J. 5 1343
[4]	Xu A G, Zhang G C, Ying Y J {2015 Acta Phys. Sin. 64 184701 (in Chinese) [许爱国, 张广财, 应阳君 2015 64 184701]
[5]	Ollila S, Denniston C, Karttunen M, Nissila T 2011 J. Chem. Phys. 134 064902
[6]	Fallah K, Khaya M, Hossein B M, Ghaderi A, Fattahi E {2012 J. Non-Newton. Flui. 177 1
[7]	Mao W, Guo Z L, Wang L 2013 Acta Phys. Sin. 62 084703 (in Chinese) [毛威, 郭照立, 王亮 2013 62 084703]
[8]	Yang T Z, Ji S D, Yang X D, Fang B 2014 Int. J. Eng. Sci. 76 47
[9]	Koido T, Furusawa T, Moriyama K 2008 J. Power. Sour. 175 127
[10]	Zhang W, Wang Y, Qian Y H 2015 Chin. Phys. B 24 064701
[11]	Qian Y, Succi S, Orszag S {1995 Annu. Rev. Comput. Phys. 195 195
[12]	Chen S, Doolen G D 1998 Annu. Rev. Fluid. Mech. 30 329
[13]	Zu Y Q, He S 2013 Phys. Rev. E 87 043301
[14]	Shu C W, Osher S {1998 J. Comput. Phys. 77 439
[15]	Duan Y L, Liu R X 2007 J. Comp. Appl. Math. 206 432
[16]	Zhang J Y, Yan G W 2008 Physica A 387 4771
[17]	Ma C F, Tang J, Chen X H {2007 Chin. J. Appl. Mech. 24 519 (in Chinese) [马昌凤, 唐嘉, 陈小红 2007 应用力学学报 24 519]
[18]	Ma C F 2005 Chin. Phys. Lett. 22 2313
[19]	He Y B, Lin X Y, Dong X L {2013 Acta Phys. Sin. 62 194701 (in Chinese) [何郁波, 林晓艳, 董晓亮 2013 62 194701]
[20]	Zhou Z Q, He Y B {2012 Pure. Appl. Math. 28 29 (in Chinese) [周志强, 何郁波 2012 纯粹数学与应用数学 28 29]
[21]	Yung K L, Lei Y M, Xu Y 2010 Chin. Phys. B 19 010503
[22]	FitzHugh R {1961 Biophys. J. 6 445
[23]	Nagumo J S, Arimoto S, Yoshizawa S 1962 Proc. IRE 50 2061
[24]	Gan C B, Matja P, Wang Q Y 2010 Chin. Phys. B 19 040508
[25]	Song Y L 2014 Chin. Phys. B 23 080504
[26]	Prager T, Neiman A B, Schimansky G L 2009 Euro. Phys. J. B 69 119
[27]	Llibre J, Valls C 2010 J. Geom. Phys. 60 1974
[28]	Lv Y, Wang W 2010 Nonlinear Anal. Real. 11 3091
[29]	Hsu C H, Yang T H, Yang C R 2009 J. Differ. Equations 247 1185
[30]	Gaiko V A 2011 Nonlinear Anal. Theor. 74 7532
[31]	Olmos D, Shizgal B {2008 Math. Comput. Simulat. 79 2258
[32]	Browne P, Nomoniat E, Mahomed F M 2008 Nonlinear. Anal. Theor. 68 1006
[33]	Kawahara T, Tanaka M 1983 Phys. Lett. A 97 311
[34]	Nucci M C, Clarkson P A 1992 Phys. Lett. A 164 49
[35]	Li H Y, Guo Y C {2006 Appl. Math. Comput. 180 524
[36]	Guo Z L, Zheng C G, Shi B C 2002 Chin. Phys. 11 366

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