搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

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

引用本文:
Citation:

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

Numerical simulation of a class of FitzHugh-Nagumo systems based on the lattice Boltzmann method

He Yu-Bo, Tang Xian-Hua, Lin Xiao-Yan
PDF
导出引用
  • 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.
    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.
      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).
    [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 物理学报 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 物理学报 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 物理学报 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

  • [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 物理学报 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 物理学报 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 物理学报 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

  • [1] 冯晶森, 闵敬春. 直通道内两相流动的格子玻尔兹曼方法模拟. 物理学报, 2023, 72(8): 084701. doi: 10.7498/aps.72.20222421
    [2] 张天鸽, 任美蓉, 崔继峰, 陈小刚, 王怡丹. 变截面微管道中高zeta势下幂律流体的旋转电渗滑移流动. 物理学报, 2022, 71(13): 134701. doi: 10.7498/aps.71.20212327
    [3] 高效伟, 丁金兴, 刘华雩. 有限线法及其在流固域间耦合传热中的应用. 物理学报, 2022, 71(19): 190201. doi: 10.7498/aps.71.20220833
    [4] 张乾毅, 韦华健, 李华兵. 基于晶格玻尔兹曼方法的多段淋巴管模型. 物理学报, 2021, 70(21): 210501. doi: 10.7498/aps.70.20210514
    [5] 张士杰, 王颖明, 王琦, 李晨宇, 李日. 基于元胞自动机-格子玻尔兹曼模型的枝晶碰撞行为模拟. 物理学报, 2021, 70(23): 238101. doi: 10.7498/aps.70.20211292
    [6] 许炜炜, 白明珠, 林强, 胡正珲. 基于个性化三维心脏-躯干模型的心磁正问题. 物理学报, 2019, 68(17): 178702. doi: 10.7498/aps.68.20190387
    [7] 冯黛丽, 冯妍卉, 石珺. 介孔复合材料声子输运的格子玻尔兹曼模拟. 物理学报, 2016, 65(24): 244401. doi: 10.7498/aps.65.244401
    [8] 周锦阳, 施娟, 陈佳民, 李华兵. 脉动流血液通栓的晶格玻尔兹曼模型. 物理学报, 2014, 63(19): 194701. doi: 10.7498/aps.63.194701
    [9] 冯鑫, 李川, 胡开群. 基于深度玻尔兹曼模型的红外与可见光图像融合. 物理学报, 2014, 63(18): 184202. doi: 10.7498/aps.63.184202
    [10] 王辉, 黄志祥, 吴先良, 任信钢, 吴博. 双色散模型的辛时域有限差分算法. 物理学报, 2014, 63(7): 070203. doi: 10.7498/aps.63.070203
    [11] 王光辉, 王林雪, 王灯山, 刘丛波, 石玉仁. K(m,n,p)方程多-Compacton相互作用的数值研究. 物理学报, 2014, 63(18): 180206. doi: 10.7498/aps.63.180206
    [12] 孙东科, 项楠, 陈科, 倪中华. 格子玻尔兹曼方法模拟弯流道中粒子的惯性迁移行为. 物理学报, 2013, 62(2): 024703. doi: 10.7498/aps.62.024703
    [13] 彭武, 何怡刚, 方葛丰, 樊晓腾. 二维泊松方程的遗传PSOR改进算法. 物理学报, 2013, 62(2): 020301. doi: 10.7498/aps.62.020301
    [14] 高加振, 谢玲玲, 谢伟苗, 高继华. FitzHugh-Nagumo系统中螺旋波的控制. 物理学报, 2011, 60(8): 080503. doi: 10.7498/aps.60.080503
    [15] 周丰茂, 孙东科, 朱鸣芳. 偏晶合金液-液相分离的格子玻尔兹曼方法模拟. 物理学报, 2010, 59(5): 3394-3401. doi: 10.7498/aps.59.3394
    [16] 刘勇, 谢勇. 分数阶FitzHugh-Nagumo模型神经元的动力学特性及其同步. 物理学报, 2010, 59(3): 2147-2155. doi: 10.7498/aps.59.2147
    [17] 尹经禅, 肖晓晟, 杨昌喜. 光纤中受激Brillouin散射动态弛豫振荡特性及其抑制方法. 物理学报, 2009, 58(12): 8316-8325. doi: 10.7498/aps.58.8316
    [18] 谭新玉, 张端明, 李智华, 关 丽, 李 莉. 纳秒脉冲激光沉积薄膜过程中的烧蚀特性研究. 物理学报, 2005, 54(8): 3915-3921. doi: 10.7498/aps.54.3915
    [19] 许友生, 李华兵, 方海平, 黄国翔. 用格子玻尔兹曼方法研究流动-反应耦合的非线性渗流问题. 物理学报, 2004, 53(3): 773-777. doi: 10.7498/aps.53.773
    [20] 赵红东, 宋殿友, 张智峰, 孙 静, 孙 梅, 武 一, 温幸饶. n型DBR中电势对垂直腔面发射激光器阈值的影响. 物理学报, 2004, 53(11): 3744-3747. doi: 10.7498/aps.53.3744
计量
  • 文章访问数:  5038
  • PDF下载量:  423
  • 被引次数: 0
出版历程
  • 收稿日期:  2016-03-20
  • 修回日期:  2016-05-25
  • 刊出日期:  2016-08-05

/

返回文章
返回