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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.
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Keywords:
- lattice Boltzmann model /
- FitzHugh-Nagumo equation /
- Chapman-Enskog expansion /
- finite difference method
[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
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[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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[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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