Numerical Solution of Nonlinear Klein-Gordon Equation Using Lattice Boltzmann Method

Qiaojie Li, Zong Ji, Zhoushun Zheng, Hongjuan Liu

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DOI: 10.4236/am.2011.212210                

Abstract

In this paper, in order to extend the lattice Boltzmann method to deal with more nonlinear equations, a one-dimensional (1D) lattice Boltzmann scheme with an amending function for the nonlinear Klein-Gordon equation is proposed. With the Taylor and Chapman-Enskog expansion, the nonlinear Klein-Gordon equation is recovered correctly from the lattice Boltzmann equation. The method is applied on some test examples, and the numerical results have been compared with the analytical solutions or the numerical solutions reported in previous studies. The L₂, L_∞ and Root-Mean-Square (RMS) errors in the solutions show the efficiency of the method computationally.

Keywords

Lattice Boltzmann, Chapman-Enskog Expansion, Nonlinear Klein-Gordon Equation

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	P. J. Caudrey, I. C. Eilbeck and J. D. Gibbon, “The Sine-Gordon Equation as a Model Classical Field Theory,” Nuovo Cimento, Vol. 25, No. 2, 1975, pp. 497-511.
[2]	R. K. Dodd, I. C. Eilbeck, J. D. Gibbon and H. C. Morris, “Solitons and Nonlinear Wave Equations,” Academic, London, 1982.
[3]	Sirendaoreji, “Auxiliary Equation Method and New Solutions of Klein-Gordon Equations,” Chaos, Solitons & Fractals, Vol. 31, No. 4, 2007, pp. 943-950. doi:10.1016/j.chaos.2005.10.048
[4]	Sirendaoreji, “A New Auxiliary Equation and Exact Travelling Wave Solutions of Nonlinear Equation,” Physics Letters A, Vol. 356, No. 2, 2006, pp. 124-130. doi:10.1016/j.physleta.2006.03.034
[5]	A. M. Wazwaz, “New Travelling Wave Solutions to the Boussinesq and Klein-Gordon Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 13, No. 5, 2008, pp. 889-901. doi:10.1016/j.cnsns.2006.08.005
[6]	M. A. Lynch, “Large Amplitude Instability in Finite Difference Approximates to the Klein-Gordon Equation,” Applied Numerical Mathematics, Vol. 31, No. 2, 1999, pp. 173-182. doi:10.1016/S0168-9274(98)00128-7
[7]	X. Li, B. Y. Guo and L. Vazquez, “A Legendre Spectral Method for Solving the Nonlinear Klein-Gordon Equation,” Mathematics Applied and Computation, Vol. 15, No. 1, 1996, pp. 19-36.
[8]	X. Li and B. Y. Guo, “A Legendre Spectral Method for Solving Nonlinear Klein-Gordon Equation,” Journal Computation of Mathematics, Vol. 15, No. 2, 1997, pp. 105-126.
[9]	M. Deghan and A. Shokri, “Numerical Solution of the Nonlinear Klein-Gordon Equation Using Radial Basis Functions,” Journal of Computational and Applied Mathematics, Vol. 230, No. 2, 2009, pp. 400-410. doi:10.1016/j.cam.2008.12.011
[10]	R. Benzi, S. Succi and M. Vergassola, “The Lattice Boltzmann Equation: Theory and Application,” Physics Reports, Vol. 222, No. 3, 1992, pp. 145-197. doi:10.1016/0370-1573(92)90090-M
[11]	S. Y. Chen and G. D. Doolen, “Lattice Boltzmann Method for Fluid Flows,” Annual Review of Fluid Mechanics, Vol. 30, No. 1, 1997, pp. 329-364. doi:10.1146/annurev.fluid.30.1.329
[12]	J. Y. Zhang, G. W. Yan and Y. F. Dong, “A New Lattice Boltzmann Model for the Laplace Equation,” Applied Mathematics and Computation, Vol. 215, No. 2, 2009, pp. 539-547.doi:10.1016/j.amc.2009.05.047
[13]	Z. H. Chai and B. C. Shi, “A Novel Lattice Boltzmann Model for the Poisson Equation,” Applied Mathematical Modelling, Vol. 32, No. 10, 2008, pp. 2050-2058. doi:10.1016/j.apm.2007.06.033
[14]	M. Hirabayashi, Y. Chen and H. Ohashi, “The Lattice BGK Model for the Poisson Equation,” JSME International Journal Series B, Vol. 44, No. 1, 2001, pp. 45-52. doi:10.1299/jsmeb.44.45
[15]	J. G. Zhou, “Lattice Boltzmann Method for Shallow Water Flows,” Springer Verlag, New York, 2004.
[16]	Z. Shen, G. Yuan and L. Shen, “Lattice Boltzmann Method for Burgers Equation,” Chinese Journal of Computational Physics, Vol. 175, No. 1, 2000, pp. 172-177.
[17]	J. Y. Zhang and G. W. Yan, “A Lattice Boltzmann Model for the Korteweg-de Vries Equation with Two Conservation Laws,” Computer Physics Communications, Vol. 180, No. 7, 2009, pp. 1054-1062. doi:10.1016/j.cpc.2008.12.027
[18]	G. W. Yan, “A Lattice Boltzmann Equation for Waves,” Journal of Computational Physics, Vol. 161, No. 1, 2000, pp. 61-69. doi:10.1006/jcph.2000.6486
[19]	J. Y. Zhang, G. W. Yan and X. Shi, “Lattice Boltzmann Model for Wave Propagation,” Physics Review E, Vol. 80, 2009, Article ID 026706. doi:10.1103/PhysRevE.80.026706
[20]	S. P. Dawson, S. Chen and G. D. Doolen, “Lattice Boltzmann Computations for Reaction-Diffusion Equation,” Journal of Chemical Physics, Vol. 98, No. 2, 1993, pp. 1514-1523. doi:10.1063/1.464316
[21]	X. Yu and B. C. Shi, “A Lattice Boltzmann Model for Reaction Dynamical Systems with Time Delay,” Applied Mathematics and Computation, Vol. 181, No. 2, 2006, pp. 958-965. doi:10.1016/j.amc.2006.02.020
[22]	S. R. Vander and M. Ernst, “Convection-Diffusion Lattice Boltzmann Scheme for Irregular Lattice,” Journal of Computational Physics, Vol. 160, No. 2, 2000, pp. 766-782. doi:10.1006/jcph.2000.6491
[23]	Z. L. Guo, B. C. Shi and N. C. Wang, “Fully Lagrangian and Lattice Boltzmann Method for the Advection-Diffusion Equation,” Journal of Scientific Computing, Vol. 14, No. 3, 1999, pp. 291-300. doi:10.1023/A:1023273603637
[24]	B.C. Shi and Z. L. Guo, “Lattice Boltzmann Model for Nonlinear Convection-Diffusion Equations,” Physics Review E, Vol. 79, 2009, Article ID 016701. doi:10.1103/PhysRevE.79.016701
[25]	Z. L. Guo, C. G. Zheng and B. C. Shi, “Non-Equilibrium Extrapolation Method for Velocity and Pressure Boundary Conditions in the Lattice Boltzmann Method,” Chinese Physics, Vo. 11, No. 4, 2002, pp. 366-374. doi:10.1088/1009-1963/11/4/310
[26]	S. Jiminez and L. Vazquez, “Analysis of Four Numerical Scheme for a Nonlinear Klein-Gordon Equation,” Applied Mathematics and Computation, Vol. 35, No. 1, 1990, pp. 61-94. doi:10.1016/0096-3003(90)90091-G