Numerical Simulation of a Class of Nonlinear Wave Equations by Lattice Boltzmann Method

Yali Duan , Linghua Kong , Min Guo

Communications in Mathematics and Statistics ›› 2017, Vol. 5 ›› Issue (1) : 13 -35.

PDF
Communications in Mathematics and Statistics ›› 2017, Vol. 5 ›› Issue (1) : 13 -35. DOI: 10.1007/s40304-016-0098-x
Article

Numerical Simulation of a Class of Nonlinear Wave Equations by Lattice Boltzmann Method

Author information +
History +
PDF

Abstract

In this paper, we develop a lattice Boltzmann model for a class of one-dimensional nonlinear wave equations, including the second-order hyperbolic telegraph equation, the nonlinear Klein–Gordon equation, the damped and undamped sine-Gordon equation and double sine-Gordon equation. By choosing properly the conservation condition between the macroscopic quantity $u_t$ and the distribution functions and applying the Chapman–Enskog expansion, the governing equation is recovered correctly from the lattice Boltzmann equation. Moreover, the local equilibrium distribution function is obtained. The results of numerical examples have been compared with the analytical solutions to confirm the good accuracy and the applicability of our scheme.

Keywords

Lattice Boltzmann method / Second-order hyperbolic telegraph equation / Klein–Gordon equation / Sine-Gordon equation / Chapman–Enskog expansion

Cite this article

Download citation ▾
Yali Duan, Linghua Kong, Min Guo. Numerical Simulation of a Class of Nonlinear Wave Equations by Lattice Boltzmann Method. Communications in Mathematics and Statistics, 2017, 5(1): 13-35 DOI:10.1007/s40304-016-0098-x

登录浏览全文

4963

注册一个新账户 忘记密码

References

[1]

Pozar D. Microwave Engineering. 1990 NewYork: Addison-Wesley

[2]

Mohebbi A, Dehghan M. High order compact solution of the one-space-dimensional linear hyperbolic equation. Numer Methods Partial Differ. Equ.. 2008, 24 5 1222-1235

[3]

Jeffrey A. Applied Partial Differential Equations. 2002 NewYork: Academic Press

[4]

Dehghan M, Ghesmati A. Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method. Eng. Anal. Bound. Elem.. 2010, 34 1 51-59

[5]

Mohanty RK. New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations. Int. J. Comput. Math.. 2009, 86 12 2061-2071

[6]

Pascal H. Pressure wave propagation in a fluid flowing through a porous medium and problems related to interpretation of Stoneley’s wave attenuation in a coustical well logging. Int. J. Eng. Sci.. 1986, 24 1553-1570

[7]

Bohme G. Non-Newtonian fluid mechanics. 1987 NewYork: North-Holland

[8]

Evans DJ, Bulut H. The numerical solution of the telegraph equation by the alternating group explicit method. Int. J. Comput. Math.. 2003, 80 1289-1297

[9]

Jordan PM, Meyer MR, Puri A. Causal implications of viscous damping in compressible fluid flows. Phys. Rev. E. 2000, 62 7918-7926

[10]

Greiner W. Relativistic Quantum Mechanics-Wave Equations. 2000 3 Berlin: Springer

[11]

Scott A. Nonlinear Science: Emergence and Dynamics of Coherent Structures. 2003 Oxford: Oxford University Press

[12]

Dauxois T, Peyrard M. Physics of Solitons. 2006 Cambridge: Cambridge University Press

[13]

Liu L, Liu H. Compact difference schemes for solving telegraphic equations with Neumann boundary conditions. Appl. Math. Comput.. 2013, 219 19 10112-10121

[14]

Rashidinia J, Mohammadi R. Tension spline approach for the numerical solution of nonlinear Klein–Gordon equation. Comput. Phys. Commun.. 2010, 181 78-91

[15]

Rashidinia J, Mohammadi R. Tension spline solution of nonlinear sine-Gordon equation. Numer Algorithms. 2011, 56 129-142

[16]

Mohebbi A, Dehghan M. High-order solution of one-dimensional sine-Gordon equation using compact finite difference and DIRKN methods. Math. Comput. Modell.. 2010, 51 537-549

[17]

Moghaderi H, Dehghan M. A multigrid compact finite differencemethod for solving the one-dimensional nonlinear sine-Gordon equation. Math. Methods Appl. Sci.. 2015, 38 3901-3922

[18]

Mohebbi A, Dehghan M. High-order solution of one-dimensional sine-Gordon equation using compact finite difference and DIRKN methods. Math. Comput. Modell.. 2010, 51 537-549

[19]

Mittal RC, Bhatia R. Numerical solution of second order one dimensional hyperbolic telegraph equation by cubic B-spline collocation method. Appl. Math. Comput.. 2013, 220 496-506

[20]

Rashidinia J, Ghasemia M, Jalilian R. Numerical solution of the nonlinear Klein–Gordon equation. J. Comput. Appl. Math.. 2010, 233 1866-1878

[21]

Sharifi S, Rashidinia J. Numerical solution of hyperbolic telegraph equation by cubic B-spline collocation method. Appl. Math. Comput.. 2016, 281 28-38

[22]

Liu W, Wu B, Sun J. Space–time spectral collocation method for the one-dimensional sine-Gordon equation. Numer. Methods Partial Differ. Equ.. 2015, 31 670-690

[23]

Shao W, Wu X. The numerical solution of the nonlinear Klein–Gordon and Sine–Gordon equations using the Chebyshev tau meshless method. Comput. Phys. Commun.. 2014, 185 1399-1409

[24]

Khuri SA, Sayfy A. A spline collocation approach for the numerical solution of a generalized nonlinear Klein–Gordon equation. Appl. Math. Comput.. 2010, 216 1047-1056

[25]

Dawson SP, Chen S, Doolen GD. Lattice Boltzmann computations for reaction–diffusion equations. J. Chem. Phys.. 1993, 2 1514-1523

[26]

Yan G. A lattice Boltzmann equation for waves. J. Comput. Phys.. 2000, 161 61-69

[27]

Zhang J, Yan G. A lattice Boltzmann model for the Korteweg–de Vries equation with two conservation laws. Comput. Phys. Commun.. 2009, 180 1054-1062

[28]

Duan Y, Liu R. Lattice Boltzmann model for two-dimensional unsteady Burgers’ equation. J. Comput. Appl. Math.. 2007, 206 432-439

[29]

Shi B, Guo Z. Lattice Boltzmann model for nonlinear convection–diffusion equations. Phy. Rev. E. 2009, 79 016701

[30]

Lai H, Ma C. Lattice Boltzmann modei for generalized nonlinear wave equation. Phys. Rev. E. 2011, 84 046708

[31]

Duan Y, Kong L. A lattice Boitzmann model for the generalized Burgers–Hulexly equation. Phys. A. 2012, 391 625-632

[32]

Duan Y, Chen X, Kong L. Lattice Boltzmann model for the compound Burgers–Korteweg–de Vries equation. Chin. J. Comput. Phys.. 2015, 32 6 639-648

[33]

Higuera F, Succi S, Benzi R. Lattice gas dynamics with enhanced collisions. Euro. Phys. Lett.. 1989, 9 345-349

[34]

Benzi R, Succi S, Vergassola M. The lattice Boltzmann equation: theory and applications. Phys. Rep.. 1992, 222 3 145-197

[35]

Qian Y, Succi S, Orszag S. Recent advances in lattice Boltzmann computing. Annu. Rev. Comput. Phys.. 1995, 3 195-242

[36]

Chen S, Doolen GD. Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech.. 1998, 30 329-364

[37]

Luo. L.: The lattice-gas and lattice Boltzmann methods: past, present and future. In: Proceedings of International Conference on Applied Computational Fluid Dynamics. October, China, Beijing, pp. 52-83 (2000)

[38]

Bhatnagar P, Gross E, Krook M. A model for collision process in gas. I: small amplitude processed in charged and neutral one component system. Phys. Rev.. 1954, 94 511-525

[39]

Dehghan M, Shokri A. A numerical method for solving the hyperbolic telegraph equation. Numer. Methods Partial Differ. Equ.. 2008, 24 1080-1093

[40]

Jang TS. A new solution procedure for the nonlinear telegraph equation. Commun. Nonlinear Sci. Numer. Simul.. 2015, 29 307-326

[41]

He B, Meng Q, Long Y, Rui W. New exact solutions of the double sine-Gordon equation using symbolic computations. Appl. Math. Comput.. 2007, 186 1334-1346

[42]

Wazwaz A-M. The tanh method and a variable separated ODE method for solving double sine-Gordon equation. Phys. Lett. A. 2006, 350 367-370

Funding

National Natural Science Foundation of China(11101399)

Provincial National Science Foundation of Anhui

National Natural Science Foundation of China(11271171)

Provincial Natural Science Foundation of Jiangxi(20161ACB20006)

Provincial Natural Science Foundation of Jiangxi(20151BAB201012)

AI Summary AI Mindmap
PDF

158

Accesses

0

Citation

Detail

Sections
Recommended

AI思维导图

/