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

Yuhua Duan1, Linghua Kong2, Min Guo3
1School of Mathematical Sciences, University of Science and Technology of China, Hefei, People’s Republic of China
2School of Mathematics and Information Science, Jiangxi Normal University, Nanchang, People’s Republic of China
3Department of Basic, North China College of Mechanics and Electrics, Changzhi, People’s Republic of China

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Tài liệu tham khảo

Pozar, D.: Microwave Engineering. Addison-Wesley, NewYork (1990)

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

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

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. 34(1), 51–59 (2010)

Mohanty, R.K.: New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations. Int. J. Comput. Math. 86(12), 2061–2071 (2009)

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. 24, 1553–1570 (1986)

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

Evans, D.J., Bulut, H.: The numerical solution of the telegraph equation by the alternating group explicit method. Int. J. Comput. Math. 80, 1289–1297 (2003)

Jordan, P.M., Meyer, M.R., Puri, A.: Causal implications of viscous damping in compressible fluid flows. Phys. Rev. E 62, 7918–7926 (2000)

Greiner, W.: Relativistic Quantum Mechanics-Wave Equations, 3rd edn. Springer, Berlin (2000)

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

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

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

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

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

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

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

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

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

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

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

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

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. 185, 1399–1409 (2014)

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

Dawson, S.P., Chen, S., Doolen, G.D.: Lattice Boltzmann computations for reaction–diffusion equations. J. Chem. Phys. 2, 1514–1523 (1993)

Yan, G.: A lattice Boltzmann equation for waves. J. Comput. Phys. 161, 61–69 (2000)

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

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

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

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

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

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

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

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

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

Chen, S., Doolen, G.D.: Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329–364 (1998)

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)

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. 94, 511–525 (1954)

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

Jang, T.S.: A new solution procedure for the nonlinear telegraph equation. Commun. Nonlinear Sci. Numer. Simul. 29, 307–326 (2015)

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

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