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

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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.

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Acknowledgements

The authors are very thankful to the reviewers for their valuable suggestions to improve the quality of the paper. This work is supported by National Natural Science Foundation of China (Nos. 11101399, 11271171, 11301234), and the Provincial Natural Science Foundation of Jiangxi (Nos. 20161ACB20006, 20142BCB23009, 20151BAB201012).

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Authors and Affiliations

  1. School of Mathematical sciences, University of Science and Technology of China, Hefei, 230026, Anhui, People’s Republic of China

    Yali Duan

  2. School of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, Jiangxi, People’s Republic of China

    Linghua Kong

  3. Department of Basic, North China College of Mechanics and Electrics, Changzhi, 046000, Shanxi, People’s Republic of China

    Min Guo

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  1. Yali Duan
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  2. Linghua Kong
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  3. Min Guo
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Correspondence to Yali Duan.

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Duan, Y., Kong, L. & Guo, M. Numerical Simulation of a Class of Nonlinear Wave Equations by Lattice Boltzmann Method. Commun. Math. Stat. 5, 13–35 (2017). https://doi.org/10.1007/s40304-016-0098-x

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  • DOI: https://doi.org/10.1007/s40304-016-0098-x

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