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

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

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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)