School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China.
Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China.
Phys Rev E. 2018 Jan;97(1-1):013304. doi: 10.1103/PhysRevE.97.013304.
In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂{t}ϕ+∑{k=1}^{m}α_{k}∂{x}^{k}Π{k}(ϕ)=0 (1≤k≤m≤6), α_{k} are constant coefficients, Π_{k}(ϕ) are some known differential functions of ϕ. As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K(n,n)-Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009)1672-179910.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009)PHYADX0378-437110.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.
本文提出了一个用于求解形式为 ∂{t}ϕ+∑{k=1}^{m}α_{k}∂{x}^{k}Π{k}(ϕ)=0 (1≤k≤m≤6)的高阶非线性偏微分方程的通用格子 Boltzmann (LB) 模型，其中 α_{k}为常数系数，Π_{k}(ϕ)为 ϕ的一些已知微分函数。作为该高阶非线性偏微分方程的一些特殊情况，经典 (m)KdV 方程、KdV-Burgers 方程、K(n,n)-Burgers 方程、Kuramoto-Sivashinsky 方程和 Kawahara 方程可以用本 LB 模型求解。与现有的 LB 模型相比，本模型最显著的特点是引入了一些合适的辅助矩，从而可以得到平衡分布函数的正确矩。此外，我们还进行了详细的Chapman-Enskog 分析，发现提出的 LB 模型可以正确地恢复高阶非线性偏微分方程。最后，进行了大量的数值模拟，发现数值结果与解析解吻合较好，并且通常本模型也比现有的 LB 模型更为精确[H. Lai 和 C. Ma，中国科学 G 辑 52, 1053 (2009)1672-179910.1007/s11433-009-0149-3; H. Lai 和 C. Ma，物理学报 A (阿姆斯特丹) 388, 1405 (2009)PHYADX0378-437110.1016/j.physa.2009.01.005]。
