求解二维对流扩散方程的格子Boltzmann方法

计算机工程与应用 ›› 2015, Vol. 51 ›› Issue (23): 68-73.

求解二维对流扩散方程的格子Boltzmann方法

彭碧涛1，郑洲顺1，2，刘红娟1，汤慧萍2，王建忠2

1.中南大学数学与统计学院，长沙 410083
2.金属多孔材料国家重点实验室，西安 710016

出版日期:2015-12-01 发布日期:2015-12-14

Lattice Boltzmann method for solving two dimensional convection-diffusion equations

PENG Bitao1, ZHENG Zhoushun1，2, LIU Hongjuan1, TANG Huiping2, WANG Jianzhong2

1.School of Mathematics and Statistics, Central South University, Changsha 410083, China
2.State Key Laboratory of Porous Metal Materials, Xi’an 710016, China

Online:2015-12-01 Published:2015-12-14

摘要/Abstract

摘要： 针对二维对流扩散方程，基于D2Q4格子速度，用Chapman-Enskog多尺度分析技术，将时间尺度取为二阶，空间尺度取为一阶，推导了各个速度方向上的平衡态分布函数所满足的条件，给出了简单且对称的平衡态分布函数表达式，所得到的平衡态分布函数能正确地恢复出二维对流扩散方程，从而构建了一种新的求解二维对流扩散方程的D2Q4格子Boltzmann（LB）模型。用所给LB模型对扩散方程和两个不同初边界条件的对流扩散方程进行了数值求解，数值实验结果表明数值解与精确解吻合较好，与相关文献结果比较边界误差要小得多，验证了模型的有效性。

关键词: 对流扩散方程, 格子Boltzmann方法, D2Q4模型, 多尺度分析技术

Abstract: In view of two-dimensional convection diffusion equation, it deduces the conditions of equilibrium distribution function must to be satisfied in every velocity directions based on D2Q4 Lattice velocity and gives the specific expressions of the equilibrium distribution function. Through Chapman-Enskog multi-scale analysis technology, taking the time scale directly to the second order and the spatial scale to the first order, the equilibrium distribution function can recover the original convection-diffusion equation, thus the new D2Q4 Lattice Boltzmann（LB） model for solving two-dimensional convection diffusion equation is constructed. Using the model, it implements a diffusion equation and two convection diffusion equation with different initial and boundary conditions, the numerical results are in good agreement with analytic solutions, furthermore the boundary error is very low compared with related document, therefore the effectiveness of the new model is verified.

Key words: convection-diffusion equation, lattice Boltzmann method, D2Q4 model, multi-scale analysis technology

彭碧涛1，郑洲顺1，2，刘红娟1，汤慧萍2，王建忠2. 求解二维对流扩散方程的格子Boltzmann方法[J]. 计算机工程与应用, 2015, 51(23): 68-73.

PENG Bitao1, ZHENG Zhoushun1，2, LIU Hongjuan1, TANG Huiping2, WANG Jianzhong2. Lattice Boltzmann method for solving two dimensional convection-diffusion equations[J]. Computer Engineering and Applications, 2015, 51(23): 68-73.

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