二维带分数阶Laplacian算子的对流扩散方程的格子Boltzmann模型研究  

LATTICE BOLTZMANN MODEL FOR THE ADVECTION AND DIFFUSION EQUATION WITH FRACTIONAL LAPLACIAN

作　　者：周彤彤 杜睿 Zhou Tongtong;Du Rui(School of Mathematics,Southeast University,Nanjing 211189,China)

机构地区：[1]东南大学数学学院,南京211189

出　　处：《数值计算与计算机应用》2023年第2期126-137,共12页Journal on Numerical Methods and Computer Applications

基　　金：江苏省自然科学基金(BK20221450);江苏省高校‘青蓝工程’和国家自然科学基金(12071073)资助.

摘　　要：带有分数阶Laplacian算子的对流扩散方程常被用来刻画自然界与社会系统中的反常扩散现象.本文提出了一种新的格子Boltzmann模型,用于求解二维带分数阶Laplacian算子的对流扩散方程.首先,基于分数阶Laplacian算子的Fourier变换和Gauss型求积公式,得到控制方程的近似方程.然后,将速度空间、时间和空间进行离散,并构造合适的平衡态分布函数和离散作用力,建立有效的格子Boltzmann-BGK模型.通过Chapman-Enskog分析,可由建立的格子Boltzmann-BGK模型恢复出宏观方程,从而证明了模型的有效性.最后,将模型应用于求解带有解析解的数值算例和Allen-Cahn方程,数值结果进一步验证了模型的正确性和有效性.Advection-diffusion equation(ADE)with fractional Laplacian has been developed to characterize some anomalous diffusion phenomena in nature and social systems.In this study,a new lattice Boltzmann model with BGK operator(LBGK)for solving 2D ADE with fractional Laplacian is proposed.First,an approximated formula of the governing equation can be obtained based on the Fourier transform of fractional Laplacian and the Gauss-Type quadrature formular.Then,by means of velocity discretization,time and space discretization,constructing appropriate equilibrium distribution function and force distribution functionan,an efficient LBGK model is developed for the ADE with fractional Laplacian.Through the Chapman-Enskog analysis,the macroscopic equation can be recovered from the present LBGK model,which proves the effectiveness of the model.Finally,the model is applied to a numerical example with analysis solution and the Allen-Cahn equation.The results further verify the correctness and effectiveness of the present model.

关 键 词：格子BOLTZMANN模型 对流扩散方程 分数阶Laplacian算子 

分 类 号：O175[理学—数学]