• 论文 • 上一篇    

空间分数阶对流扩散方程的格子Boltzmann方法

魏雪丹1, 戴厚平1, 李梦军1, 郑洲顺2   

  1. 1. 吉首大学数学与统计学院, 吉首 416000;
    2. 中南大学数学与统计学院, 长沙 410083
  • 收稿日期:2021-03-24 发布日期:2022-09-09
  • 通讯作者: 戴厚平,Email:daihouping@163.com
  • 基金资助:
    国家自然科学基金(51974377),湖南省自然科学基金(2021JJ30548),湖南省教育厅重点项目(21A0329)资助.

魏雪丹, 戴厚平, 李梦军, 郑洲顺. 空间分数阶对流扩散方程的格子Boltzmann方法[J]. 数值计算与计算机应用, 2022, 43(3): 270-280.

Wei Xuadan, Dai Houping, Li Mengjun, Zheng Zhoushun. LATTICE BOLTZMANN METHOD FOR SPATIAL FRACTIONAL CONVECTION-DIFFUSION EQUATIONS[J]. Journal on Numerica Methods and Computer Applications, 2022, 43(3): 270-280.

LATTICE BOLTZMANN METHOD FOR SPATIAL FRACTIONAL CONVECTION-DIFFUSION EQUATIONS

Wei Xuadan1, Dai Houping1, Li Mengjun1, Zheng Zhoushun2   

  1. 1. College of Mathematics and Statistics, Jishou University, Jishou 416000, China;
    2. School of Mathematics and Statistics, Central South University, Changsha 410083, China
  • Received:2021-03-24 Published:2022-09-09
本文建立了格子Boltzmann方法的D1Q3模型,数值求解了一维空间分数阶对流扩散方程.对分数阶积分部分离散化处理并进行相关收敛性分析.通过选择合适的演化方程,运用Taylor展开和Chapman-Enskog多尺度技术展开,正确恢复出宏观方程.数值实验验证了该模型的有效性.
In this paper, one dimensional spatial fractional convection-diffusion equations are numerically solved by establishing D1Q3 model of Lattice Boltzmann method. The fractional integral term is discretized and the convergence in such scheme is analyzed. The macroscopic equation which is required to be solved is correctly recovered via Taylor expansion and Chapman-Enskog technique. Finally, the numerical experiments are presented to justify the validity of the model.

MR(2010)主题分类: 

()
[1] 易倩. 分数阶对流方程及相关问题的有限差分方法[D]. 上海:上海大学, 2019.
[2] Meerschaert M M, Tadjeran C. Finite difference approximations for fractional advection-dispersion flow equations[J]. Journal of computational and applied mathematics, 2004, 172(1):65-77.
[3] Liu Q, Liu F, Turner I. Approximation of the Lévy-Feller advection-dispersion process by random walk and finite difference method[J]. Journal of Computational Physics, 2007, 222(1):57-70.
[4] Zhang Y. A finite difference method for fractional partial differential equation[J]. Applied Mathematics and Computation, 2009, 215(2):524-529.
[5] 关文绘, 曹学年. Riesz回火分数阶平流-扩散方程的隐式中点方法[J]. 数值计算与计算机应用, 2020, 41(01):42-57.
[6] 邱泽山, 曹学年. 带漂移的单侧正规化回火分数阶扩散方程的三阶数值格式[J]. 数值计算与计算机应用, 2020, 41(03):201-215.
[7] Jiang Y, Ma J. High-order finite element methods for time-fractional partial differential equations[J]. Journal of Computational and Applied Mathematics, 2011, 235(11):3285-3290.
[8] Jiang Y J, Ma J T. Moving finite element methods for time fractional partial differential equations[J]. Science China Mathematics, 2013, 56(6):1287-1300.
[9] 陈传军, 赵鑫. 一类非线性对流扩散方程两重网格特征有限元方法及误差估计[J]. 数学物理学报, 2014, 34(3):643-654.
[10] 朱晓钢, 聂玉峰, 王俊刚, 袁占斌. 分数阶对流扩散方程的特征有限元方法[J]. 计算物理, 2017, 34(04):417-424.
[11] Carella A R, Dorao C A. Least-squares spectral method for the solution of a fractional advection-dispersion equation[J]. Journal of Computational Physics, 2013, 232(1):33-45.
[12] Mohamad A A. Lattice Boltzmann Method[M]. London:Springer, 2011.
[13] Lai H L, Ma C F. Lattice Boltzmann method for the generalized Kuramoto-Sivashinsky equation[J]. Physica A, 2009, 388(8):1405-1412.
[14] Lai H L, Ma C F. The lattice Boltzmann model for the second-order Benjamin-Ono equations[J]. Journal of Statistical Mechanics:Theory and Experiment, 2010, 2010(04):P04011.
[15] Lai H L, Ma C F. Numerical study of the nonlinear combined Sine-Cosine-Gordon equation with the lattice Boltzmann method[J]. Journal of Scientific Computing, 2012, 53(3):569-585.
[16] Hu W Q, Jia S L. General propagation lattice Boltzmann model for variable-coefficient non-isospectral KdV equation[J]. Applied Mathematics Letters, 2019, 91:61-67.
[17] 赖惠林, 马昌凤. 二维对流扩散方程的格子BGK模拟[J]. 福建师范大学学报(自然科学版), 2008(05):15-18.
[18] Dellacherie S. Construction and analysis of lattice Boltzmann methods applied to a 1D convection-diffusion equation[J]. Acta Applicandae Mathematicae, 2014, 131(1):69-140.
[19] 戴厚平, 郑洲顺, 段丹丹. 一类偏微分方程的格子Boltzmann模型[J]. 计算机工程与应用, 2016, 52(03):21-26.
[20] Du R, Liu Z X. A lattice Boltzmann model for the fractional advection-diffusion equation coupled with incompressible Navier-Stokes equation[J]. Applied Mathematics Letters, 2020, 101:106074.
[21] 李德梅, 赖惠林, 甘延标, 林传栋. 一类Equal Width波方程的介观数值模拟研究[J]. 福建师范大学学报(自然科学版), 2018, 34(05):6-11.
[22] 戴厚平, 郑洲顺, 段丹丹. 变系数反应扩散方程的格子Boltzmann模型[J]. 云南大学学报(自然科学版), 2016, 38(4):524-529.
[23] Dai H P, Zheng Z S, Tan W. Lattice Boltzmann model for the Riesz space fractional reaction-diffusion[J]. Thermal Science, 2018, 22(4):1831-1843.
[24] Du R, Sun D K, Shi B C, et al. Lattice Boltzmann model for time sub-diffusion equation in Caputo sense[J]. Applied Mathematics and Computation, 2019, 358:80-90.
[25] Liang H, Zhang C, Du R, et al. Lattice Boltzmann method for fractional Cahn-Hilliard equation[J]. Communications in Nonlinear Science and Numerical Simulation, 2020, 91:105443.
[1] 张旭, 蒋艳群, 陈勋, 胡迎港. 粘性Burgers方程的高阶隐式WCNS格式[J]. 数值计算与计算机应用, 2022, 43(2): 188-201.
[2] 袁琼, 杨志伟, 付芳芳. 时空分数阶扩散方程的扩展混合有限元方法[J]. 数值计算与计算机应用, 2021, 42(3): 276-288.
[3] 刘新龙, 杨晓忠. 时间分数阶四阶扩散方程的显-隐和隐-显差分格式[J]. 数值计算与计算机应用, 2020, 41(3): 216-231.
[4] 蔡力, 袁涛, 徐文静. 一种有效的格子Boltzmann方法格点判断法[J]. 数值计算与计算机应用, 2016, 37(4): 257-264.
[5] 王风娟, 王同科. 一维抛物型方程第三边值问题的紧有限体积格式[J]. 数值计算与计算机应用, 2013, 34(1): 59-74.
[6] 李明亮, 李会元, 孙家昶. 平行六边形区域非均匀节点快速傅立叶变换[J]. 数值计算与计算机应用, 2009, 30(1): 58-69.
阅读次数
全文


摘要