求解非线性对流扩散方程的广义迁移格子Boltzmann模型
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摘要
非线性对流扩散方程是一类描述复杂运动和反应系统的基本方程,不仅能描述反应扩散过程,同时也可以描述热量和物质的传输等其他物理现象。如大气、河流污染中的污染物扩散分布、多孔介质中多层流体的流动、流体中热的传导、粒子扩散等众多现象。因此,研究对流扩散方程的数值解法研究具有很大的实用价值。Lattice Boltzmann是近几年新兴的一种建立在微观模型上高效的用于模拟流体流动的数值方法,与传统的数值方法相比,其计算简单,具有天然的并行性,并且能够方便的处理复杂边界问题,其已成功用于求解对流扩散方程。由于非线性的影响,这类问题很难求解,对于对流扩散方程而言,当扩散系数很小或对流占优问题,会出现较大的数值扩散或数值振荡等困难和精度低、不稳定等缺点。
本文针对以上问题,运用多尺度展开,通过修正一般的平衡态分布函数,将Guo等提出的用于求解流体方程的一类特殊的格子Boltzmann模型——广义迁移格子Boltmann模型应用到非线性对流扩散方程,与一般的LBGK模型相比,此模型是基于时间分裂的有限差分格式,在迁移步引入了两个自由参数,通过调整模型的自由参数,不仅可以求解对流占优的问题,同时可以提高数值稳定性,本文选用了两个有代表性的算例,实验结果表明:用广义迁移格子Boltzmann模型求的数值解与精确解吻合的很好,特别是对于对流占优和小扩散系数的问题,有着极大的优势,而且求解范围更广,稳定性也比一般的BGK模型要好。
The convection-diffusion equations are kinds of basic equations which descripe the complex movements and the response system.It can not only be used to describe the process of reactive reaction, but also other physical phenomenon of the heat quantity and substance's transport, Such as the atmosphere, the spread of the distribution of pollutants in the river, multi-layer porous media fluid flow, heat transfer fluids, particles diffusion and many other phenomena.Therefore, the research of convection-diffusion equation for the numerical solution has great practical value.Meanwhile,in the last few year, the lattice Boltzmann, is based on the microscopic models and play a great role in the simulation of fluid dynamics for the numerical method.Compared with the conventional computational fluid dynamics approach, the LBM is easy for programming,intrinsically parallel,and also easy to incorporate complicate boundary conditions which have been successfully used to solve convection-diffusion equation. Due to the hyperbolic nature of such problems, when the diffusion coefficient is small that will be a much larger numercal proliferation or numerical oscillation problems.There are problems such as low-precision, unstable or non-physical oscillations shortcomings about the convection-dominated diffusion .
To overcome the short-comings above,we use the chapman-enskog method in this paper to present a new model that based on general propagation LBK model which first proposed by Guo and is only used to flow fluid compute for a convection-diffusion equation with nonlinear convection and isotropic-diffusion terms through selecting equilibrium distribution fumction properly.Compared with the general BGK model, it is based on the time splitting of finite difference scheme,there are two free parameters in the propagation step. By adjusting the two free parameters of the model we can not only solve the problem of convection-dominated, but aslo improve the numerical stability. The experimental results show that numerical results coincide with the analytical , especially for convection-dominated problems. and improve the numerical stability of the model at the same time than the BGK model.
本文针对以上问题,运用多尺度展开,通过修正一般的平衡态分布函数,将Guo等提出的用于求解流体方程的一类特殊的格子Boltzmann模型——广义迁移格子Boltmann模型应用到非线性对流扩散方程,与一般的LBGK模型相比,此模型是基于时间分裂的有限差分格式,在迁移步引入了两个自由参数,通过调整模型的自由参数,不仅可以求解对流占优的问题,同时可以提高数值稳定性,本文选用了两个有代表性的算例,实验结果表明:用广义迁移格子Boltzmann模型求的数值解与精确解吻合的很好,特别是对于对流占优和小扩散系数的问题,有着极大的优势,而且求解范围更广,稳定性也比一般的BGK模型要好。
The convection-diffusion equations are kinds of basic equations which descripe the complex movements and the response system.It can not only be used to describe the process of reactive reaction, but also other physical phenomenon of the heat quantity and substance's transport, Such as the atmosphere, the spread of the distribution of pollutants in the river, multi-layer porous media fluid flow, heat transfer fluids, particles diffusion and many other phenomena.Therefore, the research of convection-diffusion equation for the numerical solution has great practical value.Meanwhile,in the last few year, the lattice Boltzmann, is based on the microscopic models and play a great role in the simulation of fluid dynamics for the numerical method.Compared with the conventional computational fluid dynamics approach, the LBM is easy for programming,intrinsically parallel,and also easy to incorporate complicate boundary conditions which have been successfully used to solve convection-diffusion equation. Due to the hyperbolic nature of such problems, when the diffusion coefficient is small that will be a much larger numercal proliferation or numerical oscillation problems.There are problems such as low-precision, unstable or non-physical oscillations shortcomings about the convection-dominated diffusion .
To overcome the short-comings above,we use the chapman-enskog method in this paper to present a new model that based on general propagation LBK model which first proposed by Guo and is only used to flow fluid compute for a convection-diffusion equation with nonlinear convection and isotropic-diffusion terms through selecting equilibrium distribution fumction properly.Compared with the general BGK model, it is based on the time splitting of finite difference scheme,there are two free parameters in the propagation step. By adjusting the two free parameters of the model we can not only solve the problem of convection-dominated, but aslo improve the numerical stability. The experimental results show that numerical results coincide with the analytical , especially for convection-dominated problems. and improve the numerical stability of the model at the same time than the BGK model.
引文
[1]郭照立,郑楚光,李青等.流体动力学的格子Boltzmann方法.版本(第一版).武汉:湖北科学技术出版社, 2002.
[2]郭照立,郑楚光等.格子Boltzmann方法的原理及其应用.版本(第一版).北京:科学出版社,2009.
[3] Wolfram S. Cellular automaton fluids 1.Basic Theory. J. Stat. Phys., 1986,45: 471-529.
[4] Frisch U., D’Humieres D., Hasslacher B. et al. Lattide gas hydrodynamixs in two and three dimensions. Complexs Systems, 1987, 1:649-707.
[5] McNamara G. R., Zanetti G. Use of the Boltzmann equation to simulate lattice automata. Phy. Rev. Lett., 1988, 61:2332-2338.
[6] Higuere F., Jimenez J. Boltzmann approach to lattice gas simulation. Europhys. Lett., 1989, 9:663-668.
[7] Higuere F., Succi S., Benzi R. Lattice gas dynamics with enhanced collision. Europhys. Lett ., 1989, 9:345-349.
[8] Chen H., Chen S., Matthaeus W. H. Recovery of the Navier-Stokes equation using a lattice gas Boltzmann method. Phys. Rev. A, 1992, 45:5339-5342.
[9] Koilman J. A simple lattice Boltzmann scheme for Navier-Stokes fluid flow. Europhys. Lett .,1991,15:603-607.
[10] Qian Y., D’Humieres D., Lallemand P. Lattide BGK models for Navier-Stroke equation. Europhys. Lett., 1992,17:479-487.
[11] D’Humieres D. Genetalized lattice Boltzmann equation in Rarefide gas dynamics: theory and simulation. Prog. Astronaut., 1992,159:450-458.
[12]吴大猷.理论物理(第五册)-热力学,气体运动论及统计力学.版本(第一版).北京:科学出版社,1983.
[13] Abe T. Deribation of the lattice Boltzmann method by means of the discrete oridinate method for the Boltzmann equation. J. Comput. Phys., 1997, 131:241-246.
[14] He X ., Luo L. S. On the theory of the lattice Boltzmann equation : From the Boltzmann equation to the lattice Boltzmann equation. Phys. Rev. E, 1997, 56: 6811-6817.
[15] Shan X ., He X . Discretization of the velocity space in the solution if the Boltzmannequation. Phys. Rev. L , 1998 , 80:65-68.
[16] He X., Luo L. S. Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation. Phys. Rev. E , 1997, 56:6811-6817.
[17] Ziegler D. Boundary conditions for lattice Boltzmann simulations. J. Stat. Phys., 1993, 71:1171-1177.
[18] Zou Q., He X. On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. Phys. Flu., 1997,9:1591-1597.
[19] Inamuro T., Yoshino M., Ogino F. A non-slip boundary condition for lattice Boltzmann simulations. Phys. Flu., 1995, 7:2928-2930.
[20] Skordos P. A. Initial and boundary conditions for the lattice Boltzmann method. Phys. Rev. E , 1993, 48:4823-4842.
[21] Chen S., Martimez D., Mei R. On boundary conditions for the lattice Boltzmann method. Phys. Flu., 1996, 8:2527-2536.
[22] Guo Z. L., Zheng C. G ., Shi B. C. Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method. Chin. Phys., 2002, 11:366-374.
[23] Rothman D. H. Cellular-automaton fluids: a model for flow in porous media. Geo. Phys., 1988,53:509-518
[24] Succi S., Foti E., Higuera F. Threedimensional flows in complex geometries with the lattice Boltzmann method. Europhys. Lett., 1989, 10:433-438.
[25] Adrover A., Giona M. A predictive model for permeability of correlated porous media. Chemical Engineering Jounal ., 1996, 64:7-19.
[26] Singh M., Mohanty K. Permeability of spatially correlated porous media. Chemical Engineering Science , 2000,55:5393-5403.
[27] Kim J., Lee J., Lee K. C. Nonlinear correction to Darcy’s law for a flow through periodic arrays of elliptic cylinders. Phys. A., 2001, 293:13-20.
[28] Dardis O., McCloskey J. Lattice Boltzmann scheme with real numbered solid density for the simulation of flow in porous media. Phys. Rev. E, 1998, 57:4834-4837.
[29] Spaid M. A. A., Phelan F. R. J. Lattice Bolotzmann methods for modeling microscale flow in fibrous porous media. Phys. Flu., 1997, 9:2468-2474.
[30] Spaid M. A. A., Phelan F. R. J. Modeling void formation dynamics in fibrous media with the lattice Boltzmann method. Composites Part A., 1998, 29A:749-755.
[31] Rothman D. H., Keller J. M. Immiscible cellular-automaton fluids. J. Stat. Phys.,lattice Boltzmann method on a nonuniform grid. J. Comput. Phys., 2002, 181: 675-704.
[48] Chen H. D., Kandasamy S., Orszag S. et al. Extended Boltzmann kinetix equation for turbulent flow. Science, 2003, 301:633-636.
[49] Deng B., Shi B. C., Wang N. C. A New Lattice BGK Model for the Convection Difusion with a Source Term. Chin. Phys. Lett., 2005, 22:267-270.
[50] Massaioli F., Benzi R., Succi S. Exponential tails in two-dimensional Reyleigh-Bernard convection. Europhys. Lett., 1993, 21:305-310.
[51] Bracy Elton. Convergence of Convective–Diffusive Lattice Boltzmann Methods. Slam. J. Num. Ana., 1995, 32:132-1354.
[52] Shi B. C., Deng B., Du R. et al. A new scheme for source term in LBGK model for convection-diffusion equation. Comput. Math. Appl., 2008, 55:1568-1575.
[53] Dawson S. P., Chen S. Y., Doolen G. D. et al. Lattice Boltamann computations for reaction-diffution equation. Phys., 1993, 98:1514-1523.
[54] Blaak R., Sloot P. M. Lattice dependence of reaction-diffusion in lattice Boltzmann modeling. J. Comput. Phys. Commun., 2000, 129:256-266.
[55] Yu X. M., Shi B. C. A lattice Boltzmann model for reaction dynamical systems with time delay. Appl. Math. Comput., 2006, 118:958-965.
[56] Van der Sman R. G. M. Galilean invariant lattice Boltzmann scheme for natural convection on square and rectangular lattice. Phys. Rev. E , 2006, 74:026705.
[57] Michael R., Swift, E., Orlandini W. R. et al. Lattice Boltzmann simulations of liquid-gas and binary fluid systems. Phys. Rev. E , 1996,54:5041.
[58] Guo Z. L., Shi B. C., Wang N. C. Fully Lagrangian and lattice Boltzmann methods for the advection–diffusion equation. J. Sci. Comput., 1999, 14:291.
[59] He X. Y., Li N., Goldstein B. Lattice Boltzmann simulation of diffusion–convection systems with surface chemical reaction. J. Mol. Simulat., 2000, 25:145-156.
[60] Van der Sman R. G. M., Ernst M. H. Advection–diffusion lattice Boltzmann scheme for irregular lattices. J. Comput. Phys., 2000, 160 : 766–782.
[61] Yu X. M., Shi B. C. A Lattice Bhatnagar-Gross-Krook model for a class of the generalized Burgers equations. Chin. Phys. Soc., 2006, 15: 1441–1449.
[62] Yan G. W. A lattice Boltzmann equation for waves. J. Comput. Phys., 2000, 161:61-69.
[63] Zhang X. X., Bengough Anthony G., John W. et al. A lattice BGK model for advection and anisotropic dispersion equation. Adv. Water Resource, 2002,25:1-8.lattice Boltzmann method on a nonuniform grid. J. Comput. Phys., 2002, 181: 675-704.
[48] Chen H. D., Kandasamy S., Orszag S. et al. Extended Boltzmann kinetix equation for turbulent flow. Science, 2003, 301:633-636.
[49] Deng B., Shi B. C., Wang N. C. A New Lattice BGK Model for the Convection Difusion with a Source Term. Chin. Phys. Lett., 2005, 22:267-270.
[50] Massaioli F., Benzi R., Succi S. Exponential tails in two-dimensional Reyleigh-Bernard convection. Europhys. Lett., 1993, 21:305-310.
[51] Bracy Elton. Convergence of Convective–Diffusive Lattice Boltzmann Methods. Slam. J. Num. Ana., 1995, 32:132-1354.
[52] Shi B. C., Deng B., Du R. et al. A new scheme for source term in LBGK model for convection-diffusion equation. Comput. Math. Appl., 2008, 55:1568-1575.
[53] Dawson S. P., Chen S. Y., Doolen G. D. et al. Lattice Boltamann computations for reaction-diffution equation. Phys., 1993, 98:1514-1523.
[54] Blaak R., Sloot P. M. Lattice dependence of reaction-diffusion in lattice Boltzmann modeling. J. Comput. Phys. Commun., 2000, 129:256-266.
[55] Yu X. M., Shi B. C. A lattice Boltzmann model for reaction dynamical systems with time delay. Appl. Math. Comput., 2006, 118:958-965.
[56] Van der Sman R. G. M. Galilean invariant lattice Boltzmann scheme for natural convection on square and rectangular lattice. Phys. Rev. E , 2006, 74:026705.
[57] Michael R., Swift, E., Orlandini W. R. et al. Lattice Boltzmann simulations of liquid-gas and binary fluid systems. Phys. Rev. E , 1996,54:5041.
[58] Guo Z. L., Shi B. C., Wang N. C. Fully Lagrangian and lattice Boltzmann methods for the advection–diffusion equation. J. Sci. Comput., 1999, 14:291.
[59] He X. Y., Li N., Goldstein B. Lattice Boltzmann simulation of diffusion–convection systems with surface chemical reaction. J. Mol. Simulat., 2000, 25:145-156.
[60] Van der Sman R. G. M., Ernst M. H. Advection–diffusion lattice Boltzmann scheme for irregular lattices. J. Comput. Phys., 2000, 160 : 766–782.
[61] Yu X. M., Shi B. C. A Lattice Bhatnagar-Gross-Krook model for a class of the generalized Burgers equations. Chin. Phys. Soc., 2006, 15: 1441–1449.
[62] Yan G. W. A lattice Boltzmann equation for waves. J. Comput. Phys., 2000, 161:61-69.
[63] Zhang X. X., Bengough Anthony G., John W. et al. A lattice BGK model for advection and anisotropic dispersion equation. Adv. Water Resource, 2002,25:1-8.
[64] Rasin I., Succi S., Miller W. A multi-relaxation lattice kinetic method for passive scalar diffusion. J. Comput. Phys., 2005,206:453-462.
[65] Ginzburg I. Equilibrium-type and Link-type Lattice Boltzmann models for generic advection and anisotropic-dispersion equation. Adv. Water Reso., 2005, 28:1 171-1195.
[66] Shi B. C., Guo Z. L. Lattice Boltzmann model for nonlinear convection-diffusion equations. Phys. Rev. E , 2009, 79:016701.
[67] Qian Y. Fractional propagation and the elimination of staggered invariants in lattice-BGK model. Int. Mod. Phys. C., 1997, 8:753-758.
[68] Zhang R., Chen H., Qian Y. H. et al. Effective volumertric lattice Boltzmann scheme. Phys. Rev. E , 2001, 63:056705-056711.
[69] Guo Z. L., Zheng C. G., Zhao T. S. A Lattice BGK Scheme with General Propagation. Scient. Compu., 2002, 16: 16569-585.
[70] Guo Z. L., Zhao T. S. Finite-difference-based lattice Boltzmann model for dense binary mixtures. Phys. Rev. E , 2005, 71:026701-12.
[2]郭照立,郑楚光等.格子Boltzmann方法的原理及其应用.版本(第一版).北京:科学出版社,2009.
[3] Wolfram S. Cellular automaton fluids 1.Basic Theory. J. Stat. Phys., 1986,45: 471-529.
[4] Frisch U., D’Humieres D., Hasslacher B. et al. Lattide gas hydrodynamixs in two and three dimensions. Complexs Systems, 1987, 1:649-707.
[5] McNamara G. R., Zanetti G. Use of the Boltzmann equation to simulate lattice automata. Phy. Rev. Lett., 1988, 61:2332-2338.
[6] Higuere F., Jimenez J. Boltzmann approach to lattice gas simulation. Europhys. Lett., 1989, 9:663-668.
[7] Higuere F., Succi S., Benzi R. Lattice gas dynamics with enhanced collision. Europhys. Lett ., 1989, 9:345-349.
[8] Chen H., Chen S., Matthaeus W. H. Recovery of the Navier-Stokes equation using a lattice gas Boltzmann method. Phys. Rev. A, 1992, 45:5339-5342.
[9] Koilman J. A simple lattice Boltzmann scheme for Navier-Stokes fluid flow. Europhys. Lett .,1991,15:603-607.
[10] Qian Y., D’Humieres D., Lallemand P. Lattide BGK models for Navier-Stroke equation. Europhys. Lett., 1992,17:479-487.
[11] D’Humieres D. Genetalized lattice Boltzmann equation in Rarefide gas dynamics: theory and simulation. Prog. Astronaut., 1992,159:450-458.
[12]吴大猷.理论物理(第五册)-热力学,气体运动论及统计力学.版本(第一版).北京:科学出版社,1983.
[13] Abe T. Deribation of the lattice Boltzmann method by means of the discrete oridinate method for the Boltzmann equation. J. Comput. Phys., 1997, 131:241-246.
[14] He X ., Luo L. S. On the theory of the lattice Boltzmann equation : From the Boltzmann equation to the lattice Boltzmann equation. Phys. Rev. E, 1997, 56: 6811-6817.
[15] Shan X ., He X . Discretization of the velocity space in the solution if the Boltzmannequation. Phys. Rev. L , 1998 , 80:65-68.
[16] He X., Luo L. S. Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation. Phys. Rev. E , 1997, 56:6811-6817.
[17] Ziegler D. Boundary conditions for lattice Boltzmann simulations. J. Stat. Phys., 1993, 71:1171-1177.
[18] Zou Q., He X. On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. Phys. Flu., 1997,9:1591-1597.
[19] Inamuro T., Yoshino M., Ogino F. A non-slip boundary condition for lattice Boltzmann simulations. Phys. Flu., 1995, 7:2928-2930.
[20] Skordos P. A. Initial and boundary conditions for the lattice Boltzmann method. Phys. Rev. E , 1993, 48:4823-4842.
[21] Chen S., Martimez D., Mei R. On boundary conditions for the lattice Boltzmann method. Phys. Flu., 1996, 8:2527-2536.
[22] Guo Z. L., Zheng C. G ., Shi B. C. Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method. Chin. Phys., 2002, 11:366-374.
[23] Rothman D. H. Cellular-automaton fluids: a model for flow in porous media. Geo. Phys., 1988,53:509-518
[24] Succi S., Foti E., Higuera F. Threedimensional flows in complex geometries with the lattice Boltzmann method. Europhys. Lett., 1989, 10:433-438.
[25] Adrover A., Giona M. A predictive model for permeability of correlated porous media. Chemical Engineering Jounal ., 1996, 64:7-19.
[26] Singh M., Mohanty K. Permeability of spatially correlated porous media. Chemical Engineering Science , 2000,55:5393-5403.
[27] Kim J., Lee J., Lee K. C. Nonlinear correction to Darcy’s law for a flow through periodic arrays of elliptic cylinders. Phys. A., 2001, 293:13-20.
[28] Dardis O., McCloskey J. Lattice Boltzmann scheme with real numbered solid density for the simulation of flow in porous media. Phys. Rev. E, 1998, 57:4834-4837.
[29] Spaid M. A. A., Phelan F. R. J. Lattice Bolotzmann methods for modeling microscale flow in fibrous porous media. Phys. Flu., 1997, 9:2468-2474.
[30] Spaid M. A. A., Phelan F. R. J. Modeling void formation dynamics in fibrous media with the lattice Boltzmann method. Composites Part A., 1998, 29A:749-755.
[31] Rothman D. H., Keller J. M. Immiscible cellular-automaton fluids. J. Stat. Phys.,lattice Boltzmann method on a nonuniform grid. J. Comput. Phys., 2002, 181: 675-704.
[48] Chen H. D., Kandasamy S., Orszag S. et al. Extended Boltzmann kinetix equation for turbulent flow. Science, 2003, 301:633-636.
[49] Deng B., Shi B. C., Wang N. C. A New Lattice BGK Model for the Convection Difusion with a Source Term. Chin. Phys. Lett., 2005, 22:267-270.
[50] Massaioli F., Benzi R., Succi S. Exponential tails in two-dimensional Reyleigh-Bernard convection. Europhys. Lett., 1993, 21:305-310.
[51] Bracy Elton. Convergence of Convective–Diffusive Lattice Boltzmann Methods. Slam. J. Num. Ana., 1995, 32:132-1354.
[52] Shi B. C., Deng B., Du R. et al. A new scheme for source term in LBGK model for convection-diffusion equation. Comput. Math. Appl., 2008, 55:1568-1575.
[53] Dawson S. P., Chen S. Y., Doolen G. D. et al. Lattice Boltamann computations for reaction-diffution equation. Phys., 1993, 98:1514-1523.
[54] Blaak R., Sloot P. M. Lattice dependence of reaction-diffusion in lattice Boltzmann modeling. J. Comput. Phys. Commun., 2000, 129:256-266.
[55] Yu X. M., Shi B. C. A lattice Boltzmann model for reaction dynamical systems with time delay. Appl. Math. Comput., 2006, 118:958-965.
[56] Van der Sman R. G. M. Galilean invariant lattice Boltzmann scheme for natural convection on square and rectangular lattice. Phys. Rev. E , 2006, 74:026705.
[57] Michael R., Swift, E., Orlandini W. R. et al. Lattice Boltzmann simulations of liquid-gas and binary fluid systems. Phys. Rev. E , 1996,54:5041.
[58] Guo Z. L., Shi B. C., Wang N. C. Fully Lagrangian and lattice Boltzmann methods for the advection–diffusion equation. J. Sci. Comput., 1999, 14:291.
[59] He X. Y., Li N., Goldstein B. Lattice Boltzmann simulation of diffusion–convection systems with surface chemical reaction. J. Mol. Simulat., 2000, 25:145-156.
[60] Van der Sman R. G. M., Ernst M. H. Advection–diffusion lattice Boltzmann scheme for irregular lattices. J. Comput. Phys., 2000, 160 : 766–782.
[61] Yu X. M., Shi B. C. A Lattice Bhatnagar-Gross-Krook model for a class of the generalized Burgers equations. Chin. Phys. Soc., 2006, 15: 1441–1449.
[62] Yan G. W. A lattice Boltzmann equation for waves. J. Comput. Phys., 2000, 161:61-69.
[63] Zhang X. X., Bengough Anthony G., John W. et al. A lattice BGK model for advection and anisotropic dispersion equation. Adv. Water Resource, 2002,25:1-8.lattice Boltzmann method on a nonuniform grid. J. Comput. Phys., 2002, 181: 675-704.
[48] Chen H. D., Kandasamy S., Orszag S. et al. Extended Boltzmann kinetix equation for turbulent flow. Science, 2003, 301:633-636.
[49] Deng B., Shi B. C., Wang N. C. A New Lattice BGK Model for the Convection Difusion with a Source Term. Chin. Phys. Lett., 2005, 22:267-270.
[50] Massaioli F., Benzi R., Succi S. Exponential tails in two-dimensional Reyleigh-Bernard convection. Europhys. Lett., 1993, 21:305-310.
[51] Bracy Elton. Convergence of Convective–Diffusive Lattice Boltzmann Methods. Slam. J. Num. Ana., 1995, 32:132-1354.
[52] Shi B. C., Deng B., Du R. et al. A new scheme for source term in LBGK model for convection-diffusion equation. Comput. Math. Appl., 2008, 55:1568-1575.
[53] Dawson S. P., Chen S. Y., Doolen G. D. et al. Lattice Boltamann computations for reaction-diffution equation. Phys., 1993, 98:1514-1523.
[54] Blaak R., Sloot P. M. Lattice dependence of reaction-diffusion in lattice Boltzmann modeling. J. Comput. Phys. Commun., 2000, 129:256-266.
[55] Yu X. M., Shi B. C. A lattice Boltzmann model for reaction dynamical systems with time delay. Appl. Math. Comput., 2006, 118:958-965.
[56] Van der Sman R. G. M. Galilean invariant lattice Boltzmann scheme for natural convection on square and rectangular lattice. Phys. Rev. E , 2006, 74:026705.
[57] Michael R., Swift, E., Orlandini W. R. et al. Lattice Boltzmann simulations of liquid-gas and binary fluid systems. Phys. Rev. E , 1996,54:5041.
[58] Guo Z. L., Shi B. C., Wang N. C. Fully Lagrangian and lattice Boltzmann methods for the advection–diffusion equation. J. Sci. Comput., 1999, 14:291.
[59] He X. Y., Li N., Goldstein B. Lattice Boltzmann simulation of diffusion–convection systems with surface chemical reaction. J. Mol. Simulat., 2000, 25:145-156.
[60] Van der Sman R. G. M., Ernst M. H. Advection–diffusion lattice Boltzmann scheme for irregular lattices. J. Comput. Phys., 2000, 160 : 766–782.
[61] Yu X. M., Shi B. C. A Lattice Bhatnagar-Gross-Krook model for a class of the generalized Burgers equations. Chin. Phys. Soc., 2006, 15: 1441–1449.
[62] Yan G. W. A lattice Boltzmann equation for waves. J. Comput. Phys., 2000, 161:61-69.
[63] Zhang X. X., Bengough Anthony G., John W. et al. A lattice BGK model for advection and anisotropic dispersion equation. Adv. Water Resource, 2002,25:1-8.
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