【作者】 娄钦；

【导师】 施保昌；

【作者基本信息】 华中科技大学 ， 计算数学， 2009， 硕士

【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。更多还原

【Abstract】 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.更多还原