含间断系数扩散方程的格子Boltzmann方法

发布时间：2018-06-21 12:05

本文选题：格子Boltzmann方法 + 扩散方程　；参考：《湘潭大学》2017年硕士论文

【摘要】：含间断系数扩散方程大量出现在工程应用领域,因此数值求解该类方程具有重要的意义。近三十年来,格子Boltzmann方法已成为计算流体力学领域中的一种重要的数值计算方法,因此研究含间断系数扩散方程的格子Boltzmann方法是一项非常值得开展的工作。本文主要研究内容与创新点:首先,利用Chapman-Enskog多尺度展开技术,推导了一维含光滑系数扩散方程的格子Boltzmann模型;进一步,利用解连续和流通量连续条件,提出了一种针对一维含间断系数扩散方程的格子Boltzmann方法;该方法能够有效的求解含强间断的分片常系数和变系数的扩散方程,数值实验结果验证了该方法具有二阶精度。其次,类似一维模型的推导方法,推导了二维含光滑系数扩散方程的格子Boltzmann模型,然基于该模型不易设计二维含间断系数扩散方程的算法;为此,本文提出了一种简化的格子Boltzmann模型,基于该模型的算法具有二阶收敛阶;进一步,利用解连续和流通量连续条件,提出了一种针对二维含间断系数扩散方程的格子Boltzmann方法,数值实验验证了其具有二阶精度。
[Abstract]:Diffusion equations with discontinuous coefficients are widely used in engineering applications, so it is of great significance to solve them numerically. The lattice Boltzmann method has become an important numerical method in the field of computational fluid dynamics in the past 30 years, so it is very worthwhile to study the lattice Boltzmann method with discontinuous coefficient diffusion equation. The main contents and innovations of this paper are as follows: firstly, the lattice Boltzmann model with smooth coefficient diffusion equation is derived by using Chapman-Enskog multiscale expansion technique, and further, the solution continuity and flux continuity conditions are used. A lattice Boltzmann method for one-dimensional diffusion equations with discontinuous coefficients is proposed, which can effectively solve the diffusion equations with piecewise constant coefficients and variable coefficients with strong discontinuities. The numerical results show that the method has a second-order accuracy. Secondly, the lattice Boltzmann model with smooth coefficient diffusion equation is derived, but it is difficult to design the two-dimensional diffusion equation with discontinuous coefficients based on this model. In this paper, a simplified lattice Boltzmann model is proposed. The algorithm based on the model has the order of second order convergence. Furthermore, a lattice Boltzmann method for two-dimensional diffusion equations with discontinuous coefficients is proposed by using the conditions of solution continuity and flux continuity. The second order accuracy is verified by numerical experiments.
【学位授予单位】：湘潭大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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