耦合对流扩散方程的格子Boltzmann模型及应用研究

发布时间：2018-02-20 18:39

本文关键词： 格子Boltzmann方法耦合对流扩散方程神经传导方程螺旋波波色-爱因斯坦凝聚　出处：《华中科技大学》2016年博士论文　论文类型：学位论文

【摘要】：对流扩散方程是一类描述对流和扩散现象的数学模型,在污染治理、石油输运、化学反应、疾病治疗、微流体控制等自然和工程领域有着广泛的应用。然而实际问题往往涉及多个变系数对流扩散方程的相互耦合,现有解析方法难以有效地求解此类复杂问题。随着计算机技术的快速发展,数值方法已成为研究这类复杂问题的有效手段,近年来发展起来的介观格子Botzmann方法具有本质并行、算法简单、格式通用等特点也被用于研究对流扩散方程。在单个或耦合对流扩散方程中,变系数的影响往往不可忽略。本论文首先针对单个定常与非定常变系数对流扩散方程,分别设计了相应的格子Boltzmann模型；其次,通过对源项的处理,我们分别对守恒和非守恒形式的耦合对流扩散方程构造了格子Boltzmann模型；再次,对含有时间和空间三阶混合偏导数的广义粘性波方程,我们通过变量代换将其化成耦合扩散方程,进而用格子Boltzmann模型进行求解；最后,利用发展的格子Boltzmann模型研究了由耦合对流扩散方程描述的螺旋波动力学及复方程控制的波色-爱因斯坦凝聚。本论文主要对格子Boltzmann模型构建及应用展开研究,其中,在模型构建方面,主要开展了以下四项工作：(1)基于定常变系数对流扩散方程的特点,本文提出了一个简单、通用、高效的格子Boltzmann模型。该模型可以用来处理一般定常变系数的对流扩散方程,且具有计算格式简单,收敛速度快等特点。(2)通过引入与变系数对流项相关的辅助函数,构造了一个求解非定常变系数对流扩散方程的格子Boltzmann模型。该模型能够包含常系数对流扩散方程的格子Boltzmann模型,是已有求解对流扩散方程模型的拓展。(3)针对守恒形式的n维耦合对流扩散方程和一维耦合粘性Burgers方程,利用源项的特殊处理和一维方程的特殊性,分别提出了相应的格子Boltzmann模型；对于非守恒形式的二维耦合Burgers方程,利用格子Boltzmann方法本身的特点,将对流项中变系数部分当作源项处理,构造了两个有效的格子Boltzmann模型。最后,Chapman-Enskog分析及数值实验结果验证了模型的可靠性和正确性。(4)在上述研究基础上,本论文还研究了一类三阶非线性粘性波初边值问题,通过采用变量代换将其转化为耦合扩散方程后进行求解。数值模拟结果验证了格子Boltzmann方法求解这类方程的可靠性。基于上述模型,本文也开展了以下两个方面的应用研究：(1)螺旋波问题的相关研究在心室失颤引发的心脏猝死中具有重要意义。对于耦合反应对流扩散方程控制的螺旋波问题,我们采用本文提出的求解耦合对流扩散方程的格子Boltzmann模型,模拟了螺旋波形成、在电场力作用下消亡以及在应变力作用下变形和破裂过程。(2)采用本文提出的模型研究了山Gross-Pitaevskii方程描述的波色-爱因斯坦凝聚中的聚焦及散焦问题。数值模拟结果表明,格子Boltzmann方法可以有效的研究这类问题,进一步扩展了格子Boltzmann方法的应用范围。总之,对于单个及耦合的对流扩散方程,本文从理论模型和实际应用两方面进行了研究。根据所求对流扩散方程的特点,通过源项处理提出了多个求解单个及耦合对流扩散方程的格子Boltzmann模型,并借助Chapman-Enskog分析证明了模型的正确性,同时数值模拟也表明了这些模型在具体应用上的有效性。在此基础上,我们还研究了由耦合对流扩散方程描述的螺旋波及波色-爱因斯坦凝聚,进一步拓展了格子Boltzmann方法的应用领域。
[Abstract]:The convection diffusion equation is a mathematical model describing the convection and diffusion phenomena, in pollution control, treatment of oil transportation, chemical reaction, disease, micro fluid control and other natural and engineering applications. However, problems often involve multiple convection diffusion equation with variable coefficients of mutual coupling, the existing analytical method is not effective to solve such complicated problems. With the rapid development of computer technology, numerical method has become an effective means to study this kind of complex problem, mesoscopic lattice Botzmann method developed in recent years with the nature of parallel, simple algorithm and general characteristics of format has also been used to study the convection diffusion equation. In single or coupled convection diffusion equation, influence the variable coefficient cannot always be ignored. This paper firstly studies the single steady and unsteady convection diffusion equation with variable coefficients, we design corresponding lattice Boltzman N model; secondly, according to the source, we were on the conservation and non conservation form coupled convection diffusion equation of lattice Boltzmann model; thirdly, with time and space three mixed partial derivative of the generalized viscous wave equation, we will be the formation of coupled diffusion equations by variable substitution, and lattice Boltzmann to solve the model; finally, the dynamics of spiral waves is studied by using convection diffusion equation describing the coupled lattice Boltzmann model development and complex equations of Bose Einstein condensates. This thesis focuses on the research and construction of lattice Boltzmann model and application expansion which, in the construction of model, mainly carried out the following work: four (1) characteristics of unsteady convection diffusion equation with variable coefficients based on, this paper proposes a simple, universal, efficient lattice Boltzmann model. The model can be used to deal with In general, the convection diffusion equation with variable coefficients often, and has a simple calculation format, fast convergence speed and so on. (2) through the introduction and variable coefficient convection related auxiliary function, we construct a lattice Boltzmann model for solving non steady convection diffusion equation with variable coefficients. The lattice Boltzmann model which can contain constant coefficient the convection diffusion equation, is to expand the existing model for solving the convection diffusion equation. (3) according to the conservative form of n-dimensional coupled convection diffusion equation and one-dimensional coupled viscous Burgers equation, the use of special treatment and special equation source terms, the appropriate lattice Boltzmann models are proposed for two-dimensional coupled Burgers equation; non conservation form the characteristics of the lattice Boltzmann method itself, the convection in varying coefficient partially as a processing source, construct two effective lattice Boltzmann model. Finally, Chapma N-Enskog analysis and numerical experiment results validate the model's reliability and validity. (4) on the basis of the research above, this paper also studies a class of three order nonlinear viscous wave initial boundary value problem is solved by using variable substitution into coupled diffusion equations. Simulation results verify the reliability of solving lattice Boltzmann the method of this kind of equation value. Based on the above model, the paper also researched on the following two aspects: (1) it is significance to study on spiral wave in the ventricular fibrillation problem caused by the loss of sudden cardiac death. The coupling reaction convection diffusion equation of spiral wave control problem, we use the lattice Boltzmann model to solve the coupled convection diffusion equation is presented in this paper, the simulation of the formation of spiral waves, extinction of the electric force and the strain force deformation and rupture process. (2) proposed by The model describes the mountain Gross-Pitaevskii equations of Bose Einstein condensates in the focusing and defocusing problem. The numerical simulation results show that the lattice Boltzmann method can effectively study these problems, further expands the scope of application of lattice Boltzmann method. In a word, for single and coupled convection diffusion equation, have been studied in this paper two aspects of theoretical model and practical application. According to the characteristics of the convection diffusion equation, the source processing presents a number of methods for single and coupled convection diffusion equations with lattice Boltzmann model, and using Chapman-Enskog analysis proved the correctness of the model, the numerical simulation also shows the effectiveness of the model in practical application. On this basis, we also studied the coupled convection diffusion equation describing spiral spread Bose Einstein condensation, to further expand the The application field of the lattice Boltzmann method.

【学位授予单位】：华中科技大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82

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