若干非线性偏微分方程的格子BGK模拟
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摘要
格子Boltzmann方法(LBM)是一种新兴的模拟流体和复杂物理系统的数值计算方法。不像基于宏观连续方程的传统数值方法,LBM是起源于微观模型和细观运动论的介观方法,它具有许多分子动力学的优点,如物理图像清晰、容易处理复杂边界、编程容易实现等。近年来,LBM在模拟线性和非线性偏微分方程方面取得了重要进展,但是理论部分仍有许多问题有待完善,例如如何构造出精度较高的模型和如何模拟更复杂的非线性偏微分方程。
本文首先在绪论部分简要介绍了LBM的发展历史及其应用,然后在接下来的四章中分别针对几类非线性偏微分方程,利用多尺度技术,建立了相对应的几种格子BGK模型。第一章中针对二维对流扩散方程建立D2Q4模型:第二章中针对Sine-Gordon方程建立隐式格子Boltzmann模型;第三章中针对广义KdV方程,KdV-Burgers方程,组合KdV-MKdV方程和广义Burgers-Huxley方程建立统一的具有五阶精度的格子BGK模型;第四章中针对广义Kuramoto-Sivashinsky方程建立D1Q5格子BGK模型,由于宏观方程含有四阶导数,标准的LBM无法恢复出来,因此,本模型的提出填补了这一方面的空白,拓展了LBM在模拟复杂非线性偏微分方程方面的领域。
数值结果表明所建模型均十分有效,为以后更复杂和更高维的复杂非线性偏微分方程的数值模拟积累了经验。
Lattice Boltzmann method (LBM) is a new technique for simulating fluids and modeling complex physics in fluids. Unlike conventional numerical methods based on a macroscopic continuum equation, the LBM starts from microscopic models and mesoscopic kinetic equations. It provides many of the advantages of molecular dynamics, including clear physical pictures, ease in incorporating complex boundary conditions and simplicity of programming. Recently, the LBM have been developed to simulate linear and nonlinear partial differential equations (NPDEs). However, there is a troublesome problem for solving NPDEs in many existing lattice Boltzmann models, i.e., how to derive higher accuracy and more complex nonlinear terms in NPDEs.
At the beginning of this thesis, some basis summaries of the LBM are introduced. In Chapter, 1, we construct a D2Q4 model for simulating two-dimensional convection-diffusion equation; In Chapter 2, we establish an implicit scheme of lattice BGK model for simulating Sine-Gordon equation; In Chapter 3, we propose a unified lattice BGK model, which has five order accuracy, for simulating generalized KdV equation, KdV-Burgers equation, combined KdV-MKdV equation and generalized Burgers-Huxley equation; In Chapter 4, we propose a D1Q5 lattice Boltzmann model for the generalized Kuramoto-Sivashinsky (GKS) equation which has four-order derivative. Due to the four-order derivative exists in the GKS equation, then the standard LBM cannot recover the governing evolution equations. Our presented method fill this gap and extend the application of the LBM to the higher derivative term's problem.
From the simulations, we find that the simulating results are in excellent agreement with the analytical solutions. This indicates that the present methods are efficient and flexible approachs for practical application. The present models can be extended to more complex systems.
本文首先在绪论部分简要介绍了LBM的发展历史及其应用,然后在接下来的四章中分别针对几类非线性偏微分方程,利用多尺度技术,建立了相对应的几种格子BGK模型。第一章中针对二维对流扩散方程建立D2Q4模型:第二章中针对Sine-Gordon方程建立隐式格子Boltzmann模型;第三章中针对广义KdV方程,KdV-Burgers方程,组合KdV-MKdV方程和广义Burgers-Huxley方程建立统一的具有五阶精度的格子BGK模型;第四章中针对广义Kuramoto-Sivashinsky方程建立D1Q5格子BGK模型,由于宏观方程含有四阶导数,标准的LBM无法恢复出来,因此,本模型的提出填补了这一方面的空白,拓展了LBM在模拟复杂非线性偏微分方程方面的领域。
数值结果表明所建模型均十分有效,为以后更复杂和更高维的复杂非线性偏微分方程的数值模拟积累了经验。
Lattice Boltzmann method (LBM) is a new technique for simulating fluids and modeling complex physics in fluids. Unlike conventional numerical methods based on a macroscopic continuum equation, the LBM starts from microscopic models and mesoscopic kinetic equations. It provides many of the advantages of molecular dynamics, including clear physical pictures, ease in incorporating complex boundary conditions and simplicity of programming. Recently, the LBM have been developed to simulate linear and nonlinear partial differential equations (NPDEs). However, there is a troublesome problem for solving NPDEs in many existing lattice Boltzmann models, i.e., how to derive higher accuracy and more complex nonlinear terms in NPDEs.
At the beginning of this thesis, some basis summaries of the LBM are introduced. In Chapter, 1, we construct a D2Q4 model for simulating two-dimensional convection-diffusion equation; In Chapter 2, we establish an implicit scheme of lattice BGK model for simulating Sine-Gordon equation; In Chapter 3, we propose a unified lattice BGK model, which has five order accuracy, for simulating generalized KdV equation, KdV-Burgers equation, combined KdV-MKdV equation and generalized Burgers-Huxley equation; In Chapter 4, we propose a D1Q5 lattice Boltzmann model for the generalized Kuramoto-Sivashinsky (GKS) equation which has four-order derivative. Due to the four-order derivative exists in the GKS equation, then the standard LBM cannot recover the governing evolution equations. Our presented method fill this gap and extend the application of the LBM to the higher derivative term's problem.
From the simulations, we find that the simulating results are in excellent agreement with the analytical solutions. This indicates that the present methods are efficient and flexible approachs for practical application. The present models can be extended to more complex systems.
引文
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[11]郭照立,郑楚光,李青,王能超.流体动力学的格子Boltzmann方法[M].武汉:湖北科学技术出版社,2002.
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