基于格子Boltzmann方法的一维Burgers方程的数值模拟

基于格子Boltzmann方法(LBM)的一维Burgers方程的数值解法,已有2-bit和4-bit模型。文中通过选择合适的离散速度模型构造出恰当的平衡态分布函数; 然后, 利用单松弛的格子Bhatnagar-Gross-Krook模型、Chapman-Enskog展开和多尺度技术, 提出了用于求解一维Burgers方程的3-bit的格子Boltzmann模型,即D1Q3模型,并进行了数值实验。实验结果表明,该方法的数值解与解析解吻合的程度很好,且误差比现有文献中的误差更小,从而验证了格子Boltzamnn模型的有效性。     

利用多尺度分析方法求解KdV方程和KdV—Burgers方程的格子Boltzmann模型

本文建立了用格点法解一般偏微分方程(PDE)的理论框架,构造出求解KdV方程及KdV—Burgers方程的三速格子BGK模型。引进三种时间尺度,利用多尺分析求出Boltzmann演化方程的平衡分布函数。     

一维非定常流的格子Boltzmann模拟

本文建立了一维非定常流的一般格子Ｂｏｌｔｚｍａｎｎ模型，并利用２－速度模型研究了激波管和激波的形成，反射以及相互作用。给了了不同的初始条件和边界条件的模拟结果，它与理论分析放数据分析的结果十分吻合。     

利用多尺度分析方法求解KdV方程和KdV—Burgers方程的格子Bo…

本文建立了用格点法解一般偏微分方程（ＰＤＥ）的理论框架，构造出求解ＫｄＶ方程及ＫｄＶ－Ｂｕｒｇｅｒｓ方法的三速格子ＢＧＫ模型，引进三种时间尺度，利用多尺分析求出Ｂｏｌｔｚｍａｎｎ演化方程的平衡分布函数。     

一维Tyson反应扩散系统的格子Boltzmann方法模拟

建立基于格子Boltzmann模型的一维Tyson反应扩散系统的数值求解法.利用浓度分布的Chapmann-Enskogz展开及多尺度技术,获得激励介质在反应与扩散机制同时作用的一维反应扩散方程,用于求解Tyson反应扩散的反应物和催化剂随时间的浓度空间分布值.数值结果表明本文中所提供的方法是有效的.     

二维带分数阶Laplacian算子的对流扩散方程的格子Boltzmann模型研究

带有分数阶Laplacian算子的对流扩散方程常被用来刻画自然界与社会系统中的反常扩散现象.本文提出了一种新的格子Boltzmann模型,用于求解二维带分数阶Laplacian算子的对流扩散方程.首先,基于分数阶Laplacian算子的Fourier变换和Gauss型求积公式,得到控制方程的近似方程.然后,将速度空间、时间和空间进行离散,并构造合适的平衡态分布函数和离散作用力,建立有效的格子Boltzmann-BGK模型.通过Chapman-Enskog分析,可由建立的格子Boltzmann-BGK模型恢复出宏观方程,从而证明了模型的有效性.最后,将模型应用于求解带有解析解的数值算例和Allen-Cahn方程,数值结果进一步验证了模型的正确性和有效性.     

一类偏微分方程的格子Boltzmann模型

研究了对流扩散方程、Burgers方程和Modified-Burgers方程等具有相同形式的一类偏微分方程。并且构建了带修正函数项的D1Q3格子Boltzmann模型求解这类方程。为了能准确地恢复出此宏观方程,利用Chapman-Enskog展开和多尺度分析技术,推导出了各个方向的平衡态分布函数和修正函数的具体表达式。数值计算结果表明该模型是稳定、有效的。     

求解二维对流扩散方程的格子Boltzmann方法

针对二维对流扩散方程,基于D2Q4格子速度,用Chapman-Enskog多尺度分析技术,将时间尺度取为二阶,空间尺度取为一阶,推导了各个速度方向上的平衡态分布函数所满足的条件,给出了简单且对称的平衡态分布函数表达式,所得到的平衡态分布函数能正确地恢复出二维对流扩散方程,从而构建了一种新的求解二维对流扩散方程的D2Q4格子Boltzmann（LB）模型。用所给LB模型对扩散方程和两个不同初边界条件的对流扩散方程进行了数值求解,数值实验结果表明数值解与精确解吻合较好,与相关文献结果比较边界误差要小得多,验证了模型的有效性。     

后台阶流动的非均匀格子Boltzmann方法模拟

使用非均匀格子Boltzmann方法对后台阶流动进行了数值模拟.将流体流动区域划分为不同的子区域:对于每个子区域内部,分布函数使用均匀网格计算;对于区域边界,分布函数采用嵌套网格方法进行处理.数值计算结果与其它实验、数值结果相吻合.     

格子Boltzmann算法并行性能的系统分析

从处理器映射方式、域分解方式、通信开销、通信模式、可扩展性等方面对格子Boltzmann算法的并行性能作了全面系统的分析,并提出了对该类并行程序效率定量化分析的方法。经过相应的测试与分析,说明该方法对提高大规模并行计算的效率具有指导作用。     

Lattice Boltzmann equation for axisymmetric thermal flows

A simple lattice Boltzmann equation (LBE) model for axisymmetric thermal flow is proposed in this paper. The flow field is solved by a quasi-two-dimensional nine-speed (D2Q9) LBE, while the temperature field is solved by another four-speed (D2Q4) LBE. The model is validated by a thermal flow in a pipe and some nontrivial thermal buoyancy-driven flows in vertical cylinders, including Rayleigh-Bénard convection, natural convection, and heat transfer of swirling flows. It is found that the numerical results agree excellently with analytical solution or other numerical results.     

Lattice Boltzmann Models for Anisotropic Diffusion of Images

The lattice Boltzmann method has attracted more and more attention as an alternative numerical scheme to traditional numerical methods for solving partial differential equations and modeling physical systems. The idea of the lattice Boltzmann method is to construct a simplified discrete microscopic dynamics to simulate the macroscopic model described by the partial differential equations. The use of the lattice Boltzmann method has allowed the study of a broad class of systems that would have been difficult by other means. The advantage of the lattice Boltzmann method is that it provides easily implemented fully parallel algorithms and the capability of handling complicated boundaries. In this paper, we present two lattice Boltzmann models for nonlinear anisotropic diffusion of images. We show that image feature selective diffusion (smoothing) can be achieved by making the relaxation parameter in the lattice Boltzmann equation be image feature and direction dependent. The models naturally lead to the numerical algorithms that are easy to implement. Experimental results on both synthetic and real images are described.     

A discrete Boltzmann equation model for two-phase shallow granular flows

In this paper a Discrete Boltzmann Equation model (hereinafter DBE) is proposed as solution method of the two-phase shallow granular flow equations, a complex nonlinear partial differential system, resulting from the depth-averaging procedure of mass and momentum equations of granular flows. The latter, as e.g. a debris flow, are flows of mixtures of solid particles dispersed in an ambient fluid.The reason to use a DBE, instead of a more conventional numerical model (e.g. based on Riemann solvers), is that the DBE is a set of linear advection equations, which replaces the original complex nonlinear partial differential system, while preserving the features of its solutions. The interphase drag function, an essential characteristic of any two-phase model, is accounted for easily in the DBE by adding a physically based term. In order to show the validity of the proposed approach, the following relevant benchmark tests have been considered: the 1D simple Riemann problem, the dam break problem with the wet–dry transition of the liquid phase, the dry bed generation and the perturbation of a state at rest in 2D. Results are satisfactory and show how the DBE is able to reproduce the dynamics of the two-phase shallow granular flow.     

An analysis of natural convection using the thermal finite element discrete Boltzmann equation

The lattice Boltzmann method (LBM) is the simple numerical simulator for fluids because it consists of linear equations. Excluding the higher differential term, the LBM for a temperature field is also achieved as an easy numerical simulation method. However, the LBM is hardly applied to body fitted coordinates for its formulation. It is then difficult to calculate complex lattices using the LBM. In this paper, the finite element discrete Boltzmann equation (FEDBE) is introduced to deal with this weakness of the LBM. The finite element method is applied to the discrete Boltzmann equation (DBE) of the basic equation of the LBM. For FEDBE, the simulation using complex lattices is achieved, and it will be applicable for the development in engineering fields. The natural convection in a square cavity and the Rayleigh–Bernard convection are chosen as the test problem. Each simulation model is accurate enough for the flow patterns, the temperature distribution and the Nusselt number. This method is now considered good for the flow and temperature field, and is expected to be introduced for complex lattices using the DBE.     

Accelerated Lattice Boltzmann Schemes for Steady-State Flow Simulations

A matrix formulation of the steady Lattice Boltzmann equation is presented. It is shown that the strict steady-state formulation, combined with preconditioned iterative solvers, leads to significant computational savings as compared to the standard explicit LBE scheme.     

A cache‐efficient implementation of the lattice Boltzmann method for the two‐dimensional diffusion equation

The lattice Boltzmann method is an important technique for the numerical solution of partial differential equations because it has nearly ideal scalability on parallel computers for many applications. However, to achieve the scalability and speed potential of the lattice Boltzmann technique, the issues of data reusability in cache‐based computer architectures must be addressed. Utilizing the two‐dimensional diffusion equation, , this paper examines cache optimization for the lattice Boltzmann method in both serial and parallel implementations. In this study, speedups due to cache optimization were found to be 1.9–2.5 for the serial implementation and 3.6–3.8 for the parallel case in which the domain decomposition was optimized for stride‐one access. In the parallel non‐cached implementation, the method of domain decomposition (horizontal or vertical) used for parallelization did not significantly affect the compute time. In contrast, the cache‐based implementation of the lattice Boltzmann method was significantly faster when the domain decomposition was optimized for stride‐one access. Additionally, the cache‐optimized lattice Boltzmann method in which the domain decomposition was optimized for stride‐one access displayed superlinear scalability on all problem sizes as the number of processors was increased. Copyright © 2004 John Wiley & Sons, Ltd.