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。     

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.     

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。     

随机粗糙面的一种新双站散射模型

描述了一种对改进的IEM模型的扩展双站散射模型。我们发现在非相干功率的表达式中,交叉分量和补充分量都由两项组成:一项与误差函数无关,而另一项与误差函数有关。后一项可看成是由误差函数引起的修正项。模型预测值和测量数据的良好吻合显示了这种扩展双站散射模型的有效性。     

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。     

软件可靠性J-M模型的特性分析

软件可靠性是软件质量的一个重要因素,如何提高软件可靠性已成为一个迫切的问题.本文综述了软件可靠性模型的概念和分类,详细分析了J-M模型参数方程的建立及其特性,由对方程根的判定推导出了一个切实可行的方程解法,通过与美国NASA开发的软件Smerfs的实验结果比较,可以看出方程的解法是合理的.     

KdV方程带修正函数的格子Boltzmann模拟

采用一种带修正函数的新格子Boltzmann模型模拟了KdV方程,分析了由此得出的迭代格式的单调性和稳定性,得到了格式的单调性条件。在单调性条件下,迭代格式是[L1]稳定的。数值模拟结果表明该格式是可行的。     

复杂科学与工程问题仿真的隐式微积分建模

针对现代科学与工程仿真遇到愈来愈多难以用经典微积分建模方法描述的复杂问题,在理论研究和工程实践中提出各种含有多个经验参数的唯象偏微分方程模型,或直接采用统计模型来描述.这些模型的物理意义不是很清楚且参数多,其中部分人为参数缺乏物理意义.因此,利用描述问题的基本解或统计分布构造隐式微积分控制方程.这里"隐式"是指可以不需要或难以推导出该控制方程的显式微积分表达式.该方法仅需微积分控制方程的基本解和相应的边界条件就可以进行数值仿真计算.称该方法为隐式微积分方程建模.考虑多相软物质热传导的幂律行为,采用分数阶里斯(Riesz)势核函数为基本解构造稳态问题的隐式分数阶微积分方程模型并进行数值验证.此外,以列维(Lévy)稳态统计分布的概率密度函数为基本解,构造反常扩散现象的隐式分数阶微积分方程模型.该研究的主要数值计算技术基于径向基函数的配点方法.     

结构方程模型在综合评价中的应用

目的：评价全喉切除病人生活质量,探讨影响生活质量的主要因素,确定重点关注人群。方法：使用中国头颈部癌生命质量测定量表（QLICP—HN）和喉切除术后患者家庭支持系统量表对120例全喉切除病人进行问卷调查,并通过结构方程模型构建影响病人生活质量因素初始模型,并予以验证,再根据模型参数检验、修正指数对初始进行修正,以得到最终模型。结果结构方程模型拟合参数GFI=0．927,AGFI=0．877,CFI=0．984,模型拟合较好：家庭支持情况,自身状况严重影响全喉切除病人的生活质量（P〈0．01）。结论：提高全喉切除病人生活质量的关键是大力加强家庭和社会支持;40岁以下文化程度偏低的低收入人群是重点关注人群。     

UNIFAC-SRK模型应用范围的进一步拓展

根据HV混合规则,利用UNIFAC模型、SRK状态方程及相关热力学函数,推导出UNIFAC-SRK模型的逸度系数方程。根据Nelder-Meand法,采用FORTRAN编程,估算出了18对新的基团交互作用参数,扩大了该模型的应用范围。根据估算出的基团参数对35个二元汽液平衡体系的泡点压力和汽相组成进行了预测,预测值和实验值吻合良好。     

Three-dimensional discrete-velocity BGK model for the incompressible Navier–Stokes equations

The lattice Boltzmann method (LBM) has been widely used for the simulations of the incompressible Navier–Stokes (NS) equations. The finite difference Boltzmann method (FDBM) in which the discrete-velocity Boltzmann equation is solved instead of the lattice Boltzmann equation has also been applied as an alternative method for simulating the incompressible flows. The particle velocities of the FDBM can be selected independently from the lattice configuration. In this paper, taking account of this advantage, we present the discrete velocity Boltzmann equation that has a minimum set of the particle velocities with the lattice Bharnagar–Gross–Krook (BGK) model for the three-dimensional incompressible NS equations. To recover incompressible NS equations, tensors of the particle velocities have to be isotropic up to the fifth rank. Thus, we propose to apply the icosahedral vectors that have 13 degrees of freedom to the particle velocity distributions. Validity of the proposed model (D3Q13BGK) is confirmed by numerical simulations of the shear-wave decay problem and the Taylor–Green vortex problem. With respect to numerical accuracy, computational efficiency and numerical stability, we compare the proposed model with the conventional lattice BGK models (D3Q15, D3Q19 and D3Q27) and the multiple-relaxation-time (MRT) model (D3Q13MRT) that has the same degrees of freedom as our proposal. The comparisons show that the compressibility error of the proposed model is approximately double that of the conventional lattice BGK models, but the computational efficiency of the proposed model is superior to that of the others. The linear stability of the proposed model is also superior to that of the lattice BGK models. However, in non-linear simulations, the proposed model tends to be less stable than the others.     

Application of Lattice Boltzmann Method to Image Filtering

In this paper, lattice Boltzmann D2Q5 (two dimensions and five discrete velocity directions) and D2Q9 (two dimensions and nine discrete velocity directions) models are used to solve Perona-Malik equation, which is widely used in image filtering. A set of images added three types of noise are processed using these models. Then the processed images are compared in aspects of peak signal to noise ratio (PSNR) and visual effect. The comparison show that two models have almost the same filtering effect. Simultaneously, it is validated that D2Q5 model is more efficient. Other findings are: (1) D2Q5 and D2Q9 models are more effective in dealing with some images than others; (2) salt and pepper noise is relatively difficult to remove compared with gaussian noise and speckle noise; (3) lattice Boltzmann method shows good stability in the image filtering.     

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.     

五阶精度WENO差分型格子波尔兹曼算法求解单守恒方程

给出了五阶精度WENO差分型格子波尔兹曼算法求解单守恒模型方程的计算方法.根据WENO差分格式的特点,定义了广义格子波尔兹曼分布函数,将守恒型方程的求解问题,转化成用WENO格式的差分算法对该分布函数进行求解.该方法的意义在于,将高精度高分辨率的WENO格式差分方法与近几十年发展起来的格子波尔兹曼方法相结合,从而很方便地构造出可以用于求解守恒型方程的格子波尔兹曼模型,使格子波尔兹曼方法在可压缩流领域的使用更简单.利用该方法分别构造了不同初值条件下的一维Burgers守恒型方程的求解模型,求出结果,并分析了模型的精度和稳定性.最后总结了方法的优点和不足,以及有待进一步研究解决的问题.     

Modelling solute transport in shallow water with the lattice Boltzmann method

The lattice Boltzmann method is used to investigate the solute transport in shallow water flows. Shallow water equations are solved using the lattice Boltzmann equation on a D2Q9 lattice with multiple-relaxation-time (MRT-LBM) and Bhatnagar–Gross–Krook (BGK-LBM) terms separately, and the advection–diffusion equation is also solved with a LBM-BGK on a D2Q5 lattice. Three cases: open channel flow with side discharge, shallow recirculation flow and flow in a harbour are simulated to verify the described methods. Agreements between predictions and experiments are satisfactory. In side discharge flow, the reattachment length for different ratios of side discharge velocity to main channel velocity has been studied in detail. Furthermore, the performance of MRT-LBM and BGK-LBM for these three cases has been investigated. It is found that LBM-MRT has better stability and is able to satisfactorily simulate flows with higher Reynolds number. The study shows that the lattice Boltzmann method is simple and accurate for simulating solute transport in shallow water flows, and hence it can be applied to a wide range of environmental flow problems.     

A higher-order moment method of the lattice Boltzmann model for the Korteweg–de Vries equation

In this paper, a lattice Boltzmann model for the Korteweg–de Vries (KdV) equation with higher-order accuracy of truncation error is presented by using the higher-order moment method. In contrast to the previous lattice Boltzmann model, our method has a wide flexibility to select equilibrium distribution function. The higher-order moment method bases on so-called a series of lattice Boltzmann equation obtained by using multi-scale technique and Chapman–Enskog expansion. We can also control the stability of the scheme by modulating some special moments to design the dispersion term and the dissipation term. The numerical example shows the higher-order moment method can be used to raise the accuracy of truncation error of the lattice Boltzmann scheme.     

Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves

We follow the mathematical framework proposed by Bouchut (2003) [22] and present in this contribution a dual entropy approach for determining equilibrium states of a lattice Boltzmann scheme. This method is expressed in terms of the dual of the mathematical entropy relative to the underlying conservation law. It appears as a good mathematical framework for establishing a “H-theorem” for the system of equations with discrete velocities. The dual entropy approach is used with D1Q3 lattice Boltzmann schemes for the Burgers equation. It conducts to the explicitation of three different equilibrium distributions of particles and induces naturally a nonlinear stability condition. Satisfactory numerical results for strong nonlinear shocks and rarefactions are presented. We prove also that the dual entropy approach can be applied with a D1Q3 lattice Boltzmann scheme for systems of linear and nonlinear acoustics and we present a numerical result with strong nonlinear waves for nonlinear acoustics. We establish also a negative result: with the present framework, the dual entropy approach cannot be used for the shallow water equations.     

Three-Dimensional Lattice Boltzmann Model for the Complex Ginzburg–Landau Equation

In this paper, a lattice Boltzmann model for the three-dimensional complex Ginzburg–Landau equation is proposed. The multi-scale technique and the Chapman–Enskog expansion are used to describe higher-order moments of the complex equilibrium distribution function and a series of complex partial differential equations. The modified partial differential equation of the three-dimensional complex Ginzburg–Landau equation with the third order truncation error is obtained. Based on the complex lattice Boltzmann model, some motions of the stable scroll, such as the scroll wave with a straight filament, scroll ring, and helical scroll are simulated. The comparisons between results of the lattice Boltzmann model with those obtained by the alternative direction implicit scheme are given. The numerical results show that this model can be used to simulate the three-dimensional complex Ginzburg–Landau equation.