气体分子动力论模方程及对激波结构的研究——(关于波尔兹曼方程的数学模型)

本文从“只有用非线性项才能更好地描述玻尔兹曼碰撞积分的非线性项”的基本思想出发,将逆碰撞积分写成局部麦克斯韦函数 f_e 的幂级数形式,各 f_e~n 项的系数是一速度和宏观量的函数,导出了一种新的气体分子动力论模方程.应用 Chapmann-Enskog 方法求解了本模方程,并订算出普朗特数 P_r=2/3.本模方程应用于研究一维正激波结构问题,用数值方法求得本问题的数值解答.并与实验结果作了比较,其结果良好.     

气体分子动力论模方程及对激波结构的研究(关于波尔兹曼方程的数学模型)

本文从“只有用非线性项才能更好地描述玻尔兹曼碰撞积分的非线性项”的基本思想出发,将逆碰撞积分写成局部麦克斯韦函数f_e的幂级数形式,各f_e~n项的系数是一速度和宏观量的函数,导出了一种新的气体分子动力论模方程。 应用Chapmann—Enskog方法求解了本模方程,并订算出普朗特数P_r=2/3. 本模方程应用于研究一维正激波结构问题,用数值方法求得本问题的数值解答,并与实验结果作了比较,其结果良好。     

求解定态中子输运方程的非相容DSA方法

在数值方法求解低泄漏或低俘获的粒子输运方程时,常用的源迭代法(Source iteration method,SI)收敛较慢.缓慢的迭代过程不仅效率低,并且难以确定迭代何时收敛.在已有众多的迭代加速方案中,扩散综合加速法(diffusion synthetic acceleration method,DSA)是一种有效且鲁棒的加速方法.对于一致离散DSA方法,高阶输运方程和低阶扩散算子应该满足相容性条件.然而,在处理复杂离散系统时,却很难推导出满足一致相容性条件的方法.提出了一个满足部分相容性条件的方法,即带阻尼的DSA方法.利用间断有限元方法(diffusion Galerkin method,DGA)对中子输运方程空间坐标进行离散,并利用傅里叶分析结果选择阻尼因子β.方法可用于求解定义在一维平面几何中的输运方程.傅里叶分析和数值试验表明了方法的有效性.     

带移动奇异源热方程的移动网格方法

研究了一个带若干奇异源热方程的数值求解,其源的移动由一个常微分方程描述.基于移动观察区域和区域分解思想提出了一个移动网格预估校正算法.网格方程可自然的通过并行高效求解,算法避免了跳跃信息[u]的计算而使物理方程的离散格式变得非常简单,且仍保持了空间上的二阶收敛性.数值例子验证了算法的收敛性和高效性,并模拟了非线性源函数带来的爆破现象.     

美式看跌期权定价的一种混合数值方法

本文对美式看跌期权的定价提供了一种新的混合数值方法,即快速傅里叶变换法加龙格-库塔法.首先将美式看跌期权价格所满足的Black-Scholes微分方程定解问题转化为一个标准的抛物型初、边值问题,然后通过傅里叶变换,使之转换为一个不带股价变量的常微分方程初值问题,再利用龙格-库塔法对其进行数值求解.数值实验表明,本文算法是一种快速的高精度的算法.     

定常Navier-Stokes方程的一个二步算法

建立了定常Navier-Stokes方程的一个二步算法.新算法第一步先基于P1-P0有限元配对求解一个非线性Navier-Stokes方程,第二步基于P2-P1有限元配对求解一个线性化Navier-Stokes方程.相比于经典的P2-P1有限元方法,方法可以使用较少的计算时间达到相同的收敛精度.数值分析和数值实验表明了算法的有效性.     

热传导(对流-扩散)方程源项识别的粒子群优化算法

提出了利用粒子群优化(PSO)算法反演热传导方程与对流-扩散方程源项的一种新方法,在已有文献方法的基础上,求解出这两类方程正问题的解析解,再把源项识别问题转化为最优化问题,结合粒子群优化算法寻优求解.通过数值模拟与统计检验,结果表明,此方法可快速有效地实现热传导方程与对流-扩散方程源项的识别,并可推广应用到其它数学物理方程的源项或参数的反演识别.     

求解l1极小化问题的Bregman迭代算法

在Tikhonov正则化方法的基础上将其转化为一类l1极小化问题进行求解,并基于Bregman迭代正则化构建了Bregman迭代算法,实现了l1极小化问题的快速求解.数值实验结果表明,Bregman迭代算法在快速求解算子方程的同时,有着比最小二乘法和Tikhonov正则化方法更高的求解精度.     

单调迭代结合虚拟区域法求解非线性障碍问题

讨论了二阶半线性椭圆方程障碍问题的数值求解问题.用单调迭代算法求解障碍问题,并用改进的虚拟区域法求解相关的不规则区域上具有Dirichlet边界条件的椭圆方程.在计算过程中,传统的有限元离散会导致用扩展区域规则网格计算不规则物体边界上积分的困难.为了克服此困难,给出了一种新的基于有限差分的算法,从而使得偏微分快速算法可用.算法结构简单,易于编程实现.对有扩散和增长障碍的logistic人口模型数值模拟说明算法可行且高效.     

渗流问题灰色数值模型的解法研究

灰色数值模型的求解是研究灰色数值模型的一个重要问题 .本文根据灰集合、灰数及其灰色运算规则 ,在渗流系统的基本灰色数值模型的基础上 ,分析了求解这类模型的一整套灰色数值算法 ,并对灰色数值算法、普通算法和经典数值方法的计算结果进行了全面比较 ,论证了灰色数值算法对灰信息传递的正确性和对渗流系统描述的合理性 .     

Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit

Summary. In this paper we study the numerical passage from the spatially homogeneous Boltzmann equation without cut-off to the Fokker-Planck-Landau equation in the so-called grazing collision limit. To this aim we derive a Fourier spectral method for the non cut-off Boltzmann equation in the spirit of [21,23]. We show that the kernel modes that define the spectral method have the correct grazing collision limit providing a consistent spectral method for the limiting Fokker-Planck-Landau equation. In particular, for small values of the scattering angle, we derive an approximate formula for the kernel modes of the non cut-off Boltzmann equation which, similarly to the Fokker-Planck-Landau case, can be computed with a fast algorithm. The uniform spectral accuracy of the method with respect to the grazing collision parameter is also proved. Received July 10, 2001 / Revised version received October 12, 2001 / Published online January 30, 2002     

A combination of energy method and spectral analysis for study of equations of gas motion

There have been extensive studies on the large time behavior of solutions to systems on gas motions, such as the Navier-Stokes equations and the Boltzmann equation. Recently, an approach is introduced by combining the energy method and the spectral analysis to the study of the optimal rates of convergence to the asymptotic profiles. In this paper, we will first illustrate this method by using some simple model and then we will present some recent results on the Navier-Stokes equations and the Boltzmann equation. Precisely, we prove the stability of the non-trivial steady state for the Navier-Stokes equations with potential forces and also obtain the optimal rate of convergence of solutions toward the steady state. The same issue was also studied for the Boltzmann equation in the presence of the general time-space dependent forces. It is expected that this approach can also be applied to other dissipative systems in fluid dynamics and kinetic models such as the model system of radiating gas and the Vlasov-Poisson-Boltzmann system.      

Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space

We study the hypocoercivity property for some kinetic equations in the whole space and obtain the optimal convergence rates of solutions to the equilibrium state in some function spaces. The analysis relies on the basic energy method and the compensating function introduced by Kawashima to the classical Boltzmann equation and developed by Glassey and Strauss in the relativistic setting. It is also motivated by the recent work (Duan et al., 2008 [8]) on the Boltzmann equation by combining the spectrum analysis and energy method. The advantage of the method introduced in this paper is that it can be applied to some complicated system whose detailed spectrum is not known. In fact, only some estimates through the Fourier transform on the conservative transport operator and the dissipation of the linearized operator on the subspace orthogonal to the collision invariants are needed.     

A Numerical Method for Conformal Mapping

A method is developed for constructing the conformal map ofa distorted region onto a rectangle. A discrete Fourier transformis used to map the boundary of the region onto the boundaryof the rectangle; the resulting equations may be solved usinga fast Fourier transform algorithm. The map for internal pointsmay then be constructed using a standard Laplace equation solver.The method is computationally competitive, and is applicableto field problems, for instance in fluid mechanics.     

The time-splitting Fourier spectral method for the coupled Schrödinger-Boussinesq equations

The periodic initial boundary value problem of the coupled Schrödinger-Boussinesq equations is studied by the time-splitting Fourier spectral method. A time-splitting spectral discretization for the Schrödinger-like equation is applied, while a Crank-Nicolson/leap-frog type discretization is utilized for time derivatives in the Boussinesq-like equation. Numerical tests show that the time-splitting Fourier spectral method provides high accuracy for the coupled Schrödinger-Boussinesq equations.     

Convergence to rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations

The main purpose of this paper is to study the asymptotic equivalence of the Boltzmann equation for the hard-sphere collision model to its corresponding Euler equations of compressible gas dynamics in the limit of small mean free path. When the fluid flow is a smooth rarefaction (or centered rarefaction) wave with finite strength, the corresponding Boltzmann solution exists globally in time, and the solution converges to the rarefaction wave uniformly for all time (or away from t=0) as ?→0. A decomposition of a Boltzmann solution into its macroscopic (fluid) part and microscopic (kinetic) part is adopted to rewrite the Boltzmann equation in a form of compressible Navier-Stokes equations with source terms. In this setting, the same asymptotic equivalence of the full compressible Navier-Stokes equations to its corresponding Euler equations in the limit of small viscosity and heat conductivity (depending on the viscosity) is also obtained.     

A volume integral equation method for periodic scattering problems for anisotropic Maxwell's equations

This paper presents a volume integral equation method for an electromagnetic scattering problem for three-dimensional Maxwell's equations in the presence of a biperiodic, anisotropic, and possibly discontinuous dielectric scatterer. Such scattering problem can be reformulated as a strongly singular volume integral equation (i.e., integral operators that fail to be weakly singular). In this paper, we firstly prove that the strongly singular volume integral equation satisfies a Gårding-type estimate in standard Sobolev spaces. Secondly, we rigorously analyze a spectral Galerkin method for solving the scattering problem. This method relies on the periodization technique of Gennadi Vainikko that allows us to efficiently evaluate the periodized integral operators on trigonometric polynomials using the fast Fourier transform (FFT). The main advantage of the method is its simple implementation that avoids for instance the need to compute quasiperiodic Green's functions. We prove that the numerical solution of the spectral Galerkin method applied to the periodized integral equation converges quasioptimally to the solution of the scattering problem. Some numerical examples are provided for examining the performance of the method.     

Application of a direct method for solving the Boltzmann equation in supersonic flow computation

The Boltzmann kinetic equation is used to numerically study the evolution of separated flows over a backward-facing step at low Knudsen numbers. The Boltzmann equation is solved by applying an explicit–implicit scheme. To improve the efficiency of the solution algorithm, it is parallelized with the help of MPI. The solution obtained with the kinetic equation is compared with those based on continuous medium equations. It is shown that the kinetic approach makes it possible to reproduce the distributions of surface pressure, friction coefficient, and heat transfer, as well as to obtain a flow structure close to experimental data.