随机过程动态自适应小波独立网格多尺度模拟

在随机过程数值仿真中,由多项式混沌展开谱方法得到求解展开系数的确定性偶合方程组。该方程组比相应的确定性仿真时增大许多。并且当多项式展开阶数和随机空间维数提高时,方程维数急剧增加。由于待求未知分量为表征不同尺度波动的混沌展开模,形成节点意义下的的多尺度问题,传统的网格细分自适应逼近不再适用。为此我们采用了小波的多尺度离散,并建立基于空间细化的动态自适应系统,让每个求解点上的多个未知分量有各自独立的小波网格。本文以随机对流扩散方程为例,进行了二个算例的数值实验,论证了此方法的优点。     

一维空间Riesz分数阶对流扩散方程的格子Boltzmann方法

建立格子Boltzmann方法（LBM）的D1Q3演化模型,研究一类Riesz空间分数阶对流扩散方程的数值求解问题。对分数阶微积分算子中的积分项离散化处理,得到逼近的标准对流扩散方程。结合Taylor展式和Chapman-Enskog多尺度展开技术得到模型的各个方向上的平衡态分布函数,通过D1Q3演化模型正确恢复所要求解的宏观方程。数值算例验证该方法的有效性。     

统一的对流扩散型可压缩流体力学方程与解法

流体力学的动量方程、能量方程、湍动能方程和耗散方程都具有对流扩散方程的形式,但连续方程却不是对流扩散型的。对于可压缩问题,本文通过合理的数学推导,不作任何近似、假定与简化,得到一个全新的连续方程形式．该连续方程以压力为未知变量,并具有对流扩散型形式,使得所有的流体动力学方程组都具有完全统一的方程形式,给出了这种三维对流扩散方程组的有限精确差分计算格式。对流体力学的进一步发展具有一定意义．     

对流扩散方程的涡区分离解法

对流扩散方程是流体计算中一个基本方程,常用的数值方法导至解一个高阶的代数方程组,要求较大的存贮量和较长的计算时间。本文提出一种涡区分离解法,它利用对流扩散方程的迎风性质,把涡区从对流支配区分离出来,仅在各个涡区建立代数方程组并求解。而在对流支配区,则充分利用其抛物性,只需采用显式格式进行计算。由于在各涡区建立的这些方程组阶数和带宽都较小,因此要求存贮量较小,计算速度较快。对于雷诺数较大,涡区范围较小的问题,该方法特别有效。     

改进的节块展开方法求解圆柱几何对流扩散方程

为高效求解球床高温气冷堆物理-热工耦合问题,发展改进节块展开法求解圆柱几何下的对流扩散方程.针对圆柱几何和对流扩散方程的特殊性,采用三阶多项式和指数函数作为r向横向积分方程的展开函数,在节块展开法的框架下高效求解对流扩散方程.数值验证表明,改进的节块展开方法具有固有的迎风特性,在使用粗网节块时依然能保持稳定性和较高的计算精度.     

用量纲分析法求解太阳宇宙线扩散对流方程

太阳宇宙线在行星际空间的传播,包括行星际不规则磁场中的扩散和太阳风对流这两种物理过程.应用量纲分析法可以解出现有许多能化为贝塞尔函数的方程,得到与常用分离变量法完全相同的结果.为了求出均匀无限介质中扩散对流方程的解,我们引入反映粒子扩散和对流特征的无量纲参数.在扩散为主导的情况下,解在形式上类似于以对流速度运动的源的扩散,另一对流修正项可按对流参数的幂级数展开,其系数是扩散参数的广义超几何函数组成的级数.这种解的物理概念清楚,适用于讨论中等能量（E_p≥10¹Mev）以上太阳宇宙线上升期特性.     

基于微可压液体的统一对流扩散型流体力学方程组

为了表达上的方便及求解格式的统一,通常采用统一的方程形式来表达连续方程,动量方程、能量方程、湍动能方程和耗散方程等.除了连续方程外,其他方程都可以写成对流扩散方程的形式,由于没有扩散项,连续方程比较特别,也相对不便处理.在微可压液体区,通过合理的数学推导,不作任何近似、假定与简化,本文得到一套全新的连续方程形式.该新方程以压力为未知变量,是对流扩散型的,使得所有的流体动力学方程组都具有完全统一的方程形式.     

数值求解对流扩散方程的特征有限分析方法

本文在详细分析了对流扩散过程物理特性的基础上,按对流扩散过程的物理要求应用特征方法处理对流项、以能充分描述扩散效应的有限分析方法处理扩散项,建立了一种合乎对流扩散物理要求的、无条件L_∞稳定的、数值模拟对流扩散物理现象的特征有限分析方法;并就非线性情况证明了特征有限分析方法的收敛性、给出了解的误差估计。最后的数值实验表明它能很好地模拟对流扩散过程,数值粘性小,精度高,稳定性好,并且没有伪振荡现象发生。     

基于非结构网格的不可压流动耦合求解器

采用非结构化网格有限容积法求解了不可压N-S方程组,对流项采用GAMMA格式,扩散项采用二阶中心差分格式建立离散方程,用SOAR算法处理压力与速度的耦合关系,得到了一种求解不可压N-S方程的非结构网格耦合求解器。通过方腔顶盖驱动流、后台阶绕流以及方腔自然对流等几个典型的算例,考察了求解器的计算精度及收敛特性,并与SIMPLE算法进行了比较,结果表明该求解器是有效可行的。     

求解对流扩散方程的四种差分格式的比较

利用对流扩散方程，在边界和参数存在随机扰动的情况下，考察四种差分格式的优劣，为求 解对流扩散方程提供一种可靠的差分格式，并得到通过空间加密网格的方法可以控制边界、 参数随机影响的结论. 关键词： 对流扩散方程 差分格式 随机扰动     

A method for solving stochastic equations by reduced order models and local approximations

A method is proposed for solving equations with random entries, referred to as stochastic equations (SEs). The method is based on two recent developments. The first approximates the response surface giving the solution of a stochastic equation as a function of its random parameters by a finite set of hyperplanes tangent to it at expansion points selected by geometrical arguments. The second approximates the vector of random parameters in the definition of a stochastic equation by a simple random vector, referred to as stochastic reduced order model (SROM), and uses it to construct a SROM for the solution of this equation.The proposed method is a direct extension of these two methods. It uses SROMs to select expansion points, rather than selecting these points by geometrical considerations, and represents the solution by linear and/or higher order local approximations. The implementation and the performance of the method are illustrated by numerical examples involving random eigenvalue problems and stochastic algebraic/differential equations. The method is conceptually simple, non-intrusive, efficient relative to classical Monte Carlo simulation, accurate, and guaranteed to converge to the exact solution.     

Efficient stochastic Galerkin methods for random diffusion equations

We discuss in this paper efficient solvers for stochastic diffusion equations in random media. We employ generalized polynomial chaos (gPC) expansion to express the solution in a convergent series and obtain a set of deterministic equations for the expansion coefficients by Galerkin projection. Although the resulting system of diffusion equations are coupled, we show that one can construct fast numerical methods to solve them in a decoupled fashion. The methods are based on separation of the diagonal terms and off-diagonal terms in the matrix of the Galerkin system. We examine properties of this matrix and show that the proposed method is unconditionally stable for unsteady problems and convergent for steady problems with a convergent rate independent of discretization parameters. Numerical examples are provided, for both steady and unsteady random diffusions, to support the analysis.     

Approximate solution methods for linear stochastic difference equations: I. Moments

The cumulant expansion for linear stochastic differential equations is extended to the case of linear stochastic difference equations. We consider a vector difference equation, which contains a deterministic matrix A₀ and a random perturbation matrix A₁(t). The expansion proceeds in powers of ατ_c, where τ_c is the correlation time of the fluctuations in A₁(t) and α a measure for their strength. Compared to the differential case, additional cumulants occur in the expansion. Moreover one has to distinguish between a nonsingular and a singular A₀. We also discuss a limiting situation in which the stochastic difference equation can be replaced by a stochastic differential equation. The derivation is not restricted to the case where in the limit the stochastic parameters in the difference equation are replaced by white noise.     

A path integral method for coarse-graining noise in stochastic differential equations with multiple time scales

We present a new path integral method to analyze stochastically perturbed ordinary differential equations with multiple time scales. The objective of this method is to derive from the original system a new stochastic differential equation describing the system’s evolution on slow time scales. For this purpose, we start from the corresponding path integral representation of the stochastic system and apply a multi-scale expansion to the associated path integral kernel of the corresponding Lagrangian. As a concrete example, we apply this expansion to a system that arises in the study of random dispersion fluctuations in dispersion-managed fiber-optic communications. Moreover, we show that, for this particular example, the new path integration method yields the same result at leading order as an asymptotic expansion of the associated Fokker-Planck equation.     

On closure schemes for polynomial chaos expansions of stochastic differential equations

The propagation of waves in a medium having random inhomogeneities is studied using polynomial chaos (PC) expansions, wherein environmental variability is described by a spectral representation of a stochastic process and the wave field is represented by an expansion in orthogonal random polynomials of the spectral components. A different derivation of this expansion is given using functional methods, resulting in a smaller set of equations determining the expansion coefficients, also derived by others. The connection with the PC expansion is new and provides insight into different approximation schemes for the expansion, which is in the correlation function, rather than the random variables. This separates the approximation to the wave function and the closure of the coupled equations (for approximating the chaos coefficients), allowing for approximation schemes other than the usual PC truncation, e.g. by an extended Markov approximation. For small correlation lengths of the medium, low-order PC approximations provide accurate coefficients of any order. This is different from the usual PC approximation, where, for example, the mean field might be well approximated while the wave function (which includes other coefficients) would not be. These ideas are illustrated in a geometrical optics problem for a medium with a simple correlation function.     

A dynamically adaptive wavelet approach to stochastic computations based on polynomial chaos – capturing all scales of random modes on independent grids

In stochastic computations, or uncertainty quantification methods, the spectral approach based on the polynomial chaos expansion in random space leads to a coupled system of deterministic equations for the coefficients of the expansion. The size of this system increases drastically when the number of independent random variables and/or order of polynomial chaos expansions increases. This is invariably the case for large scale simulations and/or problems involving steep gradients and other multiscale features; such features are variously reflected on each solution component or random/uncertainty mode requiring the development of adaptive methods for their accurate resolution. In this paper we propose a new approach for treating such problems based on a dynamically adaptive wavelet methodology involving space-refinement on physical space that allows all scales of each solution component to be refined independently of the rest. We exemplify this using the convection–diffusion model with random input data and present three numerical examples demonstrating the salient features of the proposed method. Thus we establish a new, elegant and flexible approach for stochastic problems with steep gradients and multiscale features based on polynomial chaos expansions.     

An improved technique on the stochastic functional approach for randomly rough surface scattering –Numerical-analytical Wiener analysis

This paper proposes an improved technique on the stochastic functional approach for randomly rough surface scattering. Its first application is made on a TE plane wave scattering from a Gaussian random surface having perfect conductivity with infinite extent. The random wavefield becomes a ‘stochastic Floquet form’ represented by a Wiener–Hermite expansion with unknown expansion coefficients called Wiener kernels. From the effective boundary condition as a model of the random surface, a series of integral equations determining the Wiener kernels are obtained. By applying a quadrature method to the first three order hierarchical equations, a matrix equation is derived. By solving that matrix equation, the exact Wiener kernels up to second order are numerically obtained. Then the incoherent scattering cross-section and the optical theorem are calculated. A prediction is that the optical theorem always holds, which is derived from previous work is confirmed in a numerical sense. It is then concluded that the improved technique is useful.     

Extended Jacobi Elliptic Function Rational Expansion Method and Its Application to （2＋1）-Dimensional Stochastic Dispersive Long Wave System

In this work, by means of a generalized method and symbolic computation, we extend the Jacobi elliptic function rational expansion method to uniformly construct a series of stochastic wave solutions for stochastic evolution equations. To illustrate the effectiveness of our method, we take the （2＋ 1）-dimensional stochastic dispersive long wave system as an example. We not only have obtained some known solutions, but also have constructed some new rational formal stochastic Jacobi elliptic function solutions.     

Two-dimensional Brownian vortices

We introduce a stochastic model of 2D Brownian vortices associated with the canonical ensemble. The point vortices evolve through their usual mutual advection but they experience in addition a random velocity and a systematic drift generated by the system as a whole. The statistical equilibrium state of this stochastic model is the Gibbs canonical distribution. We consider a single species system and a system made of two types of vortices with positive and negative circulations. At positive temperatures, like-sign vortices repel each other (“plasma” case) and at negative temperatures, like-sign vortices attract each other (“gravity” case). We derive the stochastic equation satisfied by the exact vorticity field and the Fokker-Planck equation satisfied by the N-body distribution function. We present the BBGKY-like hierarchy of equations satisfied by the reduced distribution functions and close the hierarchy by considering an expansion of the solutions in powers of 1/N, where N is the number of vortices, in a proper thermodynamic limit. For spatially inhomogeneous systems, we derive the kinetic equations satisfied by the smooth vorticity field in a mean field approximation valid for N→+∞. For spatially homogeneous systems, we study the two-body correlation function, in a Debye-Hückel approximation valid at the order O(1/N). The results of this paper can also apply to other systems of random walkers with long-range interactions such as self-gravitating Brownian particles and bacterial populations experiencing chemotaxis. Furthermore, for positive temperatures, our study provides a kinetic derivation, from microscopic stochastic processes, of the Debye-Hückel model of electrolytes.