随机对流扩散方程的数值仿真

本文针对对流一扩散随机过程在随机输入（即随机输运和源项）,作用下进行数值仿真。我们先将对流扩散随机微分方程中的随机函数采用有限项截断的多项式浑沌展开（Polynomial Chaos Expansion）展开,再由Galerkin映射法得到求解浑沌展开系数的确定性方程组。这是一个在物理空间包含多尺度解的大方程组。为此我...     

非线性系统随机过程对初始条件的敏感性

将作者所提出的基于混沌展开的动态自适应小波随机数值模拟方法进一步发展应用于对非线性随机对流-扩散Burgers方程的数值分析。不仅进一步显示了其各求解分量拥有独立的自适应小波网格特点,同时也为随机系统对干扰的敏感性分析:敏感区及其随时间的演变,提供了一个直接可应用的有效方法。数值实验的结果进一步验证了非线性系统对初始条件的敏感性,并初步揭示了输入扰动向高梯度区演变的规律。     

随机过程数值仿真的精度实验与分析

为了考核随机过程数值仿真中采用的混沌多项式展开,小波动态自适应网格等的可靠性和近似程度,本文构造了一个具有解析解的随机过程,随后进行数值仿真。比较了数值仿真结果与解析解,证明两者是相互一致的。说明所采用的数值仿真方法是成立和近似度是可以接受的。所进行比较的随机过程解析解是相对比较简单的,当对更复杂的情况需要进一步数值实验验证。在文中并讨论了影响数值仿真精度和增加数值仿真工作难度的因素。     

快速动态自适应小波配点法

动态自适应多尺度小波配点法(AWCM)能有效地模拟具有间歇性的物理现象,此方法是近几年发展起来的非常新颖的数值计算方法。为了增强该方法识别与跟踪解的奇异性的能力,并提高数值计算稳定性与计算效率,将精细时程积分算法与之相结合形成了快速动态自适应多尺度小波配点法。为了实现这一算法,给出了构造动态自适应网格配点集的新方法,构建了以小波(或尺度函数)系数为变量的时程推进公式。通过求解一维Burgers方程,证明了方法具有更加良好的数值计算性质。     

基于多项式混沌的全局敏感度分析

回顾基于多项式混沌和方差分解的全局敏感度分析方法,针对高维数随机空间和高阶多项式混沌展开面临的“维数灾难”问题,采用回归法、稀疏网格积分及基于l₁优化的稀疏重构技术（即压缩感知技术）来减少非嵌入式多项式混沌方法所需的样本配置点数目.针对几个典型响应面模型（包括Ishigami函数、Sobol函数、Corner peak函数和Morris函数）进行Sobol全局敏感度指标计算,展示多项式混沌方法在基于方差分解的全局敏感度分析中的有效性.     

随机Bonhoeffer-Van der Pol系统的随机混沌控制

讨论了具有有界随机参数的随机Bonhoeffer-Van der Pol系统的随机混沌现象，并利用噪声对其进行控制.首先运用Chebyshev多项式逼近的方法，将随机Bonhoeffer-Van der Pol系统转化为等价的确定性系统，使原系统的随机混沌控制问题转换为等价的确定性系统的确定性混沌控制问题，继而可用Lyapunov指数指标来研究等价确定性系统的确定性混沌现象和控制问题.数值结果表明，随机Bonhoeffer-Van der Pol系统的随机混沌现象与相应的确定性Bonhoeffer-Van der Pol系统极为相似.利用噪声控制法可将混沌控制到周期轨道，但是在随机参数及其强度的影响下也呈现出一些特点.     

随机参数Duffing系统中的随机混沌及其延迟反馈控制

研究了谐和激励下含有界随机参数Duffing系统(简称随机Duffing系统)中的随机混沌及其延迟反馈控制问题.借助Gegenbauer多项式逼近理论，将随机Duffing系统转化为与其等效的确定性非线性系统.这样，随机Duffing系统在谐和激励下的混沌响应及其控制问题就可借等效的确定性非线性系统来研究.分析阐明了随机混沌的主要特点，并采用Wolf算法计算等效确定性非线性系统的最大Lyapunov指数，以判别随机Duffing系统的动力学行为.数值计算表明，恰当选取不同的反馈强度和延迟时间，可分别达到抑制或诱发系统混沌的目的，说明延迟反馈技术对随机混沌控制也是十分有效的. 关键词： 随机Duffing系统 延迟反馈控制 随机混沌 Gegenbauer多项式     

隐-显积分因子间断Galerkin方法求解二维辐射扩散方程

提出一种求解二维非平衡辐射扩散方程的数值方法.空间离散上采用加权间断Galerkin有限元方法,其中数值流量的构造采用一种新的加权平均;时间离散上采用隐-显积分因子方法,将扩散系数线性化,然后用积分因子方法求解间断Galerkin方法离散后的非线性常微分方程组.数值试验中在非结构网格上求解了多介质的辐射扩散方程.结果表明:对于强非线性和强耦合的非线性扩散方程组,该方法是一种非常有效的数值算法.     

求解双曲型守恒律方程的一类自适应多分辨方法

针对双曲型守恒律方程问题,发展一种有效的自适应多分辨分析方法.通过对嵌套网格上的数值解构造离散多分辨分析,建立小波系数与多层嵌套网格点之间的对应关系.对于小波系数较大的网格点采用高精度WENO格式计算,其余区域则直接采用多项式插值.数值试验表明,该方法在保持原规则网格方法的精度和分辨率的同时,显著地减少计算的CPU时间.     

求解非线性偏微分方程的自适应小波精细积分法

以Burgers方程为例,提出了一种求解偏微分方程的自适应多层插值小波配置法,通过引入一种具有插值特性的拟Shannon小波并利用插值小波理论构造了多层自适应插值小波算子,从而在空间实现了偏微分方程的自适应离散.另外,精细时程积分方法和外推法的引入不但有助于提高求解速度和数值结果的精度,而且使时间积分步长的选取具有了自适应性.     

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.     

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.     

Analysis of stochastic bifurcation and chaos in stochastic Duffing--van der Pol system via Chebyshev polynomial approximation

The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing--van der Pol system with bounded random parameter of exponential probability density function subjected to a harmonic excitation. Firstly the stochastic system is reduced into its equivalent deterministic one, and then the responses of stochastic system can be obtained by numerical methods. Nonlinear dynamical behaviour related to stochastic period-doubling bifurcation and chaos in the stochastic system is explored. Numerical simulations show that similar to its counterpart in deterministic nonlinear system of stochastic period-doubling bifurcation and chaos may occur in the stochastic Duffing--van der Pol system even for weak intensity of random parameter. Simply increasing the intensity of the random parameter may result in the period-doubling bifurcation which is absent from the deterministic system.     

基于Laguerre多项式逼近法的随机双势阱Duffing系统的分岔和混沌研究

应用Laguerre正交多项式逼近法研究了含有随机参数的双势阱Duffing系统的分岔和混沌行为.系统参数为指数分布随机变量的非线性动力系统首先被转化为等价的确定性扩阶系统，然后通过数值方法求得其响应.数值模拟结果的比较表明，含有随机参数的双势阱Duffing系统保持着与确定性系统相类似的倍周期分岔和混沌行为，但是由于随机因素的影响，在局部小区域内随机参数系统的动力学行为会发生突变. 关键词： 双势阱Duffing系统 指数分布概率密度函数 Laguerre多项式逼近 随机分岔     

Polynomial chaos expansions for optimal control of nonlinear random oscillators

The polynomial chaos decomposition of stochastic variables and processes is implemented in conjunction with optimal polynomial control of nonlinear dynamical systems. The procedure is demonstrated on a base-excited system whereby ground motion is modeled as a stochastic process with a specified correlation function and is approximated by its Karhunen-Loeve expansion. An adaptive scheme for stochastic approximation with polynomial chaos bases is proposed which is based on a displacement-velocity norm and is applied to the identification of phase orbits of nonlinear oscillators. This approximation is then integrated in the design of an optimal polynomial controller, allowing for the efficient estimation of statistics and probability density functions of quantities of interest. Numerical investigations are carried out employing the polynomial chaos expansions and the Lyapunov asymptotic stability condition based control policy. The results reveal that the performance, as gaged by probabilistic quantities of interest, of the controlled oscillators is greatly improved. A comparative study is also presented against the classical stochastic optimal control, whereby statistical linearization based LQG is employed to design the optimal controller. It is remarked that the proposed polynomial chaos expansion is a preferred approach to the optimal control of nonlinear random oscillators.     

Stochastic period-doubling bifurcation analysis of stochastic Bonhoeffer--van der Pol system

In this paper, the Chebyshev polynomial approximation is applied to the problem of stochastic period-doubling bifurcation of a stochastic Bonhoeffer--van der Pol (BVP for short) system with a bounded random parameter. In the analysis, the stochastic BVP system is transformed by the Chebyshev polynomial approximation into an equivalent deterministic system, whose response can be readily obtained by conventional numerical methods. In this way we have explored plenty of stochastic period-doubling bifurcation phenomena of the stochastic BVP system. The numerical simulations show that the behaviour of the stochastic period-doubling bifurcation in the stochastic BVP system is by and large similar to that in the deterministic mean-parameter BVP system, but there are still some featured differences between them. For example, in the stochastic dynamic system the period-doubling bifurcation point diffuses into a critical interval and the location of the critical interval shifts with the variation of intensity of the random parameter. The obtained results show that Chebyshev polynomial approximation is an effective approach to dynamical problems in some typical nonlinear systems with a bounded random parameter of an arch-like probability density function.