求解非线性算子方程的加速迭代格式

研究非线性算子方程的近似求解方法.首先对通常的求解非线性方程加速迭代格式进行推广,得到高阶收敛速度的加速迭代格式,最后把这种加速迭代格式推广到非线性算子方程的求解中去,利用非线性算子的渐进展开,证明了这种加速格式具有三阶的收敛速度.     

向量值Padé-型逼近的递推算法在求解线性方程组上的应用

借助正交多项式,构造了向量值Padé-型逼近的一种有效的三项递推公式,并利用该公式得到了求解高阶线性方程组的一个有用的算法.对该算法的收敛速度和控制迭代步骤进行了讨论,最后用数值例子加以说明.     

构造求根迭代公式的一种方法

本文给出了构造方程求根迭代公式的一种方法,条件简单,便于应用。所得公式具有大范围收敛性,初值可任取,能在任一有限区间上求出方程的全部实根,或判断出方程无实根的情况。将这种方法应用到不同的函数类上,就可得到各种不同的具体的迭代公式。例如,应用到二次连续可微函数类上,就包含了[2],[3]的结果;应用到连续函数类上,就包含了[4]的结果。本文还给出了另外的特例,包括不需要在每步迭代中计算一阶导数和二阶导数的特例,以及不用[1]—[4]中公式求解的特例。对收敛阶也进行了讨论。     

大型线性方程组的变分迭代解法

本文提出了一种求解大型线性方程组的一种新方法——变分迭代解法．这种方法的基本思想是：先给方程一个近似的初值,然后引进若干个拉氏乘子校正其近似值,而拉氏乘子可用极值的概念最佳确定．这种方法收敛速度较快,如果只取ｎ个拉氏乘子（ｎ为方程个数）,则该方法即为Ｎｅｗｔｏｎ迭代法．     

一种基于牛顿迭代改进的新的三阶预估校正格式

基于对牛顿迭代公式的改进及预估校正迭代的思想,提出了一种求解非线性方程的新的三阶预估-校正迭代格式.迭代公式无须计算函数的导数值,且理论上证明了它至少是三阶收敛的.数值实验验证了该迭代公式的有效性.     

大型线性方程组的变分迭代解法

本文提出了一种求解大型线性方程组的一种新方法——变分迭代解法．这种方法的基本思想是：先给方程一个近似的初值,然后引进若干个拉氏乘子控正其近似值．而拉氏乘于可用垭值的概念最佳确定．这种方法收敛速度较快,如果只取,n个拉氏乘子(n为方程个数),则该方法即为Newton迭代法．     

线性多步公式的并行Newton-Raphson迭代方法

1.引言 常微分方程初值问题并行数值方法的研究,一直是并行算法研究中值得注意的问题.其原因不仅在于常微分方程初值问题是一个典型的非线性连续递推问题,也在于它在应用中的重要性,特别如实时计算的需要. [1]与[2]对两类典型的线性多步公式,Adams-Molton隐式公式和 Gear公式(即向后微分公式)进行处理,得到了一类并行算法.其基本思想是将这两类线性多步公式在一个区间上作为非线性方程进行整体迭代求解,该方法的最大特点是方程右端函数在各节点上可以并行计算,适用于多处理机系统和流水线向量机.[2]在一定的迭代初值条     

求解隐式迭代方程的多尺度配置法

用多尺度快速配置法求解病态积分方程的隐式迭代方程.在积分算子是扇形紧算子时,该方法得到了离散隐式迭代方程的近似解.采用Morozov偏差原理作为停止准则,并证明了在该准则下隐式迭代正则化方法所得近似解的收敛率.最后,用数值实验证实理论结果和说明数值方法的有效性.     

等几何分析的多重网格共轭梯度法

提高NURBS基函数阶数可以提高等几何分析的精度,同时也会降低多重网格迭代收敛速度.将共轭梯度法与多重网格方法相结合,提出了一种提高收敛速度的方法,该方法用共轭梯度法作为基础迭代算法,用多重网格进行预处理.对Poisson(泊松)方程分别用多重网格方法和多重网格共轭梯度法进行了求解,计算结果表明:等几何分析中采用高阶NURBS基函数处理三维问题时,多重网格共轭梯度法比多重网格法的收敛速度更快.     

椭圆轨道真近点角的级数计算方法

用级数展开方法得到真近点角超越方程,由迭代法求真近点角与时间的关系,讨论迭代收敛的充分条件．对于不满足迭代收敛充分条件情形,列写偏近点角超越方程,用迭代法求偏近点角变化规律,用数值积分方法求出真近点角与时间的关系,指出所有椭圆轨道都满足偏近点角迭代收敛的充分条件．讨论小偏心率椭圆轨道真近点角近似超越方程,由迭代法求真近点角与时间的关系．数值模拟结果表明该方法的有效性．     

求解方程的一类迭代公式

Using the forms of Newton iterative function, the iterative function of Newton's method to handle the problem of multiple roots and the Halley iterative function, we give a class of iterative formulae for solving equations in one variable in this paper and show that their convergence order is at least quadratic. At last we employ our methods to solve some non-linear equations and compare them with Newton's method and Halley's method. Numerical results show that our iteration schemes are convergent if we choose two suitable parametric functions λ(x) and μ(x). Therefore, our iteration schemes are feasible and effective.     

随机微分方程欧拉格式算法分析

首先给出了线性随机微分方程的欧拉格式算法,然后给出了非线性随机微分方程变步长的欧拉格式算法,接着讨论了其对初值的连续依赖性和收敛性.     

A Non-Conforming Least Squares Spectral Element Formulation for Oseen Equations with Applications to Navier-Stokes Equations

In this article, we propose a non-conforming exponentially accurate least-squares spectral element method for Oseen equations in primitive variable formulation that is applicable to both two- and three-dimensional domains. First-order reformulation is avoided, and the condition number is controlled by a suitable preconditioner for velocity components and pressure variable. A preconditioned conjugate gradient method is used to obtain the solution. Navier-Stokes equations in primitive variable formulation have been solved by solving a sequence of Oseen type iterations. For numerical test cases, similar order convergence has been achieved for all Reynolds number cases at the cost of higher iteration number for higher Reynolds number.     

Fourier-Laplace analysis of the multigrid waveform relaxation method for hyperbolic equations

The multigrid waveform relaxation (WR) algorithm has been fairly studied and implemented for parabolic equations. It has been found that the performance of the multigrid WR method for a parabolic equation is practically the same as that of multigrid iteration for the associated steady state elliptic equation. However, the properties of the multigrid WR method for hyperbolic problems are relatively unknown. This paper studies the multigrid acceleration to the WR iteration for hyperbolic problems, with a focus on the convergence comparison between the multigrid WR iteration and the multigrid iteration for the corresponding steady state equations. Using a Fourier-Laplace analysis in two case studies, it is found that the multigrid performance on hyperbolic problems no longer shares the close resemblance in convergence factors between the WR iteration for parabolic equations and the iteration for the associated steady state equations.     

非线性扰动Klein-Gordon方程初值问题的渐近理论

在二维空间中研究一类非线性扰动Klein-Gordon方程初值问题解的渐近理论. 首先利用压缩映象原理,结合一些先验估计式及Bessel函数的收敛性,根据Klein-Gordon方程初值问题的等价积分方程,在二次连续可微空间中得到了初值问题解的适定性;其次,利用扰动方法构造了初值问题的形式近似解,并得到了该形式近似解的渐近合理性;最后给出了所得渐近理论的一个应用,用渐近近似定理分析了一个具体的非线性Klein-Gordon方程初值问题解的渐近近似程度．     

Iterative Methods for a Linearly Perturbed Algebraic Matrix Riccati Equation Arising in Stochastic Control

We start with a discussion of coupled algebraic Riccati equations arising in the study of linear-quadratic optimal control problems for Markov jump linear systems. Under suitable assumptions, this system of equations has a unique positive semidefinite solution, which is the solution of practical interest. The coupled equations can be rewritten as a single linearly perturbed matrix Riccati equation with special structures. We study the linearly perturbed Riccati equation in a more general setting and obtain a class of iterative methods from different splittings of a positive operator involved in the Riccati equation. We prove some special properties of the sequences generated by these methods and determine and compare the convergence rates of these methods. Our results are then applied to the coupled Riccati equations of jump linear systems. We obtain linear convergence of the Lyapunov iteration and the modified Lyapunov iteration, and confirm that the modified Lyapunov iteration indeed has faster convergence than the original Lyapunov iteration.     

五维空间中半线性波动方程整体解的存在性定理

本文研究五维空间中半线性波动方程u_tt-△u=G（u）整体解的存在性,其中G（u）～|u|^p并且p>（3+(17)^1/2)/4.利用经典的迭代方法证明了:如果初始值很小并且紧支的,径向对称方程有一个经典整体解.     

Navier-Stokes方程与Euler方程的稳定性比较

比较了Navier-Stokes 方程和Euler方程的稳定性;并以它们的典型初值问题为例,分析了Navier-Stokes方程和Euler方程稳定性不同的原因．     

非线性发展方程的Taylor展开方法

提出了积分非线性发展方程的新方法,即Taylor展开方法．标准的Galerkin方法可以看作0-阶Taylor展开方法,而非线性Galerkin方法可以看作1-阶修正Taylor展开方法A·D2此外,证明了数值解的存在性及其收敛性．结果表明,在关于严格解的一些正则性假设下,较高阶的Taylor展开方法具有较高阶的收敛速度．最后,给出了用Taylor展开方法求解二维具有非滑移边界条件Navier-Stokes方程的具体例子．     

Exponential Convergence to Equilibrium for Kinetic Fokker-Planck Equations

A class of linear kinetic Fokker-Planck equations with a non-trivial diffusion matrix and with periodic boundary conditions in the spatial variable is considered. After formulating the problem in a geometric setting, the question of the rate of convergence to equilibrium is studied within the formalism of differential calculus on Riemannian manifolds. Under explicit geometric assumptions on the velocity field, the energy function and the diffusion matrix, it is shown that global regular solutions converge in time to equilibrium with exponential rate. The result is proved by estimating the time derivative of a modified entropy functional, as recently proposed by Villani. For spatially homogeneous solutions the assumptions of the main theorem reduce to the curvature bound condition for the validity of logarithmic Sobolev inequalities discovered by Bakry and Emery. The result applies to the relativistic Fokker-Planck equation in the low temperature regime, for which exponential trend to equilibrium was previously unknown.