热弹性动力学耦合问题的微分代数方法

对一般的热机械问题提出了一种有效的数值方法,并对二维的热弹性问题进行了测试．该方法的基本思路是将描述热机械耦合问题的偏微分方程进行降阶,使之成为一组微分代数方程,应力应变关系被写成代数方程．所得到的微分代数系统采用全隐式的向后差分公式进行求解．对该方法进行了详细的说明．为了验证该方法的有效性,将其应用于一个动态非耦合的热弹性问题的求解和一个耦合的二维热弹性问题的求解．     

预测校正公式求解指标2微分代数方程

1．前言微分代数方程（EEES）是经常出现于实际问题中的一类方程．其数值求解已成为常微分方程数值求解领域十分活跃的一个方向．目前微分代数方程求解的数值方法主要是nunge－Kutta型方法及BDF方法．Runge－Kutta型方法在网,问中有详细的介绍．Hairer等人据此编制了软件RADAU,而目前使用最广泛的软件还是PetZold等编制的DASSL．DASSL使用的方法为BDF方法,它在微分代数方程中的应用最早可以追述到Gear的开创性工作问．BDF方法一个很大的优点是刚性稳定．然而对于非刚性的微分代数方程,刚性稳定已不是主要考虑的因素．因此…     

病态代数方程求解的一种改进精细积分法

通过在病态代数方程精细积分法的基础上增加一个迭代改善算法,建立了病态代数方程求解的改进精细积分法．该方法进一步提高了病态代数方程精细积分法的精度和效率,具有良好的应用前景．算例证明了该方法在病态代数方程求解中的有效性．     

微分变换法求解二维非线性Volterra积分微分方程

为了解决二维非线性Volterra积分微分方程的求解问题,本文给出微分变换法.利用该方法将方程中的微分部分和积分部分进行变换,这样简化了原方程,进而得到非线性代数方程组,从而将原问题转换为求解非线性代数方程组的解,使得计算更简便.文中最后数值算例说明了该方法的可行性和有效性.     

在动载荷作用下框架结构大变形分析的微分代数方法

采用弧坐标首先建立了在动载荷作用下,具有不连续性条件和初始位移的框架结构大变形分析的非线性数学模型．其次, 在空间区域内, 采用微分求积单元法（DQEM）来离散非线性数学模型, 并提出了在使用DQEM来求解结构大变形分析中,多个变量具有间断性条件的有效方法,得到了一组非线性DQEM的离散化方程,它是时间域内的一组具有奇异性的非线性微分-代数方程．同时也给出了求解非线性微分-代数方程组的一个解法A·D2作为应用,求解了受集中力和分布力作用的框架和组合框架的大变形静动力学问题,并与现有结果进行了比较．数值算例表明,处理多个变量具有间断性条件的方法和求解代数-微分系统的方法是一个有效的和一般的方法,它具有较少的节点、 较小的计算工作量、 较高的精度、良好的收敛性、 操作简单以及应用广泛等优点．     

单支方法的收敛性

本文讨论用单支方法数值求解一类多刚性时滞微分代数方程的收敛性。我们获得了A－稳定的且p阶经典相容的单支方法（时滞部分用线性插值）的整体误差估计。     

波形松弛算法及其在计算流体力学中的应用

本文介绍求解线性常系数微分代数方程组的波形松弛算法, 基于Laplace积分变换得到该算法新的收敛理论. 进一步将波形松弛算法应用于求解非定常Stokes方程, 介绍并讨论了连续时间波形松弛算法CABSOR算法和离散时间波形松弛算法DABSOR算法.     

一类随机延迟微分代数系统的Euler-Maruyama方法

我们主要构造了数值求解一类1指标随机延迟微分代数系统的Euler-Maruyama方法,并且证明用该方法求解此类问题可达到1/2阶均方收敛.最后的效值试验验证了方法的有效性及所获结论的正确性.     

指标-2微分-代数系统的连续扩展Runge-Kutta迭代方法

1引言微分-代数系统包括具有约束条件的微分方程和奇异隐式微分方程,在实际应用中,如:约束力学系统、流体动力学、化学反应动力学、电子网络模拟、控制工程和机器人技术等领域就产生了诸多问题需要求解．近年来,微分-代数系统已极大地引起了许多工程师和数学工作者的关注,开展了众多相关问题的探讨,提出了许多新的算法理论[1-3]．在本文中我们对指标-2的微分-代数方程利用Runge-Kutta方法进行时间的离散和动力学迭代,研究离散迭代系统的收敛性．     

微分代数方程去奇异化分析

利用去奇异化方法讨论了拟线性微分代数方程在奇点邻域内光滑解的性质.通过尺度参数的微分同胚变换,将拟线性微分代数方程转化为相应的常微分方程,从而构造出在孤立奇点邻域内的初始微分代数方程的光滑解,给出解存在的充分条件,并进一步讨论了解的性质.     

Numerical Integration of Differential Algebraic Systems and Invariant Manifolds

The dynamics of a differential algebraic equation takes place on a lower dimensional manifold in phase space. Applying a numerical integration scheme, it is natural to ask if and how this geometric property is preserved by the discrete dynamical system. In the index-1 case answers to this question are obtained from the singularly perturbed case treated by Nipp and Stoffer, Numer. Math. 70 (1995), 245–257, for Runge-Kutta methods and in K. Nipp and D. Stoffer, Numer. Math. 74 (1996), 305–323, for linear multistep methods. As main result of this paper it is shown that also for Runge-Kutta methods and linear multistep methods applied to a index-2 problem of Hessenberg form there is a (attractive) invariant manifold for the discrete dynamical system and this manifold is close to the manifold of the differential algebraic equation.     

Asymptotic behavior of multistep runge-kutta methods for systems of delay differential equations

1. IntroductionIn order to assess the asymptotic behavior of numerical methods for DDEs, much attention has been given in the literature to the scalar case (cL [1-6]). UP to now) only partialresults (of. [7-10]) have dealt with the delay systemswhere y(t) = (yi(t), so(t),' ) yp(t))" E Cd, which is unknown for t > 0, L and M areconstat complex p x Hmatrices, T > 0 is a constat delay and W(t) 6 CP is a specifiedinitial function.In [111, C.J. Zhang and S.Z. Zhou made an investigation on…     

Geometric Properties of Runge-Kutta Discretizations for Nonautonomous Index 2 Differential Algebraic Systems

We analyze Runge-Kutta discretizations applied to nonautonomous index 2 differential algebraic equations (DAEs) in semi-explicit form. It is shown that for half-explicit and projected Runge-Kutta methods there is an attractive invariant manifold for the discrete system which is close to the invariant manifold of the DAE. The proof combines reduction techniques to autonomou index 2 differential algebraic equations with some invariant manifold results of Schropp [9]. The results support the favourable behavior of these Runge-Kutta methods applied to index 2 DAEs for t = 0.     

Stability and Dynamics of Numerical Methods for Nonlinear Ordinary Differential Equations

Stability of numerical methods for nonlinear autonomous ordinarydifferential equations is approached from the point of viewof dynamical systems. It is proved that multistep methods (withnonlinear algebraic equations exactly solved) with bounded trajectoriesalways produce correct asymptotic behaviour, but this is notthe case with Runge-Kutta. Examples are given of Runge-Kuttaschemes converging to wrong solutions in a deceptively ‘smooth’manner and a characterization of such two-stage methods is presented.PE(CE)^m schemes are examined as well, and it is demonstratedthat they, like Runge-Kutta, may lead to false asymptotics.     

Non-linear stability of a general class of differential equation methods

For a class of methods sufficiently general as to include linear multistep and Runge-Kutta methods as special cases, a concept known as algebraic stability is defined. This property is based on a non-linear test problem and extends existing results on Runge-Kutta methods and on linear multistep and one-leg methods. The algebraic stability properties of a number of particular methods in these families are studied and a generalization is made which enables estimates of error growth to be provided for certain classes of methods.     

Convergence of general linear methods on differential-algebraic systems of index 3

Many numerical methods used to solve ordinary differential equations or differential-algebraic equations can be written as general linear methods. The purpose of this paper is to extend the known convergence results for Runge-Kutta and linear multistep methods to a large class of new promising numerical schemes. The theoretical results are illustrated by some numerical experiments.     

一般线性方法的正则性

１．引言数值方法的动力特征近年引起了人们的广泛关注。其中之一就是系统的平衡态和数值方法的平衡态相一致的问题,即用一个数值方法沿定步长求解系统时,是否会出现伪平衡。不可能出现伪平衡的方法称为是正则的。ＲＫ方法和线性多步法的正则性已被众多的文献研究［２,３,４］,其它方法的正则性显然是一亟待研究的问题。本文讨论较ＲＫ方法和线性多步法远为广泛的一般线性方法的正则性。设ｆ：Ｒ~（Ｎ）→Ｒ~（Ｎ）是一充分光滑的映射,考虑求解初值问题：的一般线性方法［１］：其中步长逼近于逼近于关于微分方程真解ｙ（ｔ）在第ｎ层…     

Multistep numerical methods for functional-differential-algebraic equations

This paper deals with constructing multistep numerical methods for differential delay equations under additional algebraic constraints. Theorems on the orders of convergence of these methods are proved both for functional-differential equations with algebraic constraints and for singular functional-differential equations.     

混合方法的代数稳定性

混合方法的代数稳定性黄乘明（湖南邵阳师专）ＡＬＧＥＢＲＡＩＣＳＴＡＢＩＬＩＴＹＯＦＨＹＢＲＩＤＭＥＴＨＯＤＳ￥ＨｕａｎｇＣｈｅｎｇ－ｍｉｎｇ（ＳｈａｏｙａｎｇＴｅａｃｈｅｒｓ＇ＣＯｌｌａｇｅ）．Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒ，ｗｅｄｉｓｃｕ...     

Local error estimation for multistep collocation methods

Multistep collocation methods for initial value problems in ordinary differential equations are known to be a subclass of multistep Runge-Kutta methods and a generalisation of the well-known class of one-step collocation methods as well as of the one-leg methods of Dahlquist. In this paper we derive an error estimation method of embedded type for multistep collocation methods based on perturbed multistep collocation methods. This parallels and generalizes the results for one-step collocation methods by Nørsett and Wanner. Simple numerical experiments show that this error estimator agrees well with a theoretical error estimate which is a generalisation of an error estimate first derived by Dahlquist for one-leg methods.