辛Runge—Kutta方法的特征与构造

1 引 言 数值求解Hamilton系统时,冯康先生引进的辛方法能保持相空间辛结构,并使数值解继承Hamilton系统本身具有的许多重要特性。因而辛方法研究具有重要理论和实践意义。近年,Sanz-Serna等人深入研究了辛Runge-Kutta方法,在这方面作了系统的工作(参见[4,8—14])。 考虑s级Runge-Kutta方法     

Runge—Kutta方法的G—正交性

１Ｇ－正交矩阵微分方程考虑ＲＮ×Ｎ上常微分矩阵方程初值问题这里Ｗ：［０,＋∞）×ＲＮ×Ｎ→ＲＮ×Ｎ为一光滑的映射,Ｙ（０）ＲＮ×Ｎ为给定的初值,Ｇ为实常正定矩阵．定义１．１如果问题（１．１）的真解ｙ（ｔ）满足ＹＴ（ｔ）ＧＹ（ｔ）＝Ｇ,ｔ≥０,则称该问题真解Ｙ（ｔ）是Ｇ－正交的,以下简称该问题是Ｇ－正交的．特别地,当Ｇ＝ＩＮ时,称该问题是正交的,这里ＩＮ为Ｎ×Ｎ单位降．引理１．１［２］问题（１．１）是正交的当且仅当Ｗ（ｔ,Ｙ）＝Ｆ（ｔ,Ｙ）Ｙ,这里Ｆ;［０．＋∞）×ＲＮ×Ｎ→Ｎ×Ｎ为一反对称矩阵函…     

一类具有L稳定和B稳定的Runge—Kutta方法

在求解刚性常微分方程的数值解法中,为了使获得的结果稳定,人们往往使用具有L稳定和B稳定的数值方法,本文利用W－变换构造了一类具有L稳定和B[稳定的Runge-Kutta(RK)方法。     

多导数Runge—Kutta方法的（θ,α,β）一代数稳定性

在文献[1]中,针对Hilbert空间中的K_(σ,γ)~(p)类初值问题,本文作者建立了多导数Runge-Kutta方法的(θ,α,β)-代数稳定性准则,从而推广了李寿佛教授针对Runge-Kutta方法所提出的(θ,ρ,q)-代数稳定性理论.然而,如何判别方法具有上述稳定性仍然存在一定困难,为此本文作了若干探讨,给出了该稳定性的若干判据.     

求解刚性常微分方程的并行Rosenbrock方法

1．引言在航天工业设计与连续系统仿真领域中，许多问题都是用常微分方程来描述的，而在数值求解这些常微分方程的时候，常常会遇到刚性问题，这就需要用具有较大绝对稳定区域的隐式方法求解，而由此产生的非线性隐式方程必须采用各种类型的牛顿选代方法求解，这就使得隐式方法较之显式方法而言工作量大大提高了．文献［1，2］提出了一类并行隐式RK方法，使不同级的KI。，KZn，…，KSn在各不同处理机上同时获得，从而提高计算速度．但由于预先无法对选代次数做出准确估计，这就给方法用于实时仿真带来困难．本文构造了一类并行Rosenbroc…     

一类求解常微分方程数值解的块方法

本文讨论了一类并行计算常微分方程初值问题的带有高阶导数的块隐式混合单步方法，这种方法可以在Ｋ台处理机上并行进行数值计算，本文对方法的一般性质及收敛性进行了讨论，得知该方法的阶数为２ｌ＋１，并且指出当ｌ＝１，２时，方法是Ａ－稳定的，最后给出了一个数值例子。     

基于R—K—F方法的模糊初值问题

本在Runge-Kutta-Fehlberg方法的基础上，运用模糊仿真的近似推理规则，讨论了当初始状态具有模糊不确定性时，求微分方程数值解的问题。     

一类块隐式混合单步并行计算方法

本文讨论了一类解常微分方程初值问题的块隐式混合单步并行算法，这种算法的块数为K，精度阶为2d＋2，可在S台处理机上进行并行计算，其中K＝S·d．本文讨论了方法的一般性质，给出了方法的稳定性定理，最后给出了一个数值例子．     

具有高阶导数的块隐式混合单步并行计算方法

该文讨论了一类求解常微分方程初值问题的具有高阶导数项的块隐式混合单步并行计算方法．这种算法的块数为k，价数为（l十1）（d＋1），可以在s台处理机上进行并行计算，其中l是高阶导数的阶数，k＝s·d.该文讨论了方法的一般性质及数值稳定性，最后给数值例子。     

多步Runge-Kutta方法关于一类多延迟系统的GPk-稳定性

本文多步Ｒｕｎｇｅ－Ｋｕｔｔａ方法关于延迟微分方程系统的渐近稳定性,在本文中我们证明了在适当条件下常微多步Ｒｕｎｇｅ－Ｋｕｔｔａ方法的Ａ－稳定性等价于相应求解多延迟微分方程系统的ＧＰｋ－稳定性。     

Maximal order for second derivative general linear methods with Runge-Kutta stability

An extension of general linear methods (GLMs), so-called SGLMs (GLMs with second derivative), was introduced to the case in which second derivatives, as well as first derivatives, can be calculated. SGLMs are divided into four types, depending on the nature of the differential system to be solved and the computer architecture that is used to implement these methods. In this paper, we obtain maximal order for two types of SGLMs with Runge-Kutta stability (RKS) property. Also, we construct methods of these types which possess RKS property and A-stability. Efficiency of the constructed methods is shown by numerical experiments.     

求解延迟微分代数方程的多步Runge-Kutta方法的渐近稳定性

延迟微分代数方程(DDAEs)广泛出现于科学与工程应用领域．本文将多步Runge-Kutta方法应用于求解线性常系数延迟微分代数方程，讨论了该方法的渐近稳定性．数值试验表明该方法对求解DDAEs是有效的．     

Parallel continuous Runge-Kutta methods and vanishing lag delay differential equations

We present an explicit Runge-Kutta scheme devised for the numerical solution ofdelay differential equations (DDEs) where a delayed argument lies in the current Runge-Kutta interval. This can occur when the lag is small relative to the stepsize, and the more obvious extensions of the explicit Runge-Kutta method produce implicit equations. It transpires that the scheme is suitable forparallel implementation for solving both ODEs and more general DDEs. We associate our method with a Runge-Kutta tableau, from which the order of the method can be determined. Stability will affect the usefulness of the scheme and we derive the stability equations of the scheme when applied to the constant-coefficient test DDEu(t)=u(t) +u(t –), where the lag and the Runge-Kutta stepsizeH _n H are both constant. (The case=0 is treated separately.) In the case that 0, we consider the two distinct possibilities: (i) H and (ii)<H.In memory of Professor Leslie Fox, Balliol College, OxfordWork performed in part at The University of Auckland, New Zealand.This paper is presented as an outcome of the LMS Durham Symposium convened by Professor C.T.H. Baker on 4th–14th July 1992 with support from the SERC under Grant reference number GR/H03964.     

An improvement on the positivity results for 2-stage explicit Runge-Kutta methods

In this paper, we investigate the positivity property for a class of 2-stage explicit Runge-Kutta (RK2) methods of order two when applied to the numerical solution of special nonlinear initial value problems (IVPs) for ordinary differential equations (ODEs). We also pay particular attention to monotonicity property. We obtain new results for positivity which are important in practical applications. We provide some numerical examples to illustrate our results.     

Parallel Multi-Stage & Multi-Step Method in ODEs

In this paper, the theory of parallel multi-stage and multi-step method is discussed, which is a form of combining Runge-Kutta method with linear multi-step method that can be used for parallel computation.     

Order conditions for partitioned Runge-Kutta methods

We illustrate the use of the recent approach by P. Albrecht to the derivation of order conditions for partitioned Runge-Kutta methods for ordinary differential equations.