延迟动力系统线性θ-方法的散逸性

１．引言 科学与工程中的许多问题具有散逸性,即系统具有一有界吸引集,从任意初始条件出发的解经过有限时间后进入该吸引集并随后保持在里面．如 ２维的 Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程、Ｌｏｒｅｎｚ方程等许多重要系统都是散逸的．散逸性研究一直是动力系统研究中的重要课题（参见Ｔｅｍａｍ［７］）．当数值求解这些系统时,自然希望数值方法能保持系统的该重要特性．１９９４年, Ｈｕｍｐｈｒｉｅｓ和 Ｓｔｕａｒｔ［６］首次研究了 Ｒｕｎｇｅ－Ｋｕｔｔａ方法对有限维系统的散逸性．１９９７年Ｈｉｌｌ［２］研究了其无穷维散逸性…     

一般线性方法的线性与非线性稳定性间的关系

１．引言１９６３年,Ｄａｈｌｑｕｉｓｔ以一类线性问题为模型提出了Ａ－稳定性概念,此后有关如何判断方法是否Ａ－稳定或确定其稳定域的研究十分活跃,袁兆鼎等~［１］中对此有详细的讨论．Ｂｕｒｒａｇｅ与Ｂｕｔｃｈｅｒ［２］以一类非线性问题为模型,就一般线性方法引入了代数稳定性概念．Ｂｕｔｃｈｅｒ［４］探讨了代数稳定性与Ａ－稳定性间的内在联系．为确保代数稳定性蕴涵Ａ－稳定性,Ｂｕｔｃｈｅｒ［５］进一步要求代数稳定性定义中涉及的矩阵Ｇ是正定的。然而这样一来,正如李寿佛［６］中指出的那样,许多ＡＮ稳定且按［４…     

非线性中立型延迟积分微分方程一般线性方法的稳定性分析

本文研究求解R(α,β₁,β₂,γ)类非线性中立型延迟积分微分方程的一般线性方法的数值稳定性,获得了代数稳定的一般线性方法稳定及渐近稳定的条件,最后的数值试验验证了所获理论的正确性.       

一类求解分片延迟微分方程的线性多步法的散逸性

本文研究分片延迟微分方程本身及数值方法的散逸性问题．给出了一个关于此类问题本身散逸性的充分条件,同时得到了一类求解此类问题的线性多步法的数值散逸性结果,此结果表明所考虑的数值方法继承了方程本身的散逸性．数值试验进一步验证了理论结果的正确性．     

积分微分方程线性多步方法的散逸性

研究一类积分微分方程线性多步方法（p,σ）的散逸性．当积分项用复合求积公式逼近时,证明了线性多步方法是有限维散逸的．这说明该方法很好地继承了系统本身所具有的重要性质．这一结论为数值求解这一类微分方程提供了更多的选择．     

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…     

Volterra泛函微分方程一般线性方法的稳定性

本文研究Volterra泛函微分方程(k,p,q)-代数稳定的一般线性方法的稳定性,获得了该类方法的一系列新的稳定性结果.     

一般线性方法的散逸稳定性

1 散逸动力系统 考虑初值问题 y′(t)=f(y),y(0)=y_0∈R~N,t≥0, (1.1)这里f:R~N→R~N是满足局部Lipschitz条件的连续映射,并满足条件 Re〈u,f(u)〉≤α-β‖u‖~2 u∈R~N,(1.2)其中α≥0,β>0,〈·,·〉是R~N中标准内积,‖·‖是相应的内积范数.设y(t)是问题(1.1)-(1.2)的一个真解,则 ‖y(t)‖~2≤α/β+e~(-2βt)(‖y_0‖~2-α/β) t≥0, (1.3)及 ‖y(t)‖≤max(‖y_0‖,α/β)　 t≥0,(1.4)     

STABILITY OF GENERAL LINEAR METHODS FOR SYSTEMS OF FUNCTIONAL-DIFFERENTIAL AND FUNCTIONAL EQUATIONS

This paper is concerned with the numerical solution of functional-differential and func-tional equations which include functional-differential equations of neutral type as special cases. The adaptation of general linear methods is considered. It is proved that A-stable general linear methods can inherit the asymptotic stability of underlying linear systems.Some general results of numerical stability are also given.     

QUADRATIC INVARIANTS AND SYMPLECTIC STRUCTURE OF GENERAL LINEAR METHODS

1. IntroductionInvestigating whether a numerical method inherits some dynamical properties possessed bythe differential equation systems being integrated is an important field of numerical analysisand has received much attention in recent years See the review articlesof Sanz-Serna[9] and Section 11.16 of Hairer et. al.[2] for more detail concerning the symplectic methods. Most of the work on canonical iotegrators has dealt with one-step formulaesuch as Runge-Kutta methods and Runge'methods …     

BLOCK BIDIAGONALIZATION METHODS FOR MULTIPLE NONSYMMETRIC LINEAR SYSTEMS

1　IntroductionMany applications require the solution of large nonsymmetric linear equations withmultiple right-hand sidesAX =B ( 1 )where A is a real nonsymmetric matrix ofordern,and X=[x(1 ) ,… ,x(s) ] and B=[b(1 ) ,… ,b(s) ] are rectangular matrices of dimension n×s with s n.Most of iterative methods forthe solution of nonsymmetric linear systems with a single right-hand side may be used tosolve( 1 ) by solving the s linear systems individually.But iterative methods,such asKrylov subs…     

USING THE SKEW-SYMMETRIC ITERATIVE METHODS FOR SOLUTION OF AN INDEFINITE NONSYMMETRIC LINEAR SYSTEMS

The concept of the field of value to localize the spectrum of the iteration matrices of the skew-symmetric iterative methods is further exploited. Obtained formulas are derived to relate the fields of values of the original matrix and the iteration matrix. This allows us to determine theoretically that indefinite nonsymmetric linear systems can be solved by this class of iterative methods.     

POSITIVITY METHODS FOR DICHOTOMY OF LINEAR SKEW-PRODUCT FLOWS ON HILBERT SPACES

Abstract

In [Na-Rh] we developed a method based on positivity in order to characterize the stability of the evolution family corresponding to the nonautonomous Cauchy problem in Hilbert spaces. This method is extended to the study of hyperbolicity of linear skew-products. We also show that exponential dichotomy of a linear skew-product flow is equivalent to the existence of a Hermitian valued solution of some linear Riccati equation.     

B-CONVERGENCE PROPERTIES OF GENERAL LINEAR METHODS

In this paper, for general linear methods applied to strictly dissipative initial value problem in Hilbert spaces, we prove that algebraic stability implies B-convergence, which extends and improves the existing results on Runge-Kutta methods. Specializing our results for the case of multi-step Runge-Kutta methods, a series of B-convergence results are obtained.