一类中立型延迟积分微分方程Runge-Kutta方法的稳定性

讨论了一类非线性中立型延迟积分微分方程Runge-Kutta方法的稳定性.在适当的条件下证明了运用Runge-Kutta方法求解这类方程既是数值稳定的也是渐近稳定的.     

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…     

Stability of Runge-Kutta methods for delay differential systems with multiple delays

The stability of Runge-Kutta methods for systems of delay differentialequations (DDEs) with multiple delays is considered. The stabilityregions of explicit and implicit Runge-Kutta methods are discussedwhen they are applied to asymptotically stable linear DDEs withmultiple delays. A simple estimate on the stability regionsof explicit Runge-Kutta methods is presented. It is shown thatthe stable step-size for numerical integration of DDEs withmultiple delays can be easily selected by means of the estimate.     

STABILITY ANALYSIS OF RUNGE-KUTTA METHODS FOR NONLINEAR SYSTEMS OF PANTOGRAPH EQUATIONS

This paper is concerned with numerical stability of nonlinear systems of pantograph equations. Numerical methods based on （k, l）-algebraically stable Runge-Kutta methods are suggested. Global and asymptotic stability conditions for the presented methods are derived.     

刚性多滞量积分微分方程的Runge—Kutta方法

本文研究了求解刚性多滞量积分微分方程的Runge-Kutta方法的非线性稳定性和计算有效性.经典Runge—Kutta方法连同复合求积公式和Pouzet求积公式被改造用于求解一类刚性多滞量Volterra型积分微分方程.其分析导出了:在适当条件下,扩展的Runge-Kutta方法是渐近稳定和整体稳定的.此外,数值试验表明所给出的方法是高度有效的.     

比例延迟微分方程组具有刚性精度Runge-Kutta方法的稳定性分析

该文研究比例延迟微分方程组具有刚性精度变步长Runge-Kutta方法的渐近稳定性，给出了一类普遍意义下的变步长格式。证明当且仅当其稳定函数在无穷远点处的模小于1时，变步长Runge-Kutta方法渐近稳定。     

Stability Analysis of Runge-Kutta Methods for Non-Linear Delay Differential Equations

This paper is concerned with the numerical solution of delay differential equations(DDEs). We focus on the stability behaviour of Runge-Kutta methods for nonlinear DDEs. The new concepts of GR(l)-stability, GAR(l)-stability and weak GAR(l)-stability are further introduced. We investigate these stability properties for (k, l)-algebraically stable Runge-Kutta methods with a piecewise constant or linear interpolation procedure.     

Stability and -convergence properties of multistep Runge-Kutta methods

This paper continues earlier work by the same author concerning the stability and -convergence properties of multistep Runge-Kutta methods for the numerical solution of nonlinear stiff initial-value problems in a Hilbert space. A series of sufficient conditions and necessary conditions for a multistep Runge-Kutta method to be algebraically stable, diagonally stable, - or optimally -convergent are established, by means of which six classes of high order algebraically stable and -convergent multistep Runge-Kutta methods are constructed in a unified pattern. These methods include the class constructed by Burrage in 1987 as special case, and most of them can be regarded as extension of the Gauss, RadauIA, RadauIIA and LobattoIIIC Runge-Kutta methods. We find that the classes of multistep Runge-Kutta methods constructed in the present paper are superior in many respects to the corresponding existing one-step Runge-Kutta schemes.

     

Stability analysis of numerical methods for systems of neutral delay-differential equations

Stability analysis of some representative numerical methods for systems of neutral delay-differential equations (NDDEs) is considered. After the establishment of a sufficient condition of asymptotic stability for linear NDDEs, the stability regions of linear multistep, explicit Runge-Kutta and implicitA-stable Runge-Kutta methods are discussed when they are applied to asymptotically stable linear NDDEs. Some mentioning about the extension of the results for the multiple delay case is given.     

非线性延迟积分微分方程连续Runge-Kutta方法的稳定性分析

本文主要研究了一般形式的延迟积分微分方程,将连续Runge-Kutt,a方法用于求解该类问题,并讨论了方法的稳定性,证明了(k,l)-代数稳定的Runge-Kutta方法当0k1时对应的连续Runge-Kutta方法是渐近稳定的.最后我们通过数值试验验证了方法的有效性及所获结论的正确性.     

Dissipativity of multistep runge-kutta methods for nonlinear volterra delay-integro-differential equations

This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations.We investigate the dissipativity properties of (k,l)algebraically stable multistep Runge-Kutta methods with constrained grid and an uniform grid.The finitedimensional and infinite-dimensional dissipativity results of (k,l)-algebraically stable Runge-Kutta methods are obtained.     

Instability of Runge-Kutta methods when applied to linear systems of delay differential equations

Summary. This paper investigates the stability of Runge-Kutta methods when they are applied to the complex linear system of delay differential equations , where . We prove that no Runge-Kutta method preserves asymptotic stability. Received January 24, 2000 / Revised version received July 19, 2000 / Published online June 7, 2001     

Runge-Kutta Methods for the Numerical Solution of Stiff Semilinear Systems

This paper studies the stability and convergence properties of general Runge-Kutta methods when they are applied to stiff semilinear systems y(t) = J(t)y(t) + g(t, y(t)) with the stiffness contained in the variable coefficient linear part.We consider two assumptions on the relative variation of the matrix J(t) and show that for each of them there is a family of implicit Runge-Kutta methods that is suitable for the numerical integration of the corresponding stiff semilinear systems, i.e. the methods of the family are stable, convergent and the stage equations possess a unique solution. The conditions on the coefficients of a method to belong to these families turn out to be essentially weaker than the usual algebraic stability condition which appears in connection with the B-stability and convergence for stiff nonlinear systems. Thus there are important RK methods which are not algebraically stable but, according to our theory, they are suitable for the numerical integration of semilinear problems.This paper also extends previous results of Burrage, Hundsdorfer and Verwer on the optimal convergence of implicit Runge-Kutta methods for stiff semilinear systems with a constant coefficients linear part.     

广义中立型系统的渐近稳定性及数值分析

１．引 言 考察如下广义中立型系统：其中,Ｌ,Ｍ,Ｎ ∈ Ｃｄ×ｄ为已知矩阵,　　　为已知向量值函数,　　　　　　　　　　当ｔ＞０时为未知函数,　　　　　　　　　　　　　　　　　　　　　　　　　为常数延时量． 对于　　　　　　　　　　　　　　　　　,１９６７年,Ｂｒａｙｔｏｎ［１］基于Ｌ,Ｍ,Ｎ为实对称矩阵,以及Ｉ± Ｎ和－Ｌ± Ｍ为正定矩阵时,讨论了（１）渐近稳定的充分条件;１９８４年,Ｊａｃｋｉｅｗｉｃｚ［２］基于 Ｌ,Ｍ,Ｎ为复系数时,研究了理论解的渐近稳定性及单步方法的数值稳定性;１９８８年,Ｂ…     

Stability of Runge-Kutta methods for the generalized pantograph equation

Summary. This paper deals with stability properties of Runge-Kutta (RK) methods applied to a non-autonomous delay differential equation (DDE) with a constant delay which is obtained from the so-called generalized pantograph equation, an autonomous DDE with a variable delay by a change of the independent variable. It is shown that in the case where the RK matrix is regular stability properties of the RK method for the DDE are derived from those for a difference equation, which are examined by similar techniques to those in the case of autonomous DDEs with a constant delay. As a result, it is shown that some RK methods based on classical quadrature have a superior stability property with respect to the generalized pantograph equation. Stability of algebraically stable natural RK methods is also considered. Received May 5, 1998 / Revised version received November 17, 1998 / Published online September 24, 1999     

含分布时滞的时滞微分系统多步龙格-库塔方法的时滞相关稳定性

本文研究了一类含分布时滞的时滞微分系统的多步龙格-库塔方法的稳定性.基于辐角原理，本文给出了多步龙格-库塔方法弱时滞相关稳定性的充分条件，并通过数值算例验证了理论结果的有效性.     

Stiff微分方程初值问题的一类数值方法

1.引言 实践表明,数值积分常微分方程初值问题 dx/dt=f(t,x), (1.1) x(t_0)=x_0时,若(1.1)是Stiff的,积分过程的稳定性是一个突出的问题.用传统的数值方法,比如Euler法,Adams法或Runge-Kutta法,为了保证计算稳定,积分步长受到相当地限制.即使运算速度为 100万次/秒的计算机,计算时间也将成为重大的负担.     

积分微分方程Runge-Kutta方法的散逸性

本文针对一类积分微分方程讨论Runge-Kutta方法的散逸性,当积分项用PQ公式逼近时,证明了(k,l)-代数稳定的Runge-Kutta方法是D(l)-散逸的.     

$\mathcal{H}$-Stability of Runge-Kutta Methods with Variable Stepsize for System of Pantograph Equations

This paper deals with H-stability of Runge-Kutta methods with variable stepsize for the system of pantograph equations. It is shown that both Runge-Kutta methods with nonsingular matrix coefficient A and stiffly accurate Runge-Kutta methods are H-stable if and only if the modulus of stability function at infinity is less than 1.     

On Stability of Symplectic Algorithms

1.FundamentalDeflnitionsLemma1.Thesolutionofalinearoofinarydtherentialequationwithcon8tantcoeffcientY=AYissta6leifalleigenvalue8ofAhaven0nP6sitivercalpartsandtheeigenvalueswithnullrealpartaresingleroots0ftheminimalp0lynomial.,/P\ThelinearHamiltoniansystemcanbeden0tedasZ=JSZwhereZ=(q),J=(ELs),andtheHamiltonianfuncti0nH(z)=ty.Lemma2.Thesolution80flinearHamiltoniansy8temsarecmticallysta6leifalleigenvaluesofJShavenullrsalpartandaresinglerootsojtheminitnalp0lyno?nial.Definiti0n1.Whenthemo…