Estimation of numerically stable step-size for neutral delay-differential equations via spectral radius

We derive the estimates of numerically stable step-size for systems of neutral delay-differential equations (NDDEs), which only need to be calculated the spectral radius of the corresponding matrices. The stable step-size for numerical integration of NDDEs can be easily selected by means of the estimates. The stability regions of both linear multistep methods and explicit Runge-Kutta methods are presented.     

NONLINEAR STABILITY OF NATURAL RUNGE-KUTTA METHODS FOR NEUTRAL DELAY DIFFERENTIAL EQUATIONS

This paper first presents the stability analysis of theoretical solutions for a class of nonlinear neutral delay-differential equations (NDDEs). Then the numerical analogous results, of the natural Runge-Kutta (NRK) methods for the same class of nonlinear NDDEs, are given. In particular, it is shown that the (k, l)-algebraic stability of a RK method for ODEs implies the generalized asymptotic stability and the global stability of the induced NRK method.     

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.     

Delay-dependent stability analysis of multistep methods for delay differential equations

This paper deals with the delay-dependent stability of numerical methods for delay differential equations. First, a stability criterion of Runge-Kutta methods is extended to the case of general linear methods. Then, linear multistep methods are considered and a class of r（0）-stable methods are found. Later, some examples of r（0）-stable multistep multistage methods are given. Finally, numerical experiments are presented to confirm the theoretical results.     

Estimations on numerically stable step-size for neutral delay differential systems with multiple delays

We derive two estimations of numerically stable step-size for systems of neutral delay differential equations with multiple delays. The stable step-size for numerical integration of NDDEs with multiple delays can be easily selected by means of the logarithmic norm and the spectral radius of certain matrices. Both explicit linear multistep methods and explicit Runge-Kutta methods are considered.     

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.     

Numerical Stability and Oscillations of Runge-Kutta Methods for Differential Equations with Piecewise Constant Arguments of Advanced Type

For differential equations with piecewise constant arguments of advanced type, numerical stability and oscillations of Runge-Kutta methods are investigated. The necessary and sufficient conditions under which the numerical stability region contains the analytic stability region are given. The conditions of oscillations for the Runge-Kutta methods are obtained also. We prove that the Runge-Kutta methods preserve the oscillations of the analytic solution. Moreover, the relationship between stability and oscillations is discussed. Several numerical examples which confirm the results of our analysis are presented.     

数值求解NDDEs系统的单支方法的非线性稳定性

该文探讨了单支方法关于一类中立型延迟微分方程（ＮＤＤＥｓ）系统的整体稳定性和渐近稳定性．在适当的条件下，获得了单支方法关于ＮＤＤＥｓ系统的一些新的非线性稳定性判据．     

On the Contractivity and Asymptotic Stability of Systems of Delay Differential Equations of Neutral Type

In this paper we investigate both the contractivity and the asymptotic stability of the solutions of linear systems of delay differential equations of neutral type (NDDEs) of the form y(t) = Ly(t) + M(t)y(t – (t)) + N(t)y(t – (t)). Asymptotic stability properties of numerical methods applied to NDDEs have been recently studied by numerous authors. In particular, most of the obtained results refer to the constant coefficient version of the previous system and are based on algebraic analysis of the associated characteristic polynomials. In this work, instead, we play on the contractivity properties of the solutions and determine sufficient conditions for the asymptotic stability of the zero solution by considering a suitable reformulation of the given system. Furthermore, a class of numerical methods preserving the above-mentioned stability properties is also presented.     

Two-Stage Stochastic Runge-Kutta Methods for Stochastic Differential Equations

In this paper we discuss two-stage diagonally implicit stochastic Runge-Kutta methods with strong order 1.0 for strong solutions of Stratonovich stochastic differential equations. Five stochastic Runge-Kutta methods are presented in this paper. They are an explicit method with a large MS-stability region, a semi-implicit method with minimum principal error coefficients, a semi-implicit method with a large MS-stability region, an implicit method with minimum principal error coefficients and another implicit method. We also consider composite stochastic Runge-Kutta methods which are the combination of semi-implicit Runge-Kutta methods and implicit Runge-Kutta methods. Two composite methods are presented in this paper. Numerical results are reported to compare the convergence properties and stability properties of these stochastic Runge-Kutta methods.     

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.     

关于辛算法稳定性的若干注记

我们讨论辛算法的线性稳定性和非线性稳定性,从动力系统和计算的角度论述了研究辛算法的这两类稳定性问题的重要性,分析总结了相关重要结果.我们给出了解析方法的明确定义,证明了稳定函数是亚纯函数的解析辛方法是绝对线性稳定的.绝对线性稳定的辛方法既有解析方法（如Runge-Kutta辛方法）,也有非解析方法（如基于常数变易公式对线性部分进行指数积分而对非线性部分使用其它数值积分的方法）.我们特别回顾并讨论了R.I.McLachlan,S.K.Gray和S.Blanes,F.Casas,A.Murua等关于分裂算法的线性稳定性结果,如通过选取适当的稳定多项式函数构造具有最优线性稳定性的任意高阶分裂辛算法和高效共轭校正辛算法,这类经优化后的方法应用于诸如高振荡系统和波动方程等线性方程或者线性主导的弱非线性方程具有良好的数值稳定性.我们通过分析辛算法在保持椭圆平衡点的稳定性,能量面的指数长时间慢扩散和KAM不变环面的保持等三个方面阐述了辛算法的非线性稳定性,总结了相关已有结果.最后在向后误差分析基础上,基于一个自由度的非线性振子和同宿轨分析法讨论了辛算法的非线性稳定性,提出了一个新的非线性稳定性概念,目的是为辛算法提供一个实际可用的非线性稳定性判别法.     

A CLASS OF TWO-STEP CONTINUITY RUNGE-KUTTA METHODS FOR SOLVING SINGULAR DELAY DIFFERENTIAL EQUATIONS AND ITS STABILITY ANALYSIS

In this paper, a class of two-step continuity Runge-Kutta（TSCRK） methods for solving singular delay differential equations（DDEs） is presented. Analysis of numerical stability of this methods is given. We consider the two distinct cases： （i）τ≥ h, （ii）τ 〈 h, where the delay τ and step size h of the two-step continuity Runge-Kutta methods are both constant. The absolute stability regions of some methods are plotted and numerical examples show the efficiency of the method.     

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…     

On the practical value of the notion ofBN-stability

The oldest concept of unconditional stability of numerical integration methods for ordinary differential systems is that ofA-stability. This concept is related to linear systems having constant coefficients and has been introduced by Dahlquist in 1963. More recently, since another contribution of Dahlquist in 1975, there has been much interest in unconditional stability properties of numerical integration methods when applied to non-linear dissipative systems (G-stability,BN-stability,A-contractivity). Various classes of implicit Runge-Kutta methods have already been shown to beBN-stable. However, contrary to the property ofA-stability, when implementing such a method for practical use this unconditional stability property may be lost. The present note clarifies this for a class of diagonally implicit methods and shows at the same time that Rosenbrock's method is notBN-stable.     

The NGP-stability of Runge-Kutta methods for systems of neutral delay differential equations

Summary. This paper deals with the stability analysis of implicit Runge-Kutta methods for the numerical solutions of the systems of neutral delay differential equations. We focus on the behavior of such methods with respect to the linear test equations where ,L, M and N are complex matrices. We show that an implicit Runge-Kutta method is NGP-stable if and only if it is A-stable. Received February 10, 1997 / Revised version received January 5, 1998     

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．     

A stability property ofA-stable natural Runge-Kutta methods for systems of delay differential equations

A natural Runge-Kutta method is a special type of Runge-Kutta method for delay differential equations (DDEs); it is known that any one-step collocation method is equivalent to one of such methods. In this paper, we consider a linear constant-coefficient system of DDEs with a constant delay, and discuss the application of natural Runge-Kutta methods to the system. We show that anA-stable method preserves the asymptotic stability property of the analytical solutions of the system.     

The equivalence of algebraic stability andAN-stability

It is shown that, under certain restrictions,AN-stability is equivalent to algebraic stability for general linear methods. The restrictions have the purpose of excluding from consideration methods which can be replaced by simpler methods in various specific ways andAN-stability is to be interpreted in the strong sense. This result generalizes known results for Runge-Kutta and for one-leg methods.     

Unconditionally stable explicit methods for parabolic equations

Summary This paper discussesrational Runge-Kutta methods for stiff differential equations of high dimensions. These methods are explicit and in addition do not require the computation or storage of the Jacobian. A stability analysis (based onn-dimensional linear equations) is given. A second orderA ₀-stable method with embedded error control is constructed and numerical results of stiff problems originating from linear and nonlinear parabolic equations are presented.