Asymptotic stability of linear multistep methods for nonlinear neutral delay differential equations

This paper is concerned with the numerical solution to initial value problems of nonlinear delay differential equations of neutral type. We use A-stable linear multistep methods to compute the numerical solution. The asymptotic stability of the A-stable linear multistep methods when applied to the nonlinear delay differential equations of neutral type is investigated, and it is shown that the A-stable linear multistep methods with linear interpolation are GAS-stable. We validate our conclusions by numerical experiments.     

求解广义中立型系统的线性多步方法的NGP_G-稳定性

建立了广义中立型延迟系统理论解渐近稳定的充分条件 ,分析了用线性多步方法求解广义中立型延迟系统数值解的稳定性 ,在一定的Lagrange插值条件下 ,证明了数值求解广义中立型系统的线性多步方法NGPG_稳定的充分必要条件是线性多步方法是A_稳定的·     

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.     

Delay-Dependent Stability of Linear Multistep Methods for Neutral Systems with Distributed Delays

This paper considers the asymptotic stability of linear multistep (LM) methods for neutral systems with distributed delays. In particular, several sufficient conditions for delay-dependent stability of numerical solutions are obtained based on the argument principle. Compound quadrature formulae are used to compute the integrals. An algorithm is proposed to examine the delay-dependent stability of numerical solutions. Several numerical examples are performed to verify the theoretical results.     

一类线性多步法关于变延迟非线性中立型微分方程的渐近稳定性

1引言中立型微分方程广泛出现于生物学、物理学及工程技术等诸多领域.数值求解中立型微分方程时,数值方法的稳定性研究具有无容置疑的重要性,其中渐近稳定性的研究是其重要组成部分.对于线性中立型延迟微分方程,渐近稳定性研究已有许多重要结果,如文献[1,2,3,4,5,6]等.对于非线性中立型变延迟微分方程,数值方法的稳定性研究近几年才有进展.2000年,Bellen等在文献[7]中讨论了Runge-Kutta法求解一类特殊的中立型延迟微分     

Stability criteria for exact and discrete solutions of neutral multidelay-integro-differential equations

This paper deals with the asymptotic stability of exact and discrete solutions of neutral multidelay-integro-differential equations. Sufficient conditions are derived that guarantee the asymptotic stability of the exact solutions. Adaptations of classical Runge–Kutta and linear multistep methods are suggested for solving such systems with commensurate delays. Stability criteria are constructed for the asymptotic stability of these numerical methods and compared to the stability criteria derived for the continuous problem. It is found that, under suitable conditions, these two classes of numerical methods retain the stability of the continuous systems. Some numerical examples are given that illustrate the theoretical results. This research is supported by Fellowship F/02/019 of the Research Council of the K.U.Leuven, NSFC (No.10571066) and SRF for ROCS, SEM.     

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.     

Stability of implicit-explicit linear multistep methods for ordinary and delay differential equations

Stability properties of implicit-explicit (IMEX) linear multistep methods for ordinary and delay differential equations are analyzed on the basis of stability regions defined by using scalar test equations. The analysis is closely related to the stability analysis of the standard linear multistep methods for delay differential equations. A new second-order IMEX method which has approximately the same stability region as that of the IMEX Euler method, the simplest IMEX method of order 1, is proposed. Some numerical results are also presented which show superiority of the new method.      

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.     

多时滞微分方程数值稳定性

考虑了时滞微分方程的初值问题,分析了用线性多步法求解一类滞后型微分系统数值解的稳定性,在一定的Lagrange插值条件下,给出并证明了求解滞后型微分系统的线性多步法数值稳定的充分必要条件.     

Delay-dependent stability of linear multistep methods for DAEs with multiple delays

This paper aims to investigate the asymptotic stability of linear multistep (LM) methods for linear differential-algebraic equations (DAEs) with multiple delays. Based on the argument principle, we first establish the delay-dependent stability criteria of analytic solutions; then, we propose some practically checkable conditions for weak delay-dependent stability of numerical solutions derived by implicit LM methods. Lagrange interpolations are used to compute the delayed terms. Several numerical examples are given to illustrate the theoretical results.     

Reducible quadrature methods for Volterra integral equations of the first kind

Quadrature rules, generated by linear multistep methods for ordinary differential equations, are employed to construct a wide class of direct quadrature methods for the numerical solution of first kind Volterra integral equations. Our class covers several methods previously considered in the literature. The methods are convergent provided that both the first and second characteristic polynomial of the linear multistep method satisfy the root condition. Furthermore, the stability behaviour for fixed positive values of the stepsizeh is analyzed, and it turns out that convergence implies (fixedh) stability. The subclass formed by the backward differentiation methods up to order six is discussed and illustrated with numerical examples.     

Nonlinear Stability of General Linear Methods for Delay Differential Equations

This paper is devoted to a study of nonlinear stability of general linear methods for the numerical solution of delay differential equations in Hilbert spaces. New stability concepts are further introduced. The stability properties of (k,p,q)-algebraically stable general linear methods with piecewise constant or linear interpolation procedure are investigated. We also discuss stability of linear multistep methods viewed as a special subset of the class of general linear methods.     

Boundedness and Asymptotic Stability of Multistep Methods for Generalized Pantograph Equations

In this paper, we deal with the boundedness and the asymptotic stability of linear and one-leg multistep methods for generalized pantograph equations of neutral type, which arise from some fields of engineering. Some criteria of the boundedness and the asymptotic stability for the methods are obtained.     

广义时滞微分方程的渐近稳定性和数值分析

考虑了广义时滞微分方程的初值问题，分析了用线性多步法求解一类广义滞后型微分系统数值解的稳定性，在一定的Lagrange插值条件下，给出并证明了求解广义滞后型微分系统的线性多步法数值稳定的充分必要条件。     

Convergence of linear multistep and one-leg methods for stiff nonlinear initial value problems

To prove convergence of numerical methods for stiff initial value problems, stability is needed but also estimates for the local errors which are not affected by stiffness. In this paper global error bounds are derived for one-leg and linear multistep methods applied to classes of arbitrarily stiff, nonlinear initial value problems. It will be shown that under suitable stability assumptions the multistep methods are convergent for stiff problems with the same order of convergence as for nonstiff problems, provided that the stepsize variation is sufficiently regular.     

延迟积分微分方程线性多步方法的稳定性分析

考虑带常延迟的延迟积分微分方程线性系统零解的渐近稳定性，本文采用拉格朗日插值的线性多步方法，探讨了系统数值方法的线性稳定性。证明了所有A-稳定且强零-稳定的Pouzet型线性多步方法能够保持原线性系统的延迟不依赖稳定性。     

Inverse linear multistep methods for the numerical solution of initial value problems of second order differential equations

In this paper inverse linear multistep methods for the numerical solution of second order differential equations are presented. Local accuracy and stability of the methods are defined and discussed. The methods are applicable to a class of special second order initial value problems, not explicitly involving the first derivative. The methods are not convergent, but yield good numerical results if applied to problems they are designed for. Numerical results are presented for both the linear and nonlinear initial value problems.     

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.     

A boundary value approach to the numerical solution of initial value problems by multistep methos †

A boundary value appraoch to the numerical solution of initial value problems by means of linear multistep methods is presented. This theory is based on the study of linear difference equations when their general solution is computed by imposing boundary conditions. All the main stability and convergence properties of the obtained methods are investigated abd compared to those of the classical multistep methods. Then, as an example, new itegration formulas, called extended trapezoidal rules, are derived. For any order they have the same stability properties (in the sense of the definitions given in this paper) of the trapezoidal rule, which is the first method in this class. Some numerical examples are presented to confirm the theoretical expectations and to allow us to trust a future code based on boundary value methods.