CONVERGENCE OF PARALLEL DIAGONAL ITERATION OF RUNGE-KUTTA METHODS FOR DELAY DIFFERENTIAL EQUATIONS

Implicit Runge-Kutta method is highly accurate and stable for stiff initial value prob-lem.But the iteration technique used to solve implicit Runge-Kutta method requires lotsof computational efforts.In this paper,we extend the Parallel Diagonal Iterated Runge-Kutta(PDIRK)methods to delay differential equations(DDEs).We give the convergenceregion of PDIRK methods,and analyze the speed of convergence in three parts for theP-stability region of the Runge-Kutta corrector method.Finally,we analysis the speed-upfactor through a numerical experiment.The results show that the PDIRK methods toDDEs are efficient.     

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.     

B-Theory of Runge-Kutta methods for stiff volterra functional differential equations

B-stability andB-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra functional differential equations (VFDEs) are established which provide unified theoretical foundation for the study of Runge-Kutta methods when applied to nonlinear stiff initial value problems (IVPs) in ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs of other type which appear in practice.     

On the asymptotic stability properties of Runge-Kutta methods for delay differential equations

Summary. In this paper asymptotic stability properties of Runge-Kutta (R-K) methods for delay differential equations (DDEs) are considered with respect to the following test equation: where and is a continuous real-valued function. In the last few years, stability properties of R-K methods applied to DDEs have been studied by numerous authors who have considered regions of asymptotic stability for “any positive delay” (and thus independent of the specific value of ). In this work we direct attention at the dependence of stability regions on a fixed delay . In particular, natural Runge-Kutta methods for DDEs are extensively examined. Received April 15, 1996 / Revised version received August 8, 1996     

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.     

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…     

D-CONVERGENCE OF RUNGE-KUTTA METHODS FOR STIFF DELAY DIFFERENTIAL EQUATIONS

1. IntroductionWhen considering the applicability of numerical methods for the solution of the delay differential equation (DDE) y'(t) = f(t, y(t), y(t - T)), it is necessary to analyze the error behaviourof the methods. In fact, many papers have investigated the local and global error behaviour ofDDE solvers (cL[1,2,14]). These error analyses are based on the assumption that the fUnctionf(t,y,z) satisfies Lipschitz conditions in both the last two variables. They are suitable fornonstiff …     

A review of theoretical and numerical analysis for nonlinear stiff Volterra functional differential equations

In this review, we present the recent work of the author in comparison with various related results obtained by other authors in literature. We first recall the stability, contractivity and asymptotic stability results of the true solution to nonlinear stiff Volterra functional differential equations (VFDEs), then a series of stability, contractivity, asymptotic stability and B-convergence results of Runge-Kutta methods for VFDEs is presented in detail. This work provides a unified theoretical foundation for the theoretical and numerical analysis of nonlinear stiff problems in delay differential equations (DDEs), integro-differential equations (IDEs), delayintegro-differential equations (DIDEs) and VFDEs of other type which appear in practice.      

B-Theory of Runge-Kutta methods for stiff Volterra functional differential equations

B-stability and B-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra func-tional differential equations(VFDEs)are established which provide unified theoretical foundation for the studyof Runge-Kutta methods when applied to nonlinear stiff initial value problems(IVPs)in ordinary differentialequations(ODEs),delay differential equations(DDEs),integro-differential equatioons(IDEs)and VFDEs of     

New higher-order numerical one-step methods based on the Adomian and the modified decomposition methods

We develop new, higher-order numerical one-step methods and apply them to several examples to investigate approximate discrete solutions of nonlinear differential equations. These new algorithms are derived from the Adomian decomposition method (ADM) and the Rach-Adomian-Meyers modified decomposition method (MDM) to present an alternative to such classic schemes as the explicit Runge-Kutta methods for engineering models, which require high accuracy with low computational costs for repetitive simulations in contrast to a one-size-fits-all philosophy. This new approach incorporates the notion of analytic continuation, which extends the region of convergence without resort to other techniques that are also used to accelerate the rate of convergence such as the diagonal Padé approximants or the iterated Shanks transforms. Hence global solutions instead of only local solutions are directly realized albeit in a discretized representation. We observe that one of the difficulties in applying explicit Runge-Kutta one-step methods is that there is no general procedure to generate higher-order numeric methods. It becomes a time-consuming, tedious endeavor to generate higher-order explicit Runge-Kutta formulas, because it is constrained by the traditional Picard formalism as used to represent nonlinear differential equations. The ADM and the MDM rely instead upon Adomian’s representation and the Adomian polynomials to permit a straightforward universal procedure to generate higher-order numeric methods at will such as a 12th-order or 24th-order one-step method, if need be. Another key advantage is that we can easily estimate the maximum step-size prior to computing data sets representing the discretized solution, because we can approximate the radius of convergence from the solution approximants unlike the Runge-Kutta approach with its intrinsic linearization between computed data points. We propose new variable step-size, variable order and variable step-size, variable order algorithms for automatic step-size control to increase the computational efficiency and reduce the computational costs even further for critical engineering models.     

Stability of the Radau IA and Lobatto IIIC methods for delay differential equations

Recently, we have proved that the Radau IA and Lobatto IIIC methods are P-stable, i.e., they have an analogous stability property to A-stability with respect to scalar delay differential equations (DDEs). In this paper, we study stability of those methods applied to multidimensional DDEs. We show that they have a similar property to P-stability with respect to multidimensional equations which satisfy certain conditions for asymptotic stability of the zero solutions. The conditions are closely related to stability criteria for DDEs considered in systems theory. Received October 8, 1996 / Revised version received February 21, 1997     

Reversible methods of Runge-Kutta type for Index-2 DAEs

Summary. A new interpretation of Runge-Kutta methods for differential algebraic equations (DAEs) of index 2 is presented, where a step of the method is described in terms of a smooth map (smooth also with respect to the stepsize). This leads to a better understanding of the convergence behavior of Runge-Kutta methods that are not stiffly accurate. In particular, our new framework allows for the unified study of two order-improving techniques for symmetric Runge-Kutta methods (namely post-projection and symmetric projection) specially suited for solving reversible index-2 DAEs.Mathematics Subject Classification (1991): 65L05, 65L06     

Dissipativity of Runge-Kutta methods for Volterra functional differential equations

This paper is concerned with the numerical dissipativity of nonlinear Volterra functional differential equations (VFDEs). We give some dissipativity results of Runge-Kutta methods when they are applied to VFDEs. These results provide unified theoretical foundation for the numerical dissipativity analysis of systems in ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs), Volterra delay integro-differential equations (VDIDEs) and VFDEs of other type which appear in practice. Numerical examples are given to confirm our theoretical results.     

Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations

Summary. We consider systems of delay differential equations (DDEs) of the form with the initial condition . Recently, Torelli [10] introduced a concept of stability for numerical methods applied to dissipative nonlinear systems of DDEs (in some inner product norm), namely RN-stability, which is the straighforward generalization of the wellknown concept of BN-stability of numerical methods with respect to dissipative systems of ODEs. Dissipativity means that the solutions and corresponding to different initial functions and , respectively, satisfy the inequality , and is guaranteed by suitable conditions on the Lipschitz constants of the right-hand side function . A numerical method is said to be RN-stable if it preserves this contractivity property. After showing that, under slightly more stringent hypotheses on the Lipschitz constants and on the delay function , the solutions and are such that , in this paper we prove that RN-stable continuous Runge-Kutta methods preserve also this asymptotic stability property. Received March 29, 1996 / Revised version received August 12, 1996     

Convergence of parallel multistep hybrid methods for singular perturbation problems

Parallel multistep hybrid methods (PHMs) can be implemented in parallel with two processors, accordingly have almost the same computational speed per integration step as BDF methods of the same order with the same stepsize. But PHMs have better stability properties than BDF methods of the same order for stiff differential equations. In the present paper, we give some results on error analysis of A(α)-stable PHMs for the initial value problems of ordinary differential equations in singular perturbation form. Our convergence results are similar to those of linear multistep methods (such as BDF methods), i.e. the convergence orders are equal to their classical convergence orders, and no order reduction occurs. Some numerical examples also confirm our results.     

非线性中立型泛函微分方程Runge-Kutta 法的稳定性和收敛性

本文涉及Runge-Kutta 法变步长求解非线性中立型泛函微分方程(NFDEs) 的稳定性和收敛性.为此, 基于Volterra 泛函微分方程Runge-Kutta 方法的B- 理论, 引入了中立型泛函微分方程Runge-Kutta 方法的EB (expanded B-theory)-稳定性和EB-收敛性概念. 之后获得了Runge-Kutta 方法变步长求解此类方程的EB - 稳定性和EB- 收敛性. 这些结果对中立型延迟微分方程和中立型延迟积分微分方程也是新的.     

Discrete Jacobi sub‐equation method for nonlinear differential–difference equations

We will propose a unified algebraic method to construct Jacobi elliptic function solutions to differential–difference equations (DDEs). The solutions to DDEs in terms of Jacobi elliptic functions sn, cn and dn have a unified form and can be presented through solving the associated algebraic equations. To illustrate the effectiveness of this method, we apply the algorithm to some physically significant DDEs, including the discrete hybrid equation, semi‐discrete coupled modified Korteweg–de Vries and the discrete Klein–Gordon equation, thereby generating some new exact travelling periodic solutions to the discrete Klein–Gordon equation. A procedure is also given to determine the polynomial expansion order of Jacobi elliptic function solutions to DDEs. Copyright © 2010 John Wiley & Sons, Ltd.     

Convergence and stability results in Runge-Kutta type methods for Volterra integral equations of the second kind

Runge-Kutta type methods for Volterra integral equations of the second kind are introduced which contain additional terms in order to improve the stability behaviour. The order of convergence is given and the stability region is derived for the basic test kernelK=af.     

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.     

Global error estimates for a uniparametric family of stiffly accurate Runge-Kutta collocation methods on singularly perturbed problems

An error analysis for a newly defined uniparametric family of stiffly accurate Runge-Kutta collocation methods when applied to initial value problems for singularly perturbed differential equations is carried out. The so-called SAFERK methods possess a first internal stage of explicit type and are based on collocation nodes. Sharp convergence results are obtained for these methods through the analysis of a sequence of higher index Differential Algebraic Equations. A numerical test with the Van der Pol oscillator reveals that the proposed error estimates are realistic whenever the stepsize h is large enough compared to the stiffness parameter ε.