NP-stability of Runge-Kutta methods based on classical quadrature

Recently Bellen, Jackiewicz and Zennaro have studied stability properties of Runge-Kutta (RK) methods for neutral delay differential equations using a scalar test equation. In particular, they have shown that everyA-stable collocation method isNP-stable, i.e., the method has an analogous stability property toA-stability with respect to the test equation. Consequently, the Gauss, Radau IIA and Lobatto IIIA methods areNP-stable. In this paper, we examine the stability of RK methods based on classical quadrature by a slightly different approach from theirs. As a result, we prove that the Radau IA and Lobatto IIIC methods equipped with suitable continuous extensions are alsoNP-stable by virtue of fundamental notions related to those methods such as simplifying conditions, algebraic stability, and theW-transformation.     

A criterion forP-stability properties of Runge-Kutta methods

P-stability is an analogous stability property toA-stability with respect to delay differential equations. It is defined by using a scalar test equation similar to the usual test equation ofA-stability. EveryP-stable method isA-stable, but anA-stable method is not necessarilyP-stable. We considerP-stability of Runge-Kutta (RK) methods and its variation which was originally introduced for multistep methods by Bickart, and derive a sufficient condition for an RK method to have the stability properties on the basis of an algebraic characterization ofA-stable RK methods recently obtained by Schere and Müller. By making use of the condition we clarify stability properties of some SIRK and SDIRK methods, which are easier to implement than fully implicit methods, applied to delay differential equations.     

Stability analysis of block boundary value methods for neutral pantograph equation

This paper deals with the convergence and stability properties of block boundary value methods (BBVMs) for the neutral pantograph equation. Due to its unbounded time lags and limited computer memory, a change in the independent variable is used to transform a pantograph equation into a non-autonomous differential equation with a constant delay but variable coefficients. It is shown under the classical Lipschitz condition that a BBVM is convergent of order p if the underlying boundary value method is consistent with order p. Furthermore, it is proved under a certain condition that BBVMs can preserve the asymptotic stability of exact solutions for the neutral pantograph equation. Meanwhile, some numerical experiments are given to confirm the main conclusions.     

Runge-Kutta方法关于无穷延迟系统的稳定性分析

主要针对无穷延迟Pantograph方程构造了Runge-Kutta数值方法,并讨论了此方法在一定的条件下是p-稳定的和弱p-稳定的.     

Asymptotic Stability of Runge-Kutta Methods for the Pantograph Equations

This paper considers the asymptotic stability analysis of both exact and numerical solutions of the following neutral delay differential equation with pantograph delay.where $B,C,D\in C^{d\times d},q\in (0,1)$,and $B$ is regular. After transforming the above equation to non-automatic neutral equation with constant delay, we determine sufficient conditions for the asymptotic stability of the zero solution. Furthermore, we focus on the asymptotic stability behavior of Runge-Kutta method with variable stepsize. It is proved that a L-stable Runge-Kutta method can preserve the above-mentioned stability properties.     

The composite Milstein methods for the numerical solution of Ito stochastic differential equations

In this paper, we present the composite Milstein methods for the strong solution of Ito stochastic differential equations. These methods are a combination of semi-implicit and implicit Milstein methods. We give a criterion for choosing either the implicit or the semi-implicit scheme at each step of our numerical solution. The stability and convergence properties are investigated and discussed for the linear test equation. The convergence properties for the nonlinear case are shown numerically to be the same as the linear case. The stability properties of the composite Milstein methods are found to be more superior compared to those of the Milstein, the Euler and even better than the composite Euler method. This superiority in stability makes the methods a better candidate for the solution of stiff SDEs.     

Stability of Runge-Kutta methods for linear delay differential equations

Summary. This paper investigates the stability of Runge-Kutta methods when they are applied to the complex linear scalar delay differential equation . This kind of stability is called stability. We give a characterization of stable Runge-Kutta methods and then we prove that implicit Euler method is stable. Received November 3, 1998 / Revised version received March 23, 1999 / Published online July 12, 2000     

Invariant curves for variable step size integrators

The behaviour of one-step methods with variable step size applied to is investigated. Usually the step size for the current step depends on one or several previous steps. However, under some natural assumptions it can be shown that the step size asymptotically depends only on the locationx. This allows to introduce anx-dependent time transformation taking a variable step size method to a constant step-size method. By means of such a transformation general properties of constant step size methods carry over to variable step size methods. This is used to show that if the differential equation admits a hyperbolic periodic solution the variable step size method admits an invariant closed curve near the orbit of the periodic solution.The first author was partially supported by NSF Grant DMS87-19952 during his stay at UCLA.     

Computing stability regions--Runge-Kutta methods for delay differential equations

We discuss the practical determination of stability regionswhen various fixed-stepsize Runge-Kutta (RK) methods, combinedwith continuous extensions, are applied to the linear delaydifferential equation (DDE) y'(t)= y(t)+µ(t–) (t) with fixed delay . It is significant that the delay is not limitedto an integer multiple of the stepsize, and that we considervarious continuous extensions. The stability loci obtained in practice indicate that the standardboundary-locus technique (BLT) can fail to map the RK DDE stabilityregion correctly. The aim of this paper is to present an alternativestability boundary algorithm that overcomes the difficultiesencountered using the BLT. This new algorithm can be used forboth explicit and implicit RK methods.     

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     

Stability of collocation methods for delay differential equations with vanishing delays

We analyze the asymptotic stability of collocation solutions in spaces of globally continuous piecewise polynomials on uniform meshes for linear delay differential equations with vanishing proportional delay qt (0<q<1) (pantograph DDEs). It is shown that if the collocation points are such that the analogous collocation solution for ODEs is A-stable, then this asymptotic behaviour is inherited by the collocation solution for the pantograph DDE.     

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.     

Structure preservation of exponentially fitted Runge–Kutta methods

The preservation of some structure properties of the flow of differential systems by numerical exponentially fitted Runge–Kutta (EFRK) methods is considered. A complete characterisation of EFRK methods that preserve linear or quadratic invariants is given and, following the approach of Bochev and Scovel [On quadratic invariants and symplectic structure, BIT 34 (1994) 337–345], the sufficient conditions on symplecticity of EFRK methods derived by Van de Vyver [A fourth-order symplectic exponentially fitted integrator, Comput. Phys. Comm. 174 (2006) 255–262] are obtained. Further, a family of symplectic EFRK two-stage methods with order four has been derived. It includes the symplectic EFRK method proposed by Van de Vyver as well as a collocation method at variable nodes that can be considered as the natural collocation extension of the classical RK Gauss method. Finally, the results of some numerical experiments are presented to compare the relative merits of several fitted and nonfitted fourth-order methods in the integration of oscillatory systems.     

A new Jacobi rational–Gauss collocation method for numerical solution of generalized pantograph equations

This paper is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument. In this article, a new spectral collocation method is applied to solve the generalized pantograph equation with variable coefficients on a semi-infinite domain. This method is based on Jacobi rational functions and Gauss quadrature integration. The Jacobi rational-Gauss method reduces solving the generalized pantograph equation to a system of algebraic equations. Reasonable numerical results are obtained by selecting few Jacobi rational–Gauss collocation points. The proposed Jacobi rational–Gauss method is favorably compared with other methods. Numerical results demonstrate its accuracy, efficiency, and versatility on the half-line.     

Convergence of a class of runge-kutta methods for differential-algebraic systems of index 2

This paper deals with convergence results for a special class of Runge-Kutta (RK) methods as applied to differential-algebraic equations (DAE's) of index 2 in Hessenberg form. The considered methods are stiffly accurate, with a singular RK matrix whose first row vanishes, but which possesses a nonsingular submatrix. Under certain hypotheses, global superconvergence for the differential components is shown, so that a conjecture related to the Lobatto IIIA schemes is proved. Extensions of the presented results to projected RK methods are discussed. Some numerical examples in line with the theoretical results are included.     

A sufficient condition for GPN-stability for delay differential equations

Summary In this paper the author considers a linear test delay differential equation with non-constant coefficients related to the definitions of PN and GPN-stability for numerical methods. He defines stability properties for an ordinary differential equation together with stability properties of interpolants for numerical methods and in this way he gives sufficient conditions for GPN-stability.This work was supported by the Italian M.P.I. (funds 40%) and by C.N.R.     

Stability of some boundary value methods for the solution of initial value problems

The stability properties of three particular boundary value methods (BVMs) for the solution of initial value problems are considered. Our attention is focused on the BVMs based on the midpoint rule, on the Simpson method and on an Adams method of order 3. We investigate their BV-stability regions by considering the scalar test problem and constant stepsize. The study of the conditioning of the coefficient matrix of the discrete problem is extended to the case of variable stepsize and block ODE problems. We also analyse an appropriate choice for the stepsize for stiff problems. Numerical tests are reported to evidentiate the effectiveness of the BVMs and the differences among the BVMs considered.Work supported by the Ministero della Ricerca Scientifica, 40% project, and C.N.R. (contract of research # 92.00535.01).     

Asymptotical Stability of Numerical Methods with Constant Stepsize for Pantograph Equations

In this paper, the asymptotical stability of the analytic solution and the numerical methods with constant stepsize for pantograph equations is investigated using the Razumikhin technique. In particular, the linear pantograph equations with constant coefficients and variable coefficients are considered. The stability conditions of the analytic solutions of those equations and the numerical solutions of the θ-methods with constant stepsize are obtained. As a result Z. Jackiewicz’s conjecture is partially proved. Finally, some experiments are given. AMS subject classification (2000) 65L02, 65L05, 65L20     

Stability of implicit Runge-Kutta methods for nonlinear stiff differential equations

Under the assumption that an implicit Runge-Kutta method satisfies a certain stability estimate for linear systems with constant coefficientsl ₂-stability for nonlinear systems is proved. This assumption is weaker than algebraic stability since it is satisfied for many methods which are not evenA-stable. Some local smoothness in the right hand side of the differential equation is needed, but it may have a Jacobian and higher derivatives with large norms. The result is applied to a system derived from a strongly nonlinear parabolic equation by the method of lines.     

On the stability of Runge-Kutta methods for delay integral equations

Summary We present a class of Runge-Kutta methods for the numerical solution of a class of delay integral equations (DIEs) described by two different kernels and with a fixed delay . The stability properties of these methods are investigated with respect to a test equation with linear kernels depending on complex parameters. The results are then applied to collocation methods. In particular we obtain that any collocation method for DIEs, resulting from anA-stable collocation method for ODEs, with a stepsize which is submultiple of the delay , preserves the asymptotic stability properties of the analytic solutions.This work was supported by CNR (Italian National Council of Research)