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1.
This paper presents a new composition law for Runge-Kutta methods when applied to index-2 differential-algebraic systems. Applications of this result to the study of the order of composite methods and of symmetric methods are given. 相似文献
2.
S. Maset 《Numerische Mathematik》2002,90(3):555-562
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 相似文献
3.
R. Vermiglio 《Numerische Mathematik》1992,61(1):561-577
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) 相似文献
4.
On the asymptotic stability properties of Runge-Kutta methods for delay differential equations 总被引:5,自引:0,他引:5
Nicola Guglielmi 《Numerische Mathematik》1997,77(4):467-485
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 相似文献
5.
Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations 总被引:24,自引:0,他引:24
M. Zennaro 《Numerische Mathematik》1997,77(4):549-563
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 相似文献
6.
A stability property of A-stable collocation-based Runge-Kutta methods for neutral delay differential equations 总被引:6,自引:0,他引:6
Toshiyuki Koto 《BIT Numerical Mathematics》1996,36(4):855-859
We consider a linear homogeneous system of neutral delay differential equations with a constant delay whose zero solution is asymptotically stable independent of the value of the delay, and discuss the stability of collocation-based Runge-Kutta methods for the system. We show that anA-stable method preserves the asymptotic stability of the analytical solutions of the system whenever a constant step-size of a special form is used. 相似文献
7.
Evelyn Buckwar Martin G. Riedler 《Journal of Computational and Applied Mathematics》2011,236(6):1155-1182
In this paper we consider Runge-Kutta methods for jump-diffusion differential equations. We present a study of their mean-square convergence properties for problems with multiplicative noise. We are concerned with two classes of Runge-Kutta methods. First, we analyse schemes where the drift is approximated by a Runge-Kutta ansatz and the diffusion and jump part by a Maruyama term and second we discuss improved methods where mixed stochastic integrals are incorporated in the approximation of the next time step as well as the stage values of the Runge-Kutta ansatz for the drift. The second class of methods are specifically developed to improve the accuracy behaviour of problems with small noise. We present results showing when the implicit stochastic equations defining the stage values of the Runge-Kutta methods are uniquely solvable. Finally, simulation results illustrate the theoretical findings. 相似文献
8.
T. Koto 《BIT Numerical Mathematics》1994,34(2):262-267
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. 相似文献
9.
The infinite-delay-differential equations (IDDEs) are studied and the analytic solution of a class of nonlinear IDDEs is presented based on the characteristics of the reproducing kernel space W2[0,∞). Besides, the exact solution is represented in the form of series. It is proved that the n-term approximation un(x) converges to the exact solution u(x) of the IDDEs. Moreover, the approximate error of un(x) is monotone decreasing. The results of experiments showed that the proposed method in this paper is computationally efficient. 相似文献
10.
Many systems of ordinary differential equations are quadratic: the derivative can be expressed as a quadratic function of the dependent variable. We demonstrate that this feature can be exploited in the numerical solution by Runge-Kutta methods, since the quadratic structure serves to decrease the number of order conditions. We discuss issues related to construction design and implementation and present a number of new methods of Runge-Kutta and Runge-Kutta-Nyström type that display superior behaviour when applied to quadratic ordinary differential equations. 相似文献
11.
Ch. Lubich 《BIT Numerical Mathematics》1991,31(3):545-550
Ascher and Petzold recently introducedprojected Runge-Kutta methods for the numerical solution of semi-explicit differential-algebraic systems of index 2. Here it is shown that such a method can be regarded as the limiting case of a standard application of a Runge-Kutta method with a very small implicit Euler step added to it. This interpretation allows a direct derivation of the order conditions and of superconvergence results for the projected methods from known results for standard Runge-Kutta methods for index-2 differential-algebraic systems, and an extension to linearly implicit differential-algebraic systems. 相似文献
12.
J. C. Butcher 《BIT Numerical Mathematics》1975,15(4):358-361
A class of implicit Runge-Kutta methods is shown to possess a stability property which is a natural extension of the notion ofA-stability for non-linear systems. 相似文献
13.
14.
The present paper is devoted to a study of nonlinear stability of discontinuous Galerkin methods for delay differential equations. Some concepts, such as global and analogously asymptotical stability are introduced. We derive that discontinuous Galerkin methods lead to global and analogously asymptotical stability for delay differential equations. And these nonlinear stability properties reveal to the reader the relation between the perturbations of the numerical solution and that of the initial value or the systems. 相似文献
15.
Partitioned adaptive Runge-Kutta methods and their stability 总被引:4,自引:0,他引:4
Summary This paper deals with the solution of partitioned systems of nonlinear stiff differential equations. Given a differential system, the user may specify some equations to be stiff and others to be nonstiff. For the numerical solution of such a system partitioned adaptive Runge-Kutta methods are studied. Nonstiff equations are integrated by an explicit Runge-Kutta method while an adaptive Runge-Kutta method is used for the stiff part of the system.The paper discusses numerical stability and contractivity as well as the implementation and usage of such compound methods. Test results for three partitioned stiff initial value problems for different tolerances are presented. 相似文献
16.
The theory of positive real functions is used to provide bounds for the largest possible disk to be inscribed in the stability region of an explicit Runge-Kutta method. In particular, we show that the closed disk |+r| r can be contained in the stability region of an explicitm-stage Runge-Kutta method of order two if and only ifr m – 1. 相似文献
17.
The stability region of an explicit and consistentm-stage Runge-Kutta method cannot contain the closed disk with diameter [–2m, 0] as a proper subset. 相似文献
18.
The polynomial associated with the largest disk of stability of anm-stage explict Runge-Kutta method of orderp is unique. 相似文献
19.
S. Maset 《Numerische Mathematik》2000,87(2):355-371
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 相似文献
20.
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. 相似文献