积分微分方程Runge-Kutta方法的散逸性

本文针对一类积分微分方程讨论Runge-Kutta方法的散逸性,当积分项用PQ公式逼近时,证明了(k,l)-代数稳定的Runge-Kutta方法是D(l)-散逸的.     

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.     

Dissipativity of multistep runge-kutta methods for nonlinear volterra delay-integro-differential equations

This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations.We investigate the dissipativity properties of (k,l)algebraically stable multistep Runge-Kutta methods with constrained grid and an uniform grid.The finitedimensional and infinite-dimensional dissipativity results of (k,l)-algebraically stable Runge-Kutta methods are obtained.     

On fifth order Runge-Kutta methods

A 6 stage Runge-Kutta method is derived with the property that its order is 5 when used to solve a scalar differential equation but only 4 when used to solve a general system of differential equations. The existence of such a method underlines the necessity of carrying out theoretical analyses in a vector valued setting rather than in a one-dimensional setting as in the work of Kutta and some more recent authors.This research was supported by the New Zealand Foundation for Research, Science and Technology     

Dissipativity analysis for singular systems with time-varying delays

This paper is concerned with the dissipativity analysis problem for singular systems with time-varying delays. A delay-dependent criterion is established to guarantee the dissipativity of the underlying systems using the delay partitioning technique. All the results given in this paper are not only dependent upon the time delay, but also dependent upon the number of delay partitions. The effectiveness and the reduced conservatism of the derived results are demonstrated by two illustrative examples.     

Parallel linear system solvers for Runge-Kutta methods

If the nonlinear systems arising in implicit Runge-Kutta methods like the Radau IIA methods are iterated by (modified) Newton, then we have to solve linear systems whose matrix of coefficients is of the form I-A hJ with A the Runge-Kutta matrix and J an approximation to the Jacobian of the righthand side function of the system of differential equations. For larger systems of differential equations, the solution of these linear systems by a direct linear solver is very costly, mainly because of the LU-decompositions. We try to reduce these costs by solving the linear systems by a second (inner) iteration process. This inner iteration process is such that each inner iteration again requires the solution of a linear system. However, the matrix of coefficients in these new linear systems is of the form I - B hJ where B is similar to a diagonal matrix with positive diagonal entries. Hence, after performing a similarity transformation, the linear systems are decoupled into s subsystems, so that the costs of the LU-decomposition are reduced to the costs of s LU-decompositions of dimension d. Since these LU-decompositions can be computed in parallel, the effective LU-costs on a parallel computer system are reduced by a factor s ³ . It will be shown that matrices B can be constructed such that the inner iterations converge whenever A and J have their eigenvalues in the positive and nonpositive halfplane, respectively. The theoretical results will be illustrated by a few numerical examples. A parallel implementation on the four-processor Cray-C98/4256 shows a speed-up ranging from at least 2.4 until at least 3.1 with respect to RADAU5 applied in one-processor mode.     

A composition law for Runge-Kutta methods applied to index-2 differential-algebraic equations

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.     

Lie-Butcher theory for Runge-Kutta methods

Runge-Kutta methods are formulated via coordinate independent operations on manifolds. It is shown that there is an intimate connection between Lie series and Lie groups on one hand and Butcher's celebrated theory of order conditions on the other. In Butcher's theory the elementary differentials are represented as trees. In the present formulation they appear as commutators between vector fields. This leads to a theory for the order conditions, which can be developed in a completely coordinate free manner. Although this theory is developed in a language that is not widely used in applied mathematics, it is structurally simple. The recursion for the order conditions rests mainly on three lemmas, each with very short proofs. The techniques used in the analysis are prepared for studying RK-like methods on general Lie groups and homogeneous manifolds, but these themes are not studied in detail within the present paper.     

Maximal order for second derivative general linear methods with Runge-Kutta stability

An extension of general linear methods (GLMs), so-called SGLMs (GLMs with second derivative), was introduced to the case in which second derivatives, as well as first derivatives, can be calculated. SGLMs are divided into four types, depending on the nature of the differential system to be solved and the computer architecture that is used to implement these methods. In this paper, we obtain maximal order for two types of SGLMs with Runge-Kutta stability (RKS) property. Also, we construct methods of these types which possess RKS property and A-stability. Efficiency of the constructed methods is shown by numerical experiments.     

The stability of modified Runge-Kutta methods for the pantograph equation

In the present paper, the modified Runge-Kutta method is constructed, and it is proved that the modified Runge-Kutta method preserves the order of accuracy of the original one. The necessary and sufficient conditions under which the modified Runge-Kutta methods with the variable mesh are asymptotically stable are given. As a result, the -methods with , the odd stage Gauss-Legendre methods and the even stage Lobatto IIIA and IIIB methods are asymptotically stable. Some experiments are given.