Kevin Burrage

Non-linear stability of a general class of differential equation methods

For a class of methods sufficiently general as to include linear multistep and Runge-Kutta methods as special cases, a concept known as algebraic stability is defined. This property is based on a non-linear test problem and extends existing results on Runge-Kutta methods and on linear multistep and one-leg methods. The algebraic stability properties of a number of particular methods in these families are studied and a generalization is made which enables estimates of error growth to be provided for certain classes of methods.

A special family of Runge-Kutta methods for solving stiff differential equations

An efficient way of implementing Implicit Runge-Kutta Methods was proposed by Butcher [3]. He showed that the most efficient methods when using this implementation are those whose characteristic polynomial of the Runge-Kutta matrix has a single reals-fold zero. In this paper we will construct such a family of methods and give some results concerning their maximum attainable order and stability properties. Some consideration is also given to showing how these methods can be efficiently implemented and, in particular, how local error estimates can be obtained by the use of embedding techniques.

High order algebraically stable Runge-Kutta methods

In Burrage and Butcher [3] the concept of Algebraic Stability was introduced in the study of Runge-Kutta methods. In this paper an analysis is made of the family ofs-stage Runge-Kutta methods of order 2s—2 or more which possesses this property.

Order and stability properties of explicit multivalue methods

Abstract Recent papers by Burrage and Moss [1] and Burrage [2] have studied in some detail the order properties of implicit multivalue (or general linear) methods and certain classes of these methods were proposed as being suitable for solving stiff differential equations. In this present paper we study the order and stability of explicit multivalue methods with a view to deriving new families of methods suitablefor solving non stiff problems.

Efficiently implementable multivalue methods for solving stiff ordinary differential equations

Abstract Although there are now in existence many different types of codes solving stiff ordinary differential equations, the methods upon which these codes are based are often deficient in terms of stability or order criteria. In this paper we discuss some new research on the study of the order, stability and efficiency properties of a general class of methods called multivalue methods. New families of methods based on the extension of diagonally implicitness and singly implicitness from Runge-Kutta methods to multivalue methods are analyzed and a family of methods designed to be implemented in a variable-order variable-stepsize setting is proposed.

The dichotomy of stiffness: pragmatism versus theory

The problem of stiffness occuring in ordinary differential equations is now well understood and various concepts such as nonlinear stability, B-convergence and stiff order have been introduced as being relevant to an understanding of the behaviour of numerical methods when applied to stiff problems. However, it is often the case that some of these properties are not always attainable when methods are implemented efficiently and in this paper we will investigate this paradox in some detail. In addition, we will give a resume of some of the recent theoretical developments in the applicability of numerical methods to stiff problems and make some comparisons between existing codes based on Runge-Kutta and linear multistep methods. We will conclude this discussion with the introduction of a very general family of methods, called multivalue methods, and show how the important concepts of cheap implementation, stage order and stability apply to these methods.