J. C. Butcher

Implicit Runge-Kutta processes

Received November 1, 1962. Revised April 22, 1963. * If the function f(y) satisfies a Lipschitz condition and h is sufficiently small, then the equations defining g(1>, g(2), • • • , gw have a unique solution which may be found by iteration (see Appendix). t It will be assumed throughout that f (y) and all its derivatives exist and are continuous so that the Taylor expansions for y and y may be terminated at any term with an error of the same order as the first term omitted. 50

Coefficients for the study of Runge-Kutta integration processes

We consider a set of η first order simultaneous differential equations in the dependent variables y1, y2, …, yn and the independent variable x ⋮ No loss of gernerality results from taking the functions f1, f2, …, fn to be independent of x, for if this were not so an additional dependent variable yn+1, anc be introduced which always equals x and thus satisfies the differential equation

Stability Criteria for Implicit Runge–Kutta Methods

A comparison is made of two stability criteria. The first is a modification to nonautonomous problems of A-stability and the second is a similar modification of B-stability. It is shown that under certain mild conditions these two concepts are equivalent. A number of examples are given of methods that satisfy these new stability properties and it is also shown that the growth of errors can be estimated by an extension of this theory.

An algebraic theory of integration methods

A class of integration methods which includes Runge-Kutta methods, as well as the Picard successive approximation method, is shown to be related to a certain group which can be represented as the family of real-valued functions on the set of rooted trees. For each integration method, a group element is defined corresponding to it and it is shown that the numerical result obtained using the method is characterised by this group element. If two methods are given, then a new method may be defined in such a way that when it is applied to a given initial-value problem the result is the same as for the successive application of the given methods. It is shown that the group element for this new method is the product of the group elements corresponding to the given methods. Various properties of the group and certain of its subgroups are examined. The concept of order is defined as a relationship between group elements.

On Runge-Kutta processes of high order

An (explicit) Runge-Kutta process is a means of numerically solving the differential equation , at the point x = x 0 +h, where y , f may be vectors.

On the implementation of implicit Runge-Kutta methods

The modified Newton iterations in the implementation of ans stage implicit Runge-Kutta method for ann dimensional differential equation system require 2s3n3/3+O(n2) operations for theLU factorisations and 2s2n2+O(n) operations for the back substitutions. This paper describes a method for transforming the linear system so as to reduce these operation counts.

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 Modified Multistep Method for the Numerical Integration of Ordinary Differential Equations

Ab.stract. i t has been sho~v~, by I )ahlquis t [1] t ha t ,q~ k s t ep me thod for the numericet 15 soll~tion of an ordi t lary differential equat ion is uns table unless the order is less t h a n / :93 This paper is concerned wi th a modificati(m t~o the form of t he mul t i s tep process such t i~r higher orders can be ~ t ta ined . For k.~7 examples of such modified processes of order 2,~:÷-t :i have been found and these are given in full for k~(i .

General linear methods for ordinary differential equations

General linear methods were introduced as the natural generalizations of the classical Runge-Kutta and linear multistep methods. They have potential applications, especially for stiff problems. This paper discusses stiffness and emphasises the need for efficient implicit methods for the solution of stiff problems. In this context, a survey of general linear methods is presented, including recent results on methods with the inherent RK stability property.

A stability property of implicit Runge-Kutta methods

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.