Some new A-stable methods for ordinary differential equations
by
John Butcher
The University of Auckland

For the solution of so-called `stiff' problems, the property known as A-stability is an important indicator of the ability of a numerical method to obtain accurate results efficiently. Unfortunately, the combined requirements of A-stability and high order of accuracy are not achievable for linear multistep methods. On the other hand, Runge-Kutta methods can be A-stable of any order but suffer from other disadvantages. The new methods considered in this paper lie in the larger family of `general linear methods' and possess a property known as `inherent RK stability' which gives them a number of advantages. A similar structure, but applicable to explicit methods for non-stiff problems, has been discovered by Will Wright. We conclude the present talk by discussing a transformation that can be regarded as a step towards unifying the two numerical schemes into a single structure. In the second part of Will's talk, he will show how this process of unifying the two theories can be taken further.

Date received: September 27, 2000