The stability of some numerical schemes
by
J. C. Butcher
The University of Auckland

Numerical methods for solving ordinary differential equations have, associated with them, a ``stability matrix" M(z) whose elements are functions of a complex variable. The characteristic polynomial of this matrix, det(wI-M(z)) is a polynomial in the two variables z and w and defines a relationship between them. The set of points in the z plane for which the corresponding w values lie in the unit disc is known as the ``stability region". In particular, if the stability region contains the left half-plane, the numerical method is said to be ``A-stable" and can be reliably used to solve stiff differential equations. We will discuss known results on the A-stability of some particular methods and outline a new proof of a result formerly known as the Ehle conjecture. An apparently much more difficult proposition, known as the Butcher-Chipman conjecture, will be introduced and some partial results will be discussed.

Date received: October 8, 2001