The dichotomy of stiffness: pragmatism versus theory

摘要

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 resumé 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.