One-Step Methods for Stiff ODE and DAE IVPs

Abstract

The convergence of a method is an asymptotic result for sufficiently small step sizes; it says nothing about the quality of the solution for a specific step size τ. Here the mathematical term “sufficiently small” may hide step sizes that are much too small for any practical purposes. In 1952 C. F. Curtiss and J. O. Hirschfelder [36] observed that for certain ODEs from chemical kinetics explicit methods required absurdly small step sizes in order to achieve reasonably acceptable solutions. They found this phenomenon to be independent of the particular choice of method and hence viewed it as a characteristic property of the particular initial value problem and called such problems “stiff.” In other words, stiff problems resist approximation by means of explicit one-step methods and require for their effective solution new classes of methods that will be developed in this chapter.