$\textrm {PECE}$ algorithms for the solution of stiff systems of ordinary differential equations

Abstract:

This paper presents a study of a class of PECE algorithms consisting of an application of a predictor followed by application of one iteration of a pseudo Newton-Raphson method to a corrector. Such algorithms require precisely two evaluations of the derivative function for each forward step. Theorems 1 and 4 show that the stability properties of such algorithms compare favorably with those obtained with application of the Newton-Raphson method to the corrector iterated to convergence. A subclass of these algorithms have local truncation error of second order and some have local truncation error of third order. Theorems 2 and 3 exhibit members of this subclass wherein an estimate of the local truncation error is explicit in the algorithm at each step. Initially (in Theorem 1) these algorithms are characterized in terms of their stability properties in the limit as the interval of integration becomes indefinitely large. In Section 5, their properties for other intervals of integration are discussed through the study of some enclosure properties.