ON THE STABILITY OF PREDICTOR-CORRECTOR METHODS BETWEEN ADAMS AND BACKWARD DIFFERENCE FORMULA

Abstract

The predictor-corrector methods take the upper hand in decreasing the number of function evaluations and of derivative evaluations as well comparing to the Runger-Kutta methods and various linear multistep methods. The stability however is a traditional disease of a high order method. This problem is also the case to the predictor-corrector method. The paper discusses on the matter of stability of the k-step Adams predictor-correct method and that of the predictor-corrector methods constructed on the basis of k-step Adams-Brashfort for the predictor and the k-step or (k+1)-step backward difference formula (BDF) for the corrector with low k, say . The reason to consider the BDF corrector is from the fact of having a large portion of the absolute stability region for those methods (with ) competing to other Adams-Moulton correctors. Some awkward performances of the predictor-corrector to the stiffness are also discussed and a modified algorithm is also developed to treat the poor performance of the abovementioned methods. The main contribution of the paper is the strategy of depicting the absolute stability region of a predictor-corrector method by constructing its stability polynomial on the basis of the recurrence equation obtain form the pair of difference equations describing the predictor and corrector. On that construction, we can be able to sketch the region by the boundary locus method.