Predictor-Corrector Algorithms with Identical Regions of Stability

Abstract

It is shown that the general $PEC$ algorithm has a region of stability (absolute or relative) identical with that of a certain explicit linear multistep method $\tilde P$, whose order can be simply expressed in terms of the orders of the original predictor and corrector. Further, the region of stability of the general $P(EC)^{m + 1} $ algorithm is shown to be identical with that of the algorithm $\tilde P(EC)^m E$. It is concluded that there is little motivation for using $PEC$ algorithms, and that, among predictor-corrector algorithms employing $m + 1$ evaluations, the “best” algorithms, from the point of view of order and stability, will be found among $P(EC)^m E$ algorithms; the argument, however, takes no account of step number.