An algorithm for starting multistep methods

The present paper deals with the problem of starting multistep methods. We take into consideration an Adams-Bashforth-Moulton PECE pair, with the predictor of order q and the corrector of order q + 1. To start this method, q − 1 starting values are necessary, in addition to y0. A well-known result from theory says that the order of convergence of the whole integration is q + 1, if all those starting values are accurate of that same order. Present production codes start with a predictor of order 1 and a corrector of order 2 at the first step, and then proceed step by step, each time raising the order by 1, until all the necessary starting values have been obtained. But, in this manner, all the starting errors keep of order 3, and so the whole integration converges no faster than that order. This drawback is normally compensated for, by taking very small step sizes in the starting phase. The general algorithm we propose furnishes, at a reasonably low cost, the necessary number q − 1 of starting values, each of the appropriate order q + 1, whatever q might be; it is independent of the particular multistep formula considered, and is mainly designed to be used for high values of q (q ≥ 10), where the alternative strategies are too expensive or do not exist at all. The numerical results reported show the validity of our approach.