A special stability problem for linear multistep methods

The trapezoidal formula has the smallest truncation error among all linear multistep methods with a certain stability property. For this method error bounds are derived which are valid under rather general conditions. In order to make sure that the error remains bounded ast , even though the product of the Lipschitz constant and the step-size is quite large, one needs not to assume much more than that the integral curve is uniformly asymptotically stable in the sense of Liapunov.The preparation of this paper was partly sponsored by the Office of Naval Research and the US Army Research Office (Durham). Reproduction in whole or in part is permitted for any purpose of the US Government.