Connections Between Accuracy and Stability Properties of Linear Multistep Formulas

Abstract

This paper is concerned with stability and accuracy of families of linear k-step formulas depending on parameters, with particular emphasis on the numerical solution of stiff ordinary differential equations. An upper bound, p = k, is derived for the order of accuracy of A∞-stable formulas. Three criteria are given for A0-stability. It is shown that (1) for p = k, k arbitrary, A∞-stability implies certain necessary conditions for A0-stability and for strict stability (meaning that the extraneous roots of ρ(ζ) satisfy |ζ| < 1); (2) for p = k = 2, 3, 4, and 5, A∞-stability (for k = 5 together with another constraint) implies strict stability; and (3) for certain one-parameter classes of formulas with p = k = 3, 4, and/or 5, A∞-stability implies A0-stability. © 1975, ACM. All rights reserved.