Connections between accuracy and stability properties of linear multistep formulas

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 A 0 -stability. It is shown that (1) for p = k, k arbitrary, A ∞ -stability implies certain necessary conditions for A 0 -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 A 0 -stability.