Stiff Stability and Its Relation to $A_0 $- and $A(0)$-Stability

Necessary and sufficient conditions for an $A_0 $-stable multistep method to be stiffly stable or $A(0)$-stable respectively, are established. As an application of these results one finds that Cryer’s k-step method [1] is $A(0)$- and stiffly stable. Hence there exist $A(0)$- and stiffly stable k-step methods of order k. Moreover it is shown that to any stiff stability parameter $D > 0$ there exists a stiffly stable k-step method of order k. Plots of the stability region of Cryer’s method for $k = 1,2, \cdots ,7$ are given. The angle $\alpha $ of $A(\alpha )$-stability and D of stiff stability of Cryer’s method are listed for $k = 1,2, \cdots ,16$.