Backward Differentiation Methods

Let us now consider linear multistep methods which are not necessarily A-stable, but are -stable or stiffly stable. Since stiff stability implies -stability for some , we need, in view of the theorem, look only at implicit linear multistep methods. With the usual notation for the characteristic polynomials of a linear multistep method, the associated stability polynomial is . Both and stiff stability require that the roots of be inside the unit circle when is real and . In this limit, the roots of approach those of , and it is thus natural to choose so that its roots lie within the unit circle. In particular, the choice , which has all its roots at the origin, is appropriate. The resulting class of methods

(9.7)

are known as the of backward differentiation methods . The coefficients of order K-step method, of this class are given in the following table for .