P-stability and exponential-fitting methods for y''=f(x, y)

The definitions of a periodicity interval and P-stability, given by Lambert & Watson [J. Inst. Math. Applic. 18, 189-202 (1976)], were designed for linear multistep methods with constant coefficients. In this paper those definitions are modified so as to provide a basis for linear stability analysis of exponential-fitting methods for the special class of ordinary differential equations of second order in which the first derivative does not appear explicitly. The stability properties of several existing methods are analyzed and a new P-stable method is proposed, to establish the existence of methods to which our definition applies, and to demonstrate its relevance to stiff oscillatory problems. The work is mainly concerned with two-step methods but extensions to methods of larger step-number are also considered.