P-stable methods for periodic initial value problems of second order differential equations

A class ofP-stable finite difference methods is discussed for solving initial value problems of second order differential equations which have periodic solutions. The methods depend upon a parameterp>0, and reduce to the classical Störmer-Cowell methods forp=0. It is shown that whenp is chosen for linear problems as the square of the frequency of the periodic solution, the methods areP-stable and for some suitable choice ofp, they have extended finite interval of periodicity.