P-Stable Hybrid Super-Implicit Methods for Periodic Initial Value Problems

This paper deals with a class of symmetric (hybrid) P-stable methods for the numerical solution of special second order initial value problems (IVPs). For linear multistep methods, Lambert and Watson [5], had shown that a P-stable method is necessarily implicit and that the maximum order attainable by a P- stable method is at most two. P-stability is important in the case of 'periodic stiffness' as it is termed by Lambert and Watson [5], that is, when the solution consists of an oscillation of moderate frequency with a high frequency oscillation of small amplitude superimposed. In order to overcome the order-barrier on linear multistep P-stable methods, we developed a new type of implicit formulas of linear multistep methods. The formulas, which we call to be hybrid super-implicit, are of more implicitness than the so-called implicit formulas in the sense that they require the knowledge of functions not only at the past and present time-step but also at the future ones. In the cases when the right hand side of IVP is very complex, the super-implicit methods are preferred. Also, we have used off-step points which allow us to derive P- stable schemes of high order. We report numerical experiments to illustrate the accuracy and implementation aspects of this class of methods.