According to Lambert and Watson [6], Theorem 4 says that the linear multistep P-stable methods can not be explicit; they must be implicit and being implicit is the essential condition to obtain important feature of P-stability. Few explicit P-stable methods have been created in which they are nonlinear or at most P-stable. For the first time in literature, in this paper, we create a new family of explicit linear two-step singularly P-stable methods with phase-lag of order infinity for the numerical solution of initial value problems of second-order ordinary differential equations. Finally, the numerical results obtained by the new family for some well-known problems show its superiority in efficiency, accuracy, convergency and stability.
