Singly-implicit stabilized extended one-step methods for second-order initial-value problems with oscillating solutions

We introduce a new class of singly-implicit extended one-step methods for the numerical integration of second-order initial-value problems y″ = f(t, y), y(t0) = η0, y′(t0) = η1, with oscillating solutions. We first show that for third order, with two stages there exists a uniquely determined ‘almost’ P-stable method. We then investigate stability of the general class of fourth-order one-step methods. We first look for stabilized fourth-order methods with two stages, and show the interesting result that there exist families of two-stage fourth-order P-stable methods. We also obtain some families of three-stage fourth-order P-stable methods. The obtained methods are computationally tested on problems of practical interest.