In this paper, we present a new four-step implicit P-stable Obrechkoff method of tenth algebraic order for solving one-dimensional second-order linear periodic and oscillatory initial value problems of ordinary differential equations such as reasonable Schrödinger equation. Numerical stability and phase properties of the new methods are analyzed. Numerical experiments are carried out to show the efficiency and robustness of our new methods in comparison with the well known codes proposed in the scientific literature.
