An efficient four-step multiderivative method for the numerical solution of second-order IVPs with oscillating solutions

An explicit four-step method of 10th algebraic order is constructed and analyzed in this article for the numerical integration of initial value problems of second-order ordinary differential equations. The new method is multiderivative. It also has the most important P-stability property for problems that have one frequency. The advantage of the new method is its simplicity in implementation and, since it is explicit, it will not require any additional predictor stages. Applying our new method to the well-known problems such as Stiefel and Bettis "near periodic" problem, and Duffing's equation without damping, we found that the method has several advantages, such as simplicity, accuracy, stability, and efficiency.