A New Two Step Hybrid Singularly P−Stable Method for the Numerical Solution of Second Order IVPs with Oscillating Solutions

In this paper, a new two-step hybrid method of twelfth algebraic order is constructed and analyzed for the numerical solution of initial value problems of second-order ordinary differential equations. The proposed methods are symmetric and belong to the family of multiderivative methods. Each methods of the new family appear to be hybrid, but after implementing the hybrid terms, it will continue as a multiderivative method. Therefore, the name semihybrid is used. The consistency, convergence, stability and periodicity of the methods are investigated and analyzed. The numerical results for some quantum chemistry problems such as. undamped Duffing’s equation and orbit problems of Stiefel and Bettis indicated that the new method is superior, efficient, accurate and stable.