Numerical simulation of second-order initial-value problems using a new class of variable coefficients and two-step semi-hybrid methods

In this paper, a new family of two-step semi-hybrid schemes of the 12th algebraic order is proposed for the numerical simulation of initial-value problems of second-order ordinary differential equations. The proposed methods are symmetric and belong to the family of multiderivative methods. Each method of the new family appears to be hybrid, but after implementing the hybrid terms, it will continue as a multiderivative method. Therefore, the designation semi-hybrid is used. The consistency, convergence, stability, and periodicity of the methods are investigated and analyzed. In order to show the accuracy, consistency, convergence, and stability of the proposed family, it was tested on some well-known problems, such as the undamped Duffing's equation. The simulation results demonstrate the efficiency and advantages of the proposed method compared to the currently available methods.