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.

本文针对二阶常微分方程初值问题的数值模拟，提出了一个新的十二代数二步两步半混合格式族。所提出的方法是对称的，属于多导数方法族。新族的每种方法似乎都是混合的，但是在实现混合术语之后，它将继续作为多导数方法。因此，使用名称半混合。研究方法的一致性，收敛性，稳定性和周期性。为了显示提出的族的准确性，一致性，收敛性和稳定性，在一些众所周知的问题上进行了测试，例如未阻尼的Duffing方程。