Second Derivative Two-Step Peer Methods

This paper is devoted to extending two-step peer methods, for the numerical solution of ordinary differential equations, for the case where the second derivative of the solution is incorporated into the formula of the methods. The main features including consistency, zero-stability, and convergence of the proposed methods together with their order conditions and stability analysis are examined. Construction of explicit methods within the proposed class of the methods, possessing the Runge–Kutta stability property, is investigated, and examples of such methods up to order five are provided. The efficiency and accuracy of the constructed methods are validated through various numerical experiments conducted in both fixed and variable stepsize environments.