General linear methods for \(y^{\prime\prime} = f(y(t))\) (Q695621)

The paper introduces a general family of numerical methods which are more convenient to numerically integrating initial value problems based on special second-order ordinary differential equations of the form \[ \begin{cases} y''(t) = f(y(t)), \quad t\in [t_0, T], \\ y(t_0)= y_0 \in \mathbb{R}^d, \\ y'(t_0)= y'_0 \in \mathbb{R}^d, \end{cases} \] where the function \( f: \mathbb{R}^d \rightarrow \mathbb{R}^d \) does not explicitly depend on \( y' \) and is supposed to be smooth enough to guarantee that the problem is Hadamard well-posed. Consistency, zero-stability and convergence of the solutions of the problem are studied. The paper is mainly focused on the development of a unifying framework for the numerical solutions of special-order ordinary differential equations by considering the family of general linear methods.