Before revision:  For any } one-step time-integrators and two exemplary classes of two-step time-integrators: the parameterized Numerov methodsand the parameterized two-stage hybrid methods. The main conclusion is that the global error satisfies \|{\textbf{err}^{k+1}\|_{{\mathbf W},
After revision:  For any } one-step }^{k+1}\|_{{\mathbf W},