Abstract One-step multiple-value methods are developed which involve an accurate predictor method with higher derivatives, followed by a corrector method cast in form of an enhanced Newton–Raphson scheme. The generalized Newmark (GNpj) method may be recovered as a special case. The algorithms serve to match the accuracy of the fourth-order Runge–Kutta–Fehlberg method. Challenges to solve more reliably, accurately and efficiently non-linear differential equations are highlighted as stemming from amplitude and phase shift errors introduced by discretization in space and time – a continuous-discrete transformation. The classical stability tool of spectral radius is performed on linear systems whereas Liapunov method on nonlinear systems.