Forward–backward-difference time-integrating schemes with higher order derivatives for non-linear finite element analysis of solids and structures

• One-step multiple-value methods are developed.
• Amplitude errors are minimized and phase shift errors are controlled.
• Enhanced Newton–Raphson scheme avoids frequent calculation of a residual function.
• Generalized Newmark method of Zienkiewicz and Taylor is a subset of this algorithm.
• Hardening, softening spring-mass systems and van Pol oscillator equation are solved.

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.