A Generalized Single-Step Multi-Stage Time Integration Formulation and Novel Designs With Improved Stability and Accuracy

This paper focuses upon the single-step multi-stage time integration methods for second-order time-dependent systems. Firstly, a new and novel generalization of the Runge-Kutta (RK) and Runge-Kutta-Nyström (RKN) methods is proposed, featuring an advanced Butcher table for designing new and optimal algorithms. It encompasses not only the classical multi-stage methods as subsets, but also introduces novel designs with enhanced accuracy, stability, and numerical dissipation/dispersion properties. Secondly, to sharpen the focus on the present developments, several existing multi-stage explicit time integration methods (which are of interest and the focus of this paper) are revisited within the proposed unified mathematical framework, such that it highlights the differences, advantages, and disadvantages of various existing methods. Thirdly, the consistency analysis is rigorously demonstrated using both single-step and multi-step local truncation errors, addressing the order reduction problem observed in existing methods when applied to nonlinear dynamics problems. Finally, two sets of single-step, two-stage, third-order time-accurate schemes with controllable numerical dissipation/dispersion at the bifurcation point are presented. In contrast to existing methods, these newly proposed schemes preserve third-order time accuracy in nonlinear dynamics applications and exhibit improved stability in cases involving physical damping. Numerical examples are demonstrated to verify the theoretical analysis and the superior performance of the proposed schemes compared to existing methods.