An Efficient Adaptive Time-Integration Method for Second-Order Nonlinear Dynamics

Achieving an optimal balance among computational efficiency, robustness, and accuracy is a central challenge in simulating second-order nonlinear dynamical systems. While the existing parameterized two-sub-step composite integrator provides rigorous nonlinear stability and controllable dissipation, its fixed-step formulation limits efficiency in simulations with strongly varying dynamics. This paper presents a novel adaptive time integration method that augments the second-order base scheme with an explicit auxiliary stage for efficient error estimation. Its key innovation is a cost-free, direct error estimator, constructed by rigorously deriving the embedding coefficients via order conditions and analytically combining the implicit base stages with an extrapolated explicit stage to derive a local error estimate based on a third-order embedding without additional nonlinear iterations or matrix operations. Combined with a proportional-integral-derivative-like step-size controller, systematic numerical tests show that the proposed method achieves a significantly better computational cost-to-accuracy trade-off than highorder algebraically stable singly diagonally implicit Runge-Kutta methods. The algorithm demonstrates strong robustness in stiff and large-scale nonlinear problems while preserving the unconditional nonlinear stability and controllable dissipation of the base scheme. In summary, the proposed adaptive method offers an efficient, reliable, and self-starting tool for simulating largescale, long-duration, strongly nonlinear systems.