Accurate single-step error estimation and universality for second-order implicit and explicit time integration algorithms in computational mechanics

Abstract

This paper presents a unified single-step a posteriori error estimator for a broad class of second-order implicit and explicit time integration schemes used in solid mechanics, structural dynamics, fluid–structure interaction, and hyperbolic heat transfer and the like. The estimator is compatible with widely used classical time integration methods in compatible with widely used classical time integration methods, including the Newmark method, the Fox–Goodwin method, the TPO/Generalized- α method, the HHT- α method, the WBZ- α method, the Position Verlet method, the explicit Generalized- α method, and also newer and more modern methods such as those under the umbrella of GS4 based projections that encompass LMS method and others (see Refs. Wang et al., 2025). The single-step formulation enables direct integration with adaptive time-stepping strategies, substantially improving computational efficiency in dynamic simulations. The approach is validated on linear benchmark problems using explicit schemes and nonlinear problems using implicit schemes with Newton–Raphson iterations. Numerical examples demonstrate that the adaptive strategy effectively balances time step size and nonlinear iteration count, achieving significant reductions in CPU time compared to fixed-step simulations. The accompanying test codes are made publicly available to support reproducibility.