On an accurate A-posteriori error estimator and adaptive time stepping for the implicit and explicit composite time integration algorithms

This paper focuses on the design of an accurate and versatile A-posteriori error estimation and adaptive time stepping for general composite schemes and typical of the implicit ρ_∞-Bathe and explicit Noh-Bathe composite methods for time integration in second-order transient systems. Two novel error estimators, Disp/P3 and Disp/P4, are newly proposed based on the generalized polynomials. They have generalized compact single-step representations and work for both implicit and explicit, dissipative and non-dissipative composite schemes. Validation tests of linear and nonlinear problems show that the estimated local error reaches excellent agreement with the exact local error, and the third-order convergence rates of the local error are obtained. Besides, the Disp/P4 is more accurate than Disp/P3 as the effectivity index is almost identical to unity when random time steps are taken into consideration, and hence the Disp/P4 is highly recommended to be merged to the adaptive time stepping procedure. Three examples encompassing the wave propagation, damped/undamped spring-pendulum, and nonlinear spring-mass system are solved by implicit and explicit composite methods via the adaptive time stepping. Numerical results show the advantage of adaptive time stepping based on the proposed error estimator in contrast to constant time stepping regarding the CPU cost.