Optimal implicit single-step time integration methods with equivalence to the second-order-type linear multistep methods for structural dynamics: Accuracy analysis based on an analytical framework

This paper presents a comprehensive study of the second-order-type time integration methods family encompassing the linear multistep (LMS) methods, the single-step methods, and the LMS equivalent single-step methods for structural dynamics. An analytical accuracy framework for algorithm design, convergence, and optimization is developed, where the obstacles encountered in algorithms construction, convergence accuracy, algorithms equivalence, accuracy measurement, and accuracy optimization in traditional accuracy frameworks are well addressed. Rigorous accuracy analysis successively to the second-order-type LMS (LMS2) methods, the single-step methods, and the LMS2 equivalent single-step methods aims to reveal optimal algorithms with desirable numerical properties. First, independent implicit algorithms including the velocity type and the acceleration type LMS2 methods except the displacement type ones are newly revealed in the LMS2 methods family with second-order convergence and A-stability, and three error constants that can play essential roles in the accuracy measurement and algorithm optimization are developed. Second, a single-step methods family where the displacement, velocity, and acceleration can achieve the same order of accuracy simultaneously is revealed in a generalized 16 -parameters methods family, which contains numerous conventional single-step single-solve methods as subfamily methods. It is found that the unit consistent and non-consistent represen-tations show a significant difference in the local truncation error, stability, and convergence performances. Third, a family of accuracy-improved single-step methods with algorithmic equivalence, which is stricter than the traditional spectral equivalence, to the LMS2 methods is revealed. It is clarified that the difference between the methods with equivalence can significantly influence their convergence accuracy, and the error constants concept of the LMS2 methods can be extended to the single-step method with algorithmic equivalence. Last, an algorithm optimization strategy is developed based on the error constants and the ultimate spectral radius, and an optimal single-step method with a single parameter to control the accuracy and numerical dissipation is revealed analytically without empirical assumptions. The optimal method is second order accurate, unconditionally stable, overshooting acceptable, and equivalent to an optimal route between the Houbolt method and the trapezoidal rule, and its numerical performance is verified by the spectral accuracy and two numerical examples.