Advances and challenges in time integration methods

Time integration methods are a powerful tool for solving transient responses, which have been widely used to solve dynamic problems in aerospace, civil engineering, machinery manufacturing, and other fields. This paper reviews the advances in time integration methods in the past decades. Firstly, some classical methods, such as the series expansion method, the Runge-Kutta method and the Newmark method, are introduced. To solve the drawbacks involved in the classical methods, several time integration methods with more desirable numerical properties, including accuracy, efficiency, dissipation and stability, have been developed. Moreover, the advanced methods, including parameters methods, higher-order unconditionally stable methods, energy-conserving methods, linear multistep methods, composite methods and BN-stable methods, are introduced in this paper. Finally, the numerical properties and application scope of the existing time integration methods are compared, and some issues worthy of attention are given. © 2022 Beijing University of Aeronautics and Astronautics (BUAA). All rights reserved.