Novel Implicit Integration Algorithms With Identical Second‐Order Accuracy and Flexible Dissipation Control for First‐Order Transient Problems

Abstract

For the solutions of first‐order transient problems with optimal efficiency, conventional single‐step implicit methodologies, unaided by additional post‐processing techniques, encounter limitations in concurrently attaining identical second‐order accuracy and controllable numerical dissipation. Addressing this challenge, the present study contributes not only a comprehensive analytical framework for formulating implicit integration algorithms but also leverages the auxiliary variable and the composite sub‐step technique to propose two distinct types of implicit integration algorithms. Each type is characterized by self‐initiation, unconditional stability, identical second‐order accuracy, controllable numerical dissipation, and zero‐order overshoots. Recognizing that single‐step implicit methods can be conceptualized as composite single‐sub‐step ones, both algorithm types achieve identical effective stiffness matrices, thereby reducing the computational effort for solving linear problems. The utilization of auxiliary variables endows the proposed single‐step method with numerical attributes akin to established counterparts, while achieving identical second‐order accuracy. Conversely, the adoption of the composite sub‐step technique in the proposed two‐sub‐step methods surpasses the published algorithms by significantly reducing the error constants. This superiority persists when enforcing the same sub‐step sizes and sub‐step dissipation levels. Numerical simulations affirm that the proposed methods consistently outperform existing alternatives without incurring additional computational expenses.