A Study on Adaptive Implicit–Explicit and Explicit–Explicit Time Integration Procedures for Wave Propagation Analyses

Abstract

This study delves into the effectiveness of two time integration techniques, namely the adaptive implicit–explicit (imp–exp) and explicit–explicit (exp–exp) methods, which stand as efficient formulations for tackling intricate systems characterized by multiple time scales. The imp–exp technique combines implicit and explicit procedures by employing implicit formulations for faster components and explicit calculations for slower ones, achieving high accuracy and computational efficiency. Conversely, the exp–exp method, a variation of explicit methods with sub-cycling, excels in handling locally stiff systems by employing smaller sub-steps to resolve rapid changes while maintaining stability. For both these approaches, numerical damping may be activated by adaptive time integration parameters, allowing numerical dissipation to be locally applied, if necessary, as a function of the considered discrete model and its computed responses, enabling a highly effective numerical dissipative algorithm. Furthermore, both these techniques stand as very simple and straightforward formulations as they rely solely on single-step displacement–velocity relations, describing truly self-starting procedures, and they stand as entirely automated methodologies, requiring no effort nor expertise from the user. This work provides comparative studies of the adaptive imp–exp and exp–exp approaches to assess their accuracy and efficiency across a wide range of scenarios, with emphasis on geophysical applications characterized by multiscale problems, aiming to establish under which circumstances one approach should be preferred over the other.