A Relaxation Scheme for Solving Convolutional Forces in Adaptive Time Step ODE Solvers

Abstract Efficiently solving Ordinary Differential Equations (ODEs) is of importance in simulation-based digital-twin solutions for different marine and offshore industrial applications. Many existing engineering codes in this field adopt a uniform time-step approach in solving ODEs. For local instantaneous variations existing in the solutions, a fine time-step setup is necessary. This leads to redundant computations in global computations. An adaptive-time-step ODE solver offers potential improvement by adjusting time steps dynamically and minimizing the number of total steps required. However, this solution can still encounter inefficiencies, especially when common operations like convolution integrals are repeatedly computed within each major time step for evaluating the next state. To address this challenge, this study proposes a relaxation scheme within adaptive-time-step ODE solvers. The relaxation scheme dynamically smooth over repeated calculations with a below-threshold filtering mechanism on time consuming parts such as convolutions. This enables efficient solving within any major step. Applied to hydrodynamic problems, this method offers reduced computational complexity and improved solving speed. In addition, numerical results demonstrate that this approach maintains good accuracy for numerical emulations to marine dynamic systems. This framework could work as a foundation for further advancements in ODE solving optimization, particularly for applications where different categories of environmental loads are involved. This numerical scheme may also enable efficient time-domain simulations with multiple rigid floaters.