A Relaxation Scheme for Solving Convolutional Forces in Adaptive-Time-Step Ordinary Differential Equation Solvers

Abstract

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 simulation codes in this field adopt a uniform time-step approach in solving ODEs. The complexity of a simulation is influenced by the number of time-steps. To minimize the required number of total steps within a simulation, an adaptive-time-step explicit ODE solver can offer potential improvement by adjusting time stepping dynamically. However, this solution can still encounter inefficiencies, especially when operations like convolution integrals are repeatedly computed within each major time-step for evaluating the next state. To address this challenge, a relaxation scheme within adaptive-time-step explicit ODE solvers is proposed in this study. The relaxation scheme dynamically smooths over repeated calculations with a below-threshold filtering mechanism on time-consuming parts such as convolutions. This enables efficient solving within any major step. A case study with a floating vessel for aquacultural cultivation is performed to guide the choice of threshold value. The improved performance in calculation speed is demonstrated by comparing with results using uniform time-step solvers. The proposed relaxation scheme offers reduced computational complexity and improved solving speed. In addition, numerical results demonstrate that this approach maintains good accuracy for numerical simulations of marine dynamic systems. This study serves as a foundation for further advancements in the improvement of ODE solving, particularly for applications where different categories of environmental loads are involved. This numerical scheme may also enable efficient time-domain simulations for multiple-floater dynamics.