A multi-scale IMEX second order Runge-Kutta method for 3D hydrodynamic ocean models

Understanding complex physical phenomena often involves dealing with partial differential equations (PDEs) where different phenomena exhibit distinct timescales. Fast terms, associated with short characteristic times, coexist with slower ones requiring relatively longer time steps for resolution. The challenge becomes more manageable when, despite the varying characteristic times of fast and slow terms, the computational cost associated with faster terms is significantly lower than that of slower terms. Additionally, slower terms can also exhibit two distinct longer characteristic times, adding complexity to the system and resulting in a total of three characteristic timescales. In this paper, an innovative split second-order IMEX (IMplicit-EXplicit) temporal scheme is introduced to address this temporal complexity. It is used to solve the primitive equation ocean model. Extremely short times are handled explicitly with small time steps, while longer timescales are managed explicitly and semi-implicitly using larger time steps. The decision to solve a portion of the slower terms semi-implicitly is due to the fact that it does not significantly increase the total computational cost, allowing for greater flexibility in the time step without imposing a substantial burden on the overall computational efficiency. This strategy enables efficient management of the various temporal scales present in the equations, thereby optimizing computational resources. The proposed scheme is applied to solve 3D hydrodynamics equations encompassing three time scale: fast terms representing wave phenomena, slow terms describing horizontal aspects and stiff terms for vertical ones. Furthermore, the scheme is designed to respect crucial physical properties, namely global and local conservation. The obtained results on different test cases demonstrate the robustness and efficiency of the IMEX approach in simulating these complex systems.