High-Order Multirate Explicit Time-Stepping Schemes for the Baroclinic-Barotropic Split Dynamics in Primitive Equations
Ocean motion can be modeled effectively by primitive equations, and the splitting of baroclinic-barotropic dynamics is a commonly used approach for efficiently solving primitive equations. Based on such mode splitting, scientists have developed two multirate explicit time-stepping schemes (SSPRK2-SE and SSPRK3-SE) based on the classic strong stability preserving Runge-Kutta (SSPRK) framework. In the proposed schemes, the 3D (slow) baroclinic mode is advanced with a large time step, and the 2D (fast) barotropic mode is predicted with a small time step and later corrected with interpolated barotropic forces. Although different time steps are used for each of the two mode solves to achieve good efficiency, high-order temporal accuracies are still able to be maintained. In addition, the proposed schemes reconcile the sea surface height (SSH) perturbations to avoid the numerical inconsistencies in SSH due to the mode splitting and overdetermination in the discrete setting.
The proposed SSPRK-SE schemes have the potential to make considerable impact in the ocean modeling community. They not only achieve better efficiency than time integration of the original unsplit dynamics system in primitive equations, but also can obtain better temporal accuracies than the split-explicit scheme currently used in MPAS-Ocean. Furthermore, the proposed schemes are easy to implement and naturally inherit excellent parallel scalability of explicit time stepping as demonstrated by various numerical tests performed on the NERSC Cori cluster.
To treat the multiple time scales of ocean dynamics efficiently, the baroclinic-barotropic dynamics splitting technique has been widely employed for solving primitive equations in ocean modeling. This work proposes and analyzes two high-order multirate explicit time-stepping schemes (SSPRK2-SE and SSPRK3-SE) for such split system. The proposed schemes allow for a large time step for advancing the 3D baroclinic mode and a small time step for predicting and correcting the 2D barotropic mode, so that each of the two mode solves only need to satisfy their respective CFL conditions to maintain overall numerical stability while still achieving high-order accuracies. Meanwhile, we also reconcile the sea surface height perturbations with carefully calculated flux deficits and velocity adjustments at all stages of each time step to resolve the numerical inconsistency issue due to the mode splitting. Two benchmark tests drawn from MPAS-Ocean are finally used to numerically demonstrate the excellent performance of the proposed schemes in terms of accuracy, efficiency, and parallel scalability.