Toward a fully parallel multigrid in time algorithm in PETSc environment: A case study in ocean models

We consider linear systems that arise from the discretization of evolutionary models. Typically, solution algorithms are based on a time-stepping approach, solving for one time step after the other. Parallelism is limited to the spatial dimension only. Because time is sequential in nature, the idea of simultaneously solving along time steps is not intuitive. One approach to achieve parallelism in time direction is MGRIT algorithm [7], based on multigrid reduction (MGR) techniques. Here we refer to this approach as MGR-1D. Other kind of approach is the space-time multigrid, where time is simply another dimension in the grid. Analougsly, we refer to this approach as MGR-4D. In this work, motivated by the need of maximizing the availability of new algorithms to climate science, we propose a new parallel approach that mixes both the MGR-1D idea and classical space multigrid methods. We refer to it as the MGR3D+1 approach. Moreover, we discuss their implementation in the high performance scientific library PETSc, as starting point to develope more efficient and scalable algorithms in ocean models.