This paper investigates the use of overlapping Schwarz domain decomposition preconditioners to solve fully implicit three-dimensional ocean circulation models efficiently. These are essential for understanding climate-relevant ocean dynamics but pose formidable computational challenges due to their spatial and temporal dimensions combined with ill-conditioned systems typical of realistic ocean climate models.

The paper begins by describing an established ocean model which solves the primitive equations coupled with temperature and salinity convection, subject to wind stress, heat, and salt flux forcing. The model is discretized using a second-order control volume scheme on a hybrid B-C Lorenz grid. This results in a large non-linear system whose Jacobian features tightly coupled momentum, mass, and tracer blocks. Solving this system relies on Newton-Krylov iterations, whose convergence depends heavily on effective preconditioning.

The paper presents numerical experiments. It first examines a Laplace system to systematically study how the overlap size and number of subdomains affect solver convergence. Results confirm that increased overlap reduces iterations and that two-level Schwarz methods with coarse corrections outperform one-level approaches, especially as mesh refinement and subdomain counts increase.

Building on these insights, the study extends to a fully implicit ocean model that solves the primitive equations with coupled temperature and salinity dynamics under wind, heat, and salt flux forcing. Applying a one-level Schwarz preconditioner, it becomes clear that as temperature and salinity forcing intensify, the Jacobian’s conditioning deteriorates sharply which leads to a steep rise in GMRES iterations. Additional experiments isolating individual tracers indicate that the temperature exerts a particularly strong influence on convergence, though buoyancy interactions complicate this attribution.

The work concludes that while one-level Schwarz methods improve performance on simpler problems, fully implicit ocean models under strong tracer forcing require advanced two-level Schwarz preconditioners with robust coarse spaces to maintain efficiency and scalability. Future work is encouraged to incorporate coarse grid corrections explicitly to better handle these challenges and enable more detailed and computationally feasible ocean simulations.