Algebraic Multigrid and Schur Complement Strategies within a Multilayer Spectral Element Ocean Model

We present and compare three methods for accelerating the filtering process used in the multilayered Spectral Element Ocean Model (SEOM). The methods consist of a Schur complement preconditioner, a lumping of small entries and an algebraic multigrid (AMG) algorithm, and a algebraic multigrid with patch smoothing algorithm. 1. The shallow water equations, ocean model, and filtering process The shallow water equations approximate the equations of fluid motion well whenever the fluid’s density is homogeneous and its depth is much smaller than a characteristic horizontal distance. The shallow water equations can be written in the vector form: ut + g∇ζ = F (1.1) ζt +∇ · [(h+ ζ) u] = Q (1.2) where u = (u, v) is the velocity vector, ζ is the sea surface displacement (which, due to hydrostaticity, also stands for the pressure), g is the gravitational acceleration, h is the resting depth of the fluid, Q is a mass source/sink term. The vector F = (f, f) is a generalized forcing term for the momentum equations that includes the Coriolis force, non-linear advection, viscous dissipation, and wind forcing. For simplicity, we assume no-slip boundary conditions. The shallow water equations are often used to model the circulation in coastal areas and in shallow bodies of water. They reduce the complicated set of the 3D equations to a stacked set of 2D ones while still representing a large part of the dynamics. The shallow water equations also arise frequently in the solution of the 3D primitive hydrostatic equations if the top surface of the fluid is a free boundary and can move up and down. The presence of the free surface allows the propagation of gravity waves at the speed of √ gh. The gravity wave speed can greatly exceed the advective velocity of the fluid in the deep part of the ocean and results in a very restrictive CFL limit in order to maintain stability in 3D ocean models [8, 9]. 1991 Mathematics Subject Classification. 68W10, 65Y05, 47N40, 76D33. This research has been partially supported by NSF grant ACR-9721388. c ©0000 (copyright holder)