We discretize the shallow water equations with an Adams-Bashford scheme combined with the Crank-Nicholson scheme for the time derivatives and spectral elements for the discretization in space. The resulting coupled system of equations will be reduced to a Schur complement system with a special structure of the Schur complement. This system can be solved with a preconditioned conjugate gradients, where the matrix-vector product is only implicitly given. We derive an overlapping block preconditioner based on additive Schwarz methods for preconditioning the reduced system.
This research has been partially supported by NSF grants DMS-9707040, ACR-9721388, ACR-9814651, CCR- 9902022, and CCR-9988165, and National Computational Science Alliance grant OCE980001 (utilizing the University of Illinois Origin 2000 and the University of New Mexico Los Lobos systems).
Douglas, Craig C.; Haase, Gundolf; and Iskandarani, Mohamed, "An Additive Schwarz Preconditioner for the Spectral Element Ocean Model Formulation of the Shallow Water Equations" (2003). Computer Science Faculty Publications. 10.