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Domain decomposition for the shallow water equations. (English) Zbl 0830.76061

Keyes, David E. (ed.) et al., Domain decomposition methods in scientific and engineering computing. Proceedings of the 7th international conference on domain decomposition, October 27-30, 1993, Pennsylvania State University, PA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 180, 485-490 (1994).
The study deals with the numerical solution of the shallow water equations (SEWs). They are obtained from the three-dimensional Euler equations for a shallow layer of inviscid liquid on a sphere rotating with constant angular speed, reduced under the assumptions of constant density, hydrostatic balance in the radial direction and the Taylor- Proudman hypothesis. The unknown fields are vertically averaged velocities in the longitudinal and latitudinal directions (momentum equations) and free-surface potential. Domain decomposition (Krylov- Schwarz) methods are suggested for the parallel implicit solution of the unsteady geopotential equation that arises when semi-implicit, semi- Lagrangian (SISL) methods are employed in the long-time integration of the SWEs. SISL methods permit timesteps on the scale of Rossby wave dynamics and thus satisfy the stability bound of an explicit method. However, the price of the semi-implicitness is a global elliptic problem on a multiply-connected semi-periodic domain with variable coefficients that become singular at the poles of latitude-longitude coordinate systems. Elliptic solvers based on domain decomposition offer flexibility in discretization and good algebraic convergence properties. Demonstrating the applicability of elliptic-based domain decomposition precondionness to the SWEs opens the door to a variety of parallel implicit models in long-time integration geophysics applications.
For the entire collection see [Zbl 0809.00026].
Reviewer: J.Siekmann (Essen)

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76U05 General theory of rotating fluids
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
86A05 Hydrology, hydrography, oceanography
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