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**Robust multigrid solution of the shallow water balance equations.**
*(English)*
Zbl 0919.76063

Summary: Balance equation models describing accurate, gravity-wave-free states on the so-called “slow manifold” of the primitive equations are of wide and growing interest, both theoretical and practical, for geophysical fluid dynamics. As a particular example with only two spatial dimensions, the shallow water balance equations are a coupled, highly nonlinear, nonsymmetric system of partial differential equations, for which only ad hoc solvers of limited robustness have previously been developed. Two multigrid algorithms are presented, one explicit and one implicit in time, which are shown by analysis and numerical examples to be efficient and robust solution techniques for this system. These examples include modons and shallow water turbulence at finite Rossby number. It is found that, in some regimes of physical parameters, quite large time steps can be taken with the implicit solver, with little loss of accuracy or efficiency. This is interpreted as due to significantly slower evolutionary rates of the dominant patterns compared to parcel trajectory rates.

### MSC:

76M20 | Finite difference methods applied to problems in fluid mechanics |

76B65 | Rossby waves (MSC2010) |

86A05 | Hydrology, hydrography, oceanography |

65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |