A multigrid solver for semi‐implicit global shallow‐water models

Abstract

The multigrid principle produces fast solvers for systems of algebraic equations, particularly those that arise from discretizing elliptic boundary-value problems. A multigrid solver is developed for the discretized two-dimensional elliptic equation on the sphere that arises from a semi-implicit time discretization of the global shallow-water equations. We experiment with different formulations of the semi-implicit scheme that result in variable- IntroductionAn efficient discretization of the atmospheric flow equations should permit a choice of time step dependent on the physical properties of the solution, rather than on more stringent computational properties of the numerical scheme. Such a discretization inevitably requires time-implicit treatment of at least those terms in the model equations that give rise to fast gravity-wave propagation. Fully or partially implicit discretization of the evolution equations leads to large, multi-dimensional systems of algebraic equations to be solved at each time step. Highly efficient implementations of semi-and even fully implicit schemes have been achieved with the aid of splitting or factorization techniques, by which the implicit operator is reduced to simple one-dimensional operators that can easily be inverted by direct solvers (Bates, 1984;Cohn et al., 1985). However, the additional truncation error that is due to simplifying the implicit operator may be unacceptably large for time steps that are physically reasonable (Yakimiw and Robert, 1986;Tanguay and Robert, 1986). The semi-implicit scheme in its original formulation for a grid-point model (Kwizak and Robert, 1971) involves the solution of scalar Helmholtz equations. The combination of this scheme with Lagrangian timeintegration techniques enhances the stability of the method (Robert, 1981;1982) and hence its efficiency, as larger time steps are permitted. Semi-Lagrangian semiimplicit schemes that use three time levels and that are second-order accurate in time (Robert et al., 1985) and similar schemes that use two time levels but are first-order accurate in time (McDonald and Bates, 1988) also lead to scalar Helmholtz equations. These can be solved very efficiently by direct, so-called fast solvers based on reduction methods that rely on particular features of the equations such as separability (Sweet, 1977).Recently developed second-order accurate semi-Lagrangian methods that use only two time levels (Temperton and Staniforth, 1987;Bates, 1988) require the solution of more complicated, non-separable elliptic equations. Direct solution methods are inappropriate for such problems. Present implementations employ iterative solution methods that are quite expensive, since they require several calls to a fast, direct solver at each time step. (Temperton and Staniforth, 1987). Variable-coefficient elliptic equations also arise in models that feature variable grid spacing. Such models have been advocated (Staniforth and Mitchell, 1978;Temperton and Staniforth, 1987) as a way to expand the domain of a limit...