Multigrid Methods for Elliptic Problems: A Review

Fulton, Scott R.; Ciesielski, Paul E.; Schubert, Wayne H.

doi:10.1175/1520-0493(1986)114<0943:mmfepa>2.0.co;2

Cited by 96 publications

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“…Gauss-Seidel, SOR) in which the solution estimate is updated point by point, are inadequate. This is typical for anisotropic problems, which can be demonstrated by a simple local mode analysis (Fulton et al, 1986;Stuben and Trottenberg, 1982;Brandt, 1984). In such problems line relaxation should be used, in which the values along an entire line of grid points are updated simultaneously (e.g.…”

Section: Relaxation Schemementioning

confidence: 92%

“…Furthermore, these solvers achieve optimal efficiency in the sense that the total operation count is a linear function of the number of unknowns. An excellent explanation of the basic concepts and techniques of multigrid methods for elliptic problems has appeared in the meteorological literature in Fulton et al (1986).…”

mentioning

confidence: 99%

“…For a more complete treatment, see Fulton et al (1986), Stuben and Trottenberg (1982), Brandt (1984) and McCormick (1987). Section 3 contains our formulations of the semi-implicit scheme and the elliptic equations that result.…”

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confidence: 99%

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

1990

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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...

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Section: Relaxation Schemementioning

confidence: 92%

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confidence: 99%

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

1990

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“…The Helmholtz problem [(44)] is solved using a geometric multigrid method (e.g., Fulton et al 1986). The grid is coarsened only in the horizontal direction.…”

Section: Helmholtz Solvermentioning

confidence: 99%

“…Methods Fluids). See, for example, Fulton et al (1986) for a clear introduction to multigrid methods in the context of atmospheric modeling. Such methods require only local (rather than global) data communication at each smoother iteration.…”

Section: Introductionmentioning

confidence: 99%

A Semi-Implicit Version of the MPAS-Atmosphere Dynamical Core

Sandbach

Thuburn

Vassilev

et al. 2015

Monthly Weather Review

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An important question for atmospheric modeling is the viability of semi-implicit time integration schemes on massively parallel computing architectures. Semi-implicit schemes can provide increased stability and accuracy. However, they require the solution of an elliptic problem at each time step, creating concerns about their parallel efficiency and scalability. Here, a semi-implicit (SI) version of the Model for Prediction Across Scales (MPAS) is developed and compared with the original model version, which uses a split Runge-Kutta (SRK3) time integration scheme. The SI scheme is based on a quasi-Newton iteration toward a Crank-Nicolson scheme. Each Newton iteration requires the solution of a Helmholtz problem; here, the Helmholtz problem is derived, and its solution using a geometric multigrid method is described. On two standard test cases, a midlatitude baroclinic wave and a small-planet nonhydrostatic gravity wave, the SI and SRK3 versions produce almost identical results. On the baroclinic wave test, the SI version can use somewhat larger time steps (about 60%) than the SRK3 version before losing stability. The SI version costs 10%-20% more per step than the SRK3 version, and the weak and strong scalability characteristics of the two versions are very similar for the processor configurations the authors have been able to test (up to 1920 processors). Because of the spatial discretization of the pressure gradient in the lowest model layer, the SI version becomes unstable in the presence of realistic orography. Some further work will be needed to demonstrate the viability of the SI scheme in this case.

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Framework for improvement by vertical enhancement: A simple approach to improve representation of low and high‐level clouds in large‐scale models

Yamaguchi

Feingold

Larson

2017

J Adv Model Earth Syst

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Low and high clouds of shallow extent, especially stratocumulus and even more so for high‐level cirrus clouds that reside where vertical resolution is particularly coarse, are poorly represented in large‐scale models such as global climate models and weather forecasting models. This adversely affects, among others, estimation of cloud feedbacks for climate prediction and weather forecasts. Here we address vertical resolution as a reason for the failure of these models to adequately represent shallow clouds. We introduce a new methodology, the Framework for Improvement by Vertical Enhancement (FIVE). FIVE computes selected processes on a one‐dimensional vertical grid with local high resolution in the boundary layer and near the tropopause. In addition to the host model, variables on the locally high‐resolution grid are predicted in parallel so that high‐resolution information is retained. By exchanging tendencies with one another, the host model and high‐resolution field are always synchronized. The methodology is demonstrated for drizzling stratocumulus capped by a sharp inversion. First, FIVE is applied to a single‐column model to identify the cause of biases associated with computing an assigned process at low resolution. Second, a two‐dimensional regional model coupled with FIVE is shown to produce results comparable to those performed with high vertical resolution. FIVE is thus expected to represent low clouds more realistically and hence reduce the low‐cloud bias in large‐scale models. Finally, we propose a number of methods that will be developed and tested to further optimize FIVE.

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Multigrid Methods for Elliptic Problems: A Review

Cited by 96 publications

References 0 publications

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

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

A Semi-Implicit Version of the MPAS-Atmosphere Dynamical Core

Framework for improvement by vertical enhancement: A simple approach to improve representation of low and high‐level clouds in large‐scale models

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