A solution to the trilemma of the moist Charney–Phillips staggering

Bendall, Thomas M.; Wood, Nigel; Thuburn, John; Cotter, Colin J.

doi:10.1002/qj.4406

Cited by 8 publications

(8 citation statements)

References 34 publications

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“…However, the idea is quite general. For example, it has also been tested with a sub-stepped Eulerian method of lines advection scheme similar to that discussed by Bendall et al (2023).…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…However, for such iterative-implicit schemes the expense of advection is compounded because the advection terms then need to be computed multiple times per time step. To mitigate this expense, the GEM, ENDGame, and GungHo iterative solvers take two inner iterations within two outer iterations, with the advection terms recomputed only in the outer iterations-see Bendall et al (2023) for the GungHo case. However, the effectiveness of the inner iterations, in which advection terms are not recomputed, is questionable.…”

Section: Introductionmentioning

confidence: 99%

“…To mitigate this expense, the GEM, ENDGame, and GungHo iterative solvers take two inner iterations within two outer iterations, with the advection terms recomputed only in the outer iterations—see Bendall et al . (2023) for the GungHo case. However, the effectiveness of the inner iterations, in which advection terms are not recomputed, is questionable.…”

Section: Introductionmentioning

confidence: 99%

“…A significant challenge with iterative algorithms like GEM, ENDGame, and GungHo is that, even if the underlying advection scheme is consistent and conservative, the solution produced by the algorithm might only approach consistency and conservation as the algorithm is iterated towards convergence. Unless the iterative updates are explicitly formulated to ensure consistency and conservation, there can be noticeable consistency or conservation errors when using a small number of solver iterations (Bendall et al ., 2023).…”

Section: Introductionmentioning

confidence: 99%

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Consistent, conservative, and efficient advection updatesfor iterative‐implicit atmospheric solvers

Thuburn

2024

Quart J Royal Meteoro Soc

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Atmospheric model dynamical cores that iterate towards a Crank–Nicolson‐like implicit time‐stepping scheme are attractive for operational prediction because their excellent stability properties permit the use of long time steps. However, the long‐time‐step advection schemes used in such models are relatively expensive, and that expense is compounded by the need to compute the advection terms multiple times in the iterative solver. Moreover, unless care is taken in the design of the solver, desirable properties of an advection scheme, such as conservation, consistency, and boundedness, might only be achieved in the unaffordable limit of solver convergence. Here, a modification to such iterative solvers is proposed, similar to the previously published SLIC scheme, in which full advection calculations are made only once per time step, with cheap advection updates made at each solver iteration. This modification significantly reduces the cost of such iterative solvers. It is shown here that the cheap advection updates and the solver back‐substitution calculations can be formulated in such a way that the advection remains conservative, consistent, and bounded no matter how many solver iterations are taken, and not only at solver convergence. The proposed approach is demonstrated in shallow‐water model simulations.

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“…However, the idea is quite general. For example, it has also been tested with a sub-stepped Eulerian method of lines advection scheme similar to that discussed by Bendall et al (2023).…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Consistent, conservative, and efficient advection updatesfor iterative‐implicit atmospheric solvers

Thuburn

2024

Quart J Royal Meteoro Soc

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“…This produced the first atmospheric simulations using compatible finite elements with moist physics. Bendall, Wood, Thuburn and Cotter (2022) then showed how to achieve this framework whilst conserving mass and total moisture.…”

Section: Recovered Space Schemesmentioning

confidence: 99%

Compatible finite element methods for geophysical fluid dynamics

Cotter¹

2023

Acta Numerica

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This article surveys research on the application of compatible finite element methods to large-scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa’s C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge–Helmholtz decomposition, which has been used to exclude the possibility of several types of spurious oscillations from linear equations of geophysical flow. This means that compatible finite element spaces provide a useful framework for building dynamical cores. In this article we introduce the main concepts of compatible finite element spaces, and discuss their wave propagation properties. We survey some methods for discretizing the transport terms that arise in dynamical core equation systems, and provide some example discretizations, briefly discussing their iterative solution. Then we focus on the recent use of compatible finite element spaces in designing structure preserving methods, surveying variational discretizations, Poisson bracket discretizations and consistent vorticity transport.

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A mixed finite‐element, finite‐volume, semi‐implicit discretisation for atmospheric dynamics: Spherical geometry

Melvin,

Shipway,

Wood

et al. 2024

Quart J Royal Meteoro Soc

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Our previously described reformulation of the Met Office's dynamical core for weather and climate prediction is extended to spherical domains using a cubed‐sphere mesh. We update the semi‐implicit mixed finite‐element formulation to be suitable for spherical domains. In particular, the finite‐volume transport scheme is extended to take account of non‐uniform, non‐orthogonal meshes and uses an advective‐then‐flux formulation so that increment from the transport scheme is linear in the divergence. The resulting model is then applied to a standard set of dry dynamical core tests and compared with the existing semi‐implicit semi‐Lagrangian dynamical core currently used in the Met Office's operational model.

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A solution to the trilemma of the moist Charney–Phillips staggering

Abstract: The version presented here may differ from the published version. If citing, you are advised to consult the published version for pagination, volume/issue and date of publication

Cited by 8 publications

References 34 publications

Consistent, conservative, and efficient advection updatesfor iterative‐implicit atmospheric solvers

Consistent, conservative, and efficient advection updatesfor iterative‐implicit atmospheric solvers

Compatible finite element methods for geophysical fluid dynamics

A mixed finite‐element, finite‐volume, semi‐implicit discretisation for atmospheric dynamics: Spherical geometry

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