Improving the accuracy of discretisations of the vector transport equation on the lowest-order quadrilateral Raviart-Thomas finite elements

Bendall, Thomas M.; Wimmer, Golo A.

doi:10.1016/j.jcp.2022.111834

Cited by 3 publications

(12 citation statements)

References 37 publications

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“…A probably more stable approach is to use the form (4.15) for the velocity equation, substituting ¯ in for , and in for . This was done using quadrilateral 0 elements in Bendall and Wimmer (2023). Another approach is to replace the implicit midpoint rule update for +1 and +1 given ¯ , instead using an explicit transport step (or several substeps).…”

Section: Rotating Shallow Water Equations On the Spherementioning

confidence: 99%

“…This can facilitate more sophisticated transport schemes with limiters that are hard to implement in implicit schemes. This was also done using quadrilateral 0 elements in Bendall and Wimmer (2023).…”

Section: Rotating Shallow Water Equations On the Spherementioning

confidence: 99%

“…This means that if we use the above scheme then it will only be first order accurate. Bendall and Wimmer (2023) looked at using an auxiliary field ∈ V 0 ℎ with = − 0 , i.e. approximates ∇ ⊥ ⋅ if the solution domain Ω has no boundary.…”

mentioning

confidence: 99%

“…Modifying the test function according to the SUPG approach is ungainly here, because of the surface terms in ′ . Instead, Bendall and Wimmer (2023) used a residual based approach, writing ⟨ ,…”

mentioning

confidence: 99%

“…Bendall and Wimmer (2023) showed in numerical experiments (including a nonlinear rotating shallow water equation test case on the sphere) that this discretisation produces second order accurate solutions using lowest order RT quadrilateral elements on a cubed sphere grid, whilst (4.18) only produces first order accurate solutions. A predecessor of this scheme was considered in Kent, Melvin and Wimmer (2022), but without the consistent definition of and .…”

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confidence: 99%

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Compatible finite element methods for geophysical fluid dynamics

Cotter¹

2023

Preprint

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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 which are 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 discretising the transport terms that arise in dynamical core equation systems, and provide some example discretisations, briefly discussing their iterative solution. Then we focus on the recent use of compatible finite element spaces in designing structure preserving methods,surveying variational discretisations, Poisson bracket discretisations, and consistent vorticity transport.

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Section: Rotating Shallow Water Equations On the Spherementioning

confidence: 99%

Section: Rotating Shallow Water Equations On the Spherementioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Compatible finite element methods for geophysical fluid dynamics

Cotter¹

2023

Preprint

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Physics–dynamics–chemistry coupling across different meshes in LFRic‐Atmosphere: Formulation and idealised tests

Brown,

Bendall,

Boutle

et al. 2024

Quart J Royal Meteoro Soc

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The main components of an atmospheric model for numerical weather prediction are the “dynamical core,” which describes the resolved flow, and the “physical parametrisation,” which capture the effects of non‐fluid and non‐resolved fluid processes. Additionally, models used for air quality or climate applications may include a component that represents the evolution of chemicals and aerosols within the atmosphere. Though, traditionally, all these components use the same mesh with the same grid spacing, we present a formulation for the different components to use a series of nested meshes, with different horizontal grid spacings. This gives the model greater flexibility in the allocation of computational resources, so that resolution can be targeted to those parts that provide the greatest benefits in accuracy. The formulation presented here concerns the methods for mapping fields between meshes and is designed for the compatible finite‐element discretisation used by LFRic‐Atmosphere, the Met Office's next‐generation atmosphere model. Key properties of the formulation include the consistent and conservative transport of tracers on a mesh that is coarser than the dynamical core, and the handling of moisture to ensure mass conservation without generation of unphysical negative values. Having presented the formulation, it is then demonstrated through a series of idealised test cases that show the feasibility of this approach.

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Compatible finite element methods for geophysical fluid dynamics

Cotter¹

2023

Acta Numerica

View full text Add to dashboard Cite

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.

show abstract

Improving the accuracy of discretisations of the vector transport equation on the lowest-order quadrilateral Raviart-Thomas finite elements

Cited by 3 publications

References 37 publications

Compatible finite element methods for geophysical fluid dynamics

Compatible finite element methods for geophysical fluid dynamics

Physics–dynamics–chemistry coupling across different meshes in LFRic‐Atmosphere: Formulation and idealised tests

Compatible finite element methods for geophysical fluid dynamics

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