A mixed finite‐element, finite‐volume, semi‐implicit discretization for atmospheric dynamics: Cartesian geometry

Melvin, Thomas; Benacchio, Tommaso; Shipway, Ben J.; Wood, Nigel; Thuburn, John; Cotter, Colin J.

doi:10.1002/qj.3501

Cited by 45 publications

(115 citation statements)

References 34 publications

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“…Driven by its negative buoyancy, the initial perturbation moves downwards, impacts the bottom boundary and travels sideways developing vortices (Figure 3). The numerical solution converges with increasing spatial resolution (Figure 4), and the final perturbation amplitude and front position agree with published results ( Table 1, for comparison see, e.g., [17] and the similar table in [30]). The final minimum potential temperature perturbation at 25 m resolution agrees with the result in [30] up to the third decimal digit.…”

Section: Density Currentsupporting

confidence: 87%

“…Examples of operational dynamical cores using semi-implicit timeintegrations strategies are the ECMWF 1 's IFS [19], that discretizes the hydrostatic primitive equations, and the UK Met Office's ENDGame [9,50]. In particular, ENDGame uses a double-loop structure in the implicit solver entailing four solves per time step in its operational incarnation, a strategy carried over in recent developments [30], and allowing non-operational configurations to run stably and second-order accurately without additional numerical damping (for operational forecasts, a small amount of off-centering is usually employed for safety reasons). By contrast, many other semi-implicit or time-split explicit discretizations resort to off-centering, divergence damping [10], or otherwise artificial diffusion in order to quell numerical instabilities.…”

Section: Related Numerical Schemes In the Literaturementioning

confidence: 99%

“…However, as noted in [49], whose model does not use pertubations, large deviations from the background state may question the assumptions underpinning the resulting system. In [50] and [30], the authors use the model state computed at the previous time step as evolving background profile.…”

Section: Related Numerical Schemes In the Literaturementioning

confidence: 99%

See 2 more Smart Citations

A Semi-Implicit Compressible Model for Atmospheric Flows with Seamless Access to Soundproof and Hydrostatic Dynamics

Benacchio

Klein

2019

Monthly Weather Review

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We introduce a second-order numerical scheme for compressible atmospheric motions at small to planetary scales. The collocated finite volume method treats the advection of mass, momentum, and mass-weighted potential temperature in conservation form while relying on Exner pressure for the pressure gradient term. It discretises the rotating compressible equations by evolving full variables rather than perturbations around a background state, and operates with time steps constrained by the advection speed only. Perturbation variables are only used as auxiliary quantities in the formulation of the elliptic problem. Borrowing ideas on forward-in-time differencing, the algorithm reframes the authors' previously proposed schemes into a sequence of implicit midpoint, advection, and implicit trapezoidal steps that allows for a time integration unconstrained by the internal gravity wave speed. Compared with existing approaches, results on a range of benchmarks of nonhydrostaticand hydrostatic-scale dynamics are competitive. The test suite includes a new planetary-scale inertia-gravity wave test highlighting the properties of the scheme and its large time step capabilities. In the hydrostatic-scale cases the model is run in pseudoincompressible and hydrostatic mode with simple switching within a uniform discretization framework. The differences with the compressible runs return expected relative magnitudes. By providing seamless access to soundproof and hydrostatic dynamics, the developments represent a necessary step towards an all-scale blended multimodel solver. 2

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Section: Density Currentsupporting

confidence: 87%

Section: Related Numerical Schemes In the Literaturementioning

confidence: 99%

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A Semi-Implicit Compressible Model for Atmospheric Flows with Seamless Access to Soundproof and Hydrostatic Dynamics

Benacchio

Klein

2019

Monthly Weather Review

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“…This test case is especially challenging for our higher order spectral element formulation, firstly since we have not applied any sort of upwinding or monotonicity preservation method to either the continuity or temperature equation, and secondly because unlike other models we formulate our energy equation as a flux form equation for the density weighted potential temperature, Θ = ρθ, from which the potential temperature, θ, is then diagnosed. This is in contrast to a material form of the potential temperature advection, in which upwinding is directly applied in the flux reconstruction [12]. For this reason we configure our model with slightly higher resolution than that specified [40].…”

Section: Non-hydrostatic Gravity Wavementioning

confidence: 99%

A mixed mimetic spectral element model of the 3D compressible Euler equations on the cubed sphere

Lee

Palha

2020

Journal of Computational Physics

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A model of the three-dimensional rotating compressible Euler equations on the cubed sphere is presented. The model uses a mixed mimetic spectral element discretization which allows for the exact exchanges of kinetic, internal and potential energy via the compatibility properties of the chosen function spaces. A Strang carryover dimensional splitting procedure is used, with the horizontal dynamics solved explicitly and the vertical dynamics solved implicitly so as to avoid the CFL restriction of the vertical sound waves. The function spaces used to represent the horizontal dynamics are discontinuous across vertical element boundaries, such that each horizontal layer is solved independently so as to avoid the need to invert a global 3D mass matrix, while the function spaces used to represent the vertical dynamics are similarly discontinuous across horizontal element boundaries, allowing for the serial solution of the vertical dynamics independently for each horizontal element. The model is validated against standard test cases for baroclinic instability within an otherwise hydrostatically and geostrophically balanced atmosphere, and a nonhydrostatic gravity wave as driven by a temperature perturbation.

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“…Pairings including the RT or BDM spaces such as RT k × DG k−1 or BDM k × DG k−1 fall within the set of compatible mixed spaces ideal for geophysical fluids (Cotter and Shipton, 2012;Natale et al, 2016;Melvin et al, 2018). In particular, the lowestorder RT method (RT 1 × DG 0 ) on a structured quadrilateral grid (such as the latitude-longitude grid used many operational dynamical cores) corresponds to the Arakawa C-grid finite difference discretization.…”

Section: Semi-implicit Finite Element Discretizationmentioning

confidence: 99%

Slate: extending Firedrake's domain-specific abstraction to hybridized solvers for geoscience and beyond

et al. 2020

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Within the finite element community, discontinuous Galerkin (DG) and mixed finite element methods have become increasingly popular in simulating geophysical flows. However, robust and efficient solvers for the resulting saddle-point and elliptic systems arising from these discretizations continue to be an on-going challenge. One possible approach for addressing this issue is to employ a method known as hybridization, where the discrete equations are transformed such that classic static condensation and local post-processing methods can be employed. However, it is challenging to implement hybridization as performant parallel code within complex models, whilst maintaining separation of concerns between applications scientists and software experts. In this paper, we introduce a domain-specific abstraction within the Firedrake finite element library that permits the rapid execution of these hybridization techniques within a code-generating framework. The resulting framework composes naturally with Firedrake's solver environment, allowing for the implementation of hybridization and static condensation as runtime-configurable preconditioners via the Python interface to PETSc, petsc4py. We provide examples derived from second order elliptic problems and geophysical fluid dynamics. In addition, we demonstrate that hybridization shows great promise for improving the performance of solvers for mixed finite element discretizations of equations related to large-scale geophysical flows.

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A mixed finite‐element, finite‐volume, semi‐implicit discretization for atmospheric dynamics: Cartesian geometry

Cited by 45 publications

References 34 publications

A Semi-Implicit Compressible Model for Atmospheric Flows with Seamless Access to Soundproof and Hydrostatic Dynamics

A Semi-Implicit Compressible Model for Atmospheric Flows with Seamless Access to Soundproof and Hydrostatic Dynamics

A mixed mimetic spectral element model of the 3D compressible Euler equations on the cubed sphere

Slate: extending Firedrake's domain-specific abstraction to hybridized solvers for geoscience and beyond

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