Energy‐ and enstrophy‐conserving schemes for the shallow‐water equations, based on mimetic finite elements

McRae, Andrew T. T.; Cotter, Colin J.

doi:10.1002/qj.2291

Cited by 58 publications

(85 citation statements)

References 61 publications

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“…We note that the stream function/vorticity SUPG scheme can be re‐interpreted as a mixed velocity–pressure scheme, since if ψ ∈ V 0 then ∇ ⊥ ψ spans the divergence‐free subspace of V 1 and, in the absence of boundaries, ω ∈ V 0 can be obtained by solving

{(γ, ω)}_{Ω} = - {(\nabla^{⊥} γ, bold-italicu)}_{Ω}

for all test functions γ in V 0 . Then we obtain an equivalent formulation that is an approximation of the incompressible Euler equations in Lie derivative form, with

\begin{matrix} alignleft \\ align-1 a (u; u, v) = & align-2 {(v, u^{⊥} ω)}_{Ω}, \\ align-1 & align-2 \end{matrix}

\begin{matrix} alignleft \\ align-1 s (u; u, v) = & align-2 {(v, - u^{⊥} τ_{s} R (ω))}_{Ω} . \end{matrix}

This formulation allows us to extend the energy‐conserving SUPG approach to compatible finite‐element method discretizations of the shallow‐water equations in velocity–height formulation, following McRae and Cotter (2014). The equivalent discretization is

\begin{matrix} alignleft \\ align-1 {(\partial_{t} u, v)}_{Ω} + {(v, F^{⊥} q)}_{Ω} & align-2 \\ align-1 + {(v, - F^{⊥} τ_{s} (q_{t} + u \cdot \nabla q))}_{Ω} & align-2 \\ align-1 - \end{matrix}

…”

Section: Finite‐element Frameworkmentioning

confidence: 99%

Scale‐selective dissipation in energy‐conserving finite‐element schemes for two‐dimensional turbulence

Natale

Cotter

2017

Quart J Royal Meteoro Soc

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We analyze the multiscale properties of energy‐conserving upwind‐stabilized finite‐element discretizations of the two‐dimensional incompressible Euler equations. We focus our attention on two particular methods: the Lie derivative discretization introduced by Natale and Cotter and the Streamline Upwind/Petrov–Galerkin (SUPG) discretization of the vorticity advection equation. Such discretizations provide control on enstrophy by modelling different types of scale interactions. We quantify the performance of the schemes in reproducing the non‐local energy backscatter that characterizes two‐dimensional turbulent flows.

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{(γ, ω)}_{Ω} = - {(\nabla^{⊥} γ, bold-italicu)}_{Ω}

for all test functions γ in V 0 . Then we obtain an equivalent formulation that is an approximation of the incompressible Euler equations in Lie derivative form, with

\begin{matrix} alignleft \\ align-1 a (u; u, v) = & align-2 {(v, u^{⊥} ω)}_{Ω}, \\ align-1 & align-2 \end{matrix}

\begin{matrix} alignleft \\ align-1 s (u; u, v) = & align-2 {(v, - u^{⊥} τ_{s} R (ω))}_{Ω} . \end{matrix}

\begin{matrix} alignleft \\ align-1 {(\partial_{t} u, v)}_{Ω} + {(v, F^{⊥} q)}_{Ω} & align-2 \\ align-1 + {(v, - F^{⊥} τ_{s} (q_{t} + u \cdot \nabla q))}_{Ω} & align-2 \\ align-1 - \end{matrix}

…”

Section: Finite‐element Frameworkmentioning

confidence: 99%

Scale‐selective dissipation in energy‐conserving finite‐element schemes for two‐dimensional turbulence

Natale

Cotter

2017

Quart J Royal Meteoro Soc

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“…In the previous section we introduced bracket (2.3.17) -(2.3.19), which is based on the variational scheme (2.2.5) as given in [12] and extends it to include upwinding in the depth and velocity fields.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…In this section, we extend the energy conserving space discretisation for the rotating shallow water equations presented in [12], by introducing an upwind formulation for the depth field, and further using the energy conserving velocity field upwinding as presented for the incompressible Euler equations in [17].…”

Section: Energy Conserving Formulation For Rotating Shallow Water Equmentioning

confidence: 99%

“…An energy conserving space discretisation of the rotating shallow water equations in boundary-free domains was presented in [12]. It is based on the Hamiltonian framework as reviewed above, together with a compatible finite element discretisation.…”

Section: Existing Discretised Formulation Without Upwinding Termsmentioning

confidence: 99%

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Energy conserving upwinded compatible finite element schemes for the rotating shallow water equations

Wimmer

Cotter

Bauer

2020

Journal of Computational Physics

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We present an energy conserving space discretisation of the rotating shallow water equations using compatible finite elements. It is based on an energy and enstrophy conserving Hamiltonian formulation as described in McRae and Cotter (2014), and extends it to include upwinding in the velocity and depth advection to increase stability. Upwinding for velocity in an energy conserving context was introduced for the incompressible Euler equations in Natale and Cotter (2017), while upwinding in the depth field in a Hamiltonian finite element context is newly described here. The energy conserving property is validated by coupling the spatial discretisation to an energy conserving time discretisation. Further, the discretisation is demonstrated to lead to an improved field development with respect to stability when upwinding in the depth field is included.

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Shallow‐water equations on a spherical multiple‐cell grid

2017

Quart J Royal Meteoro Soc

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The shallow‐water equations (SWEs) are discretized on a spherical multiple‐cell (SMC) grid and validated with classical tests, including steady zonal flow over flat or hill floor, the Rossby–Haurwitz wave and the unstable zonal jet. The numerical schemes follow the conventional latitude–longitude (lat‐lon) grid ones at low latitudes but switch to a fixed‐reference direction system to define vector variables at high latitudes for reduction of polar curvature errors. Semi‐implicit schemes are used for both the Coriolis terms and the potential energy gradients. A C‐grid mass‐conserving advection–diffusion scheme is used for the thickness variable and mixed A‐ and D‐grid schemes are applied on the momentum equation. Tests demonstrate that the two reference directions work fine on the SMC grid and the SWEs model is stable as long as numerical noises are suppressed with enough smoothing. This implies that other reduced grids could be used for dynamical models if the vector polar problem is properly solved. The unstructured SMC grid also supports multi‐resolutions in mesh refinement style and a refined area is included to show grid refinement effects. For steady smooth flows, the refinement is almost unnoticeable while in unstable jet case, it may stir up instability ripples in the jet flow unless strong average is applied.

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Energy‐ and enstrophy‐conserving schemes for the shallow‐water equations, based on mimetic finite elements

Cited by 58 publications

References 61 publications

Scale‐selective dissipation in energy‐conserving finite‐element schemes for two‐dimensional turbulence

Scale‐selective dissipation in energy‐conserving finite‐element schemes for two‐dimensional turbulence

Energy conserving upwinded compatible finite element schemes for the rotating shallow water equations

Shallow‐water equations on a spherical multiple‐cell grid

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