Imperial College London


Faculty of Natural SciencesDepartment of Mathematics

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8 results found

Bendall TM, Cotter CJ, Shipton J, 2019, The 'recovered space' advection scheme for lowest-order compatible finite element methods, Journal of Computational Physics, Vol: 390, Pages: 342-358, ISSN: 0021-9991

We present a new compatible finite element advection scheme for the compressible Euler equations. Unlike the discretisations described in Cotter and Kuzmin (2016) and Shipton et al. (2018), the discretisation uses the lowest-order family of compatible finite element spaces, but still retains second-order numerical accuracy. This scheme obtains this second-order accuracy by first ‘recovering’ the function in higher-order spaces, before using the discontinuous Galerkin advection schemes of Cotter and Kuzmin (2016). As well as describing the scheme, we also present its stability properties and a strategy for ensuring boundedness. We then demonstrate its properties through some numerical tests, before presenting its use within a model solving the compressible Euler equations.

Journal article

Shipton J, Gibson T, Cotter CJ, 2018, Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere, Journal of Computational Physics, Vol: 375, Pages: 1121-1137, ISSN: 0021-9991

We describe a compatible finite element discretisation for the shallow water equations on the rotating sphere, concentrating on integrating consistent upwind stabilisation into the framework. Although the prognostic variables are velocity and layer depth, the discretisation has a diagnostic potential vorticity that satisfies a stable upwinded advection equation through a Taylor–Galerkin scheme; this provides a mechanism for dissipating enstrophy at the gridscale whilst retaining optimal order consistency. We also use upwind discontinuous Galerkin schemes for the transport of layer depth. These transport schemes are incorporated into a semi-implicit formulation that is facilitated by a hybridisation method for solving the resulting mixed Helmholtz equation. We demonstrate that our discretisation achieves the expected second order convergence and provide results from some standard rotating sphere test problems.

Journal article

Yamazaki H, Shipton J, Cullen MJP, Mitchell L, Cotter CJet al., 2017, Vertical slice modelling of nonlinear Eady waves using a compatible finite element method, Journal of Computational Physics, Vol: 343, Pages: 130-149, ISSN: 1090-2716

A vertical slice model is developed for the Euler–Boussinesq equations with a constant temperature gradient in the direction normal to the slice (the Eady–Boussinesq model). The model is a solution of the full three-dimensional equations with no variation normal to the slice, which is an idealised problem used to study the formation and subsequent evolution of weather fronts. A compatible finite element method is used to discretise the governing equations. To extend the Charney–Phillips grid staggering in the compatible finite element framework, we use the same node locations for buoyancy as the vertical part of velocity and apply a transport scheme for a partially continuous finite element space. For the time discretisation, we solve the semi-implicit equations together with an explicit strong-stability-preserving Runge–Kutta scheme to all of the advection terms. The model reproduces several quasi-periodic lifecycles of fronts despite the presence of strong discontinuities. An asymptotic limit analysis based on the semi-geostrophic theory shows that the model solutions are converging to a solution in cross-front geostrophic balance. The results are consistent with the previous results using finite difference methods, indicating that the compatible finite element method is performing as well as finite difference methods for this test problem. We observe dissipation of kinetic energy of the cross-front velocity in the model due to the lack of resolution at the fronts, even though the energy loss is not likely to account for the large gap on the strength of the fronts between the model result and the semi-geostrophic limit solution.

Journal article

Natale A, Shipton J, Cotter CJ, 2016, Compatible finite element spaces for geophysical fluid dynamics, Dynamics and Statistics of the Climate System, Vol: 1, ISSN: 2059-6987

Compatible finite elements provide a framework for preserving important structures in equations of geophysical uid dynamics, and are becoming important in their use for building atmosphere and ocean models. We survey the application of compatible finite element spaces to geophysical uid dynamics, including the application to the nonlinear rotating shallow water equations, and the three-dimensional compressible Euler equations. We summarise analytic results about dispersion relations and conservation properties, and present new results on approximation properties in three dimensions on the sphere, and on hydrostatic balance properties.

Journal article

Maddison JR, Marshall DP, Shipton J, 2015, On the dynamical influence of ocean eddy potential vorticity fluxes, Ocean Modelling, Vol: 92, Pages: 169-182, ISSN: 1463-5011

The impact of eddy potential vorticity fluxes on the dynamical evolution of the flow is obscured by the presence of large and dynamically-inert rotational fluxes. However, the decomposition of eddy potential vorticity fluxes into rotational and divergent components is non-unique in a bounded domain and requires the imposition of an additional boundary condition. Here it is proposed to invoke a one-to-one correspondence between divergent eddy potential vorticity fluxes and non-divergent eddy momentum tendencies in the quasi-geostrophic residual-mean equations in order to select a unique divergent eddy potential vorticity flux. The divergent eddy potential vorticity flux satisfies a zero tangential component boundary condition. In a simply connected domain, the resulting divergent eddy potential vorticity flux satisfies a powerful optimality condition: it is the horizontally oriented divergent flux with minimum L2 norm. Hence there is a well-defined sense in which this approach removes as much of the dynamically inactive eddy potential vorticity flux as possible, and extracts an underlying dynamically active divergent eddy potential vorticity flux. It is shown that this approach leads to a divergent eddy potential vorticity flux which has an intuitive physical interpretation, via a direct relationship to the resulting forcing of the mean circulation.

Journal article

Cotter CJ, Shipton J, 2012, Mixed finite elements for numerical weather prediction, J. Comp. Phys.

Journal article

Ham DA, Farrell PE, Gorman GJ, Maddison JR, Wilson CR, Kramer SC, Shipton J, Collins GS, Cotter CJ, Piggott MDet al., 2009, Spud 1.0: generalising and automating the user interfaces of scientific computer models, Geoscientific Model Development, Vol: 2, Pages: 33-42

The interfaces by which users specify the scenarios to be simulated by scientific computer models are frequently primitive, under-documented and ad-hoc text files which make using the model in question difficult and error-prone and significantly increase the development cost of the model. In this paper, we present a model-independent system, Spud, which formalises the specification of model input formats in terms of formal grammars. This is combined with an automated graphical user interface which guides users to create valid model inputs based on the grammar provided, and a generic options reading module, libspud, which minimises the development cost of adding model options. Together, this provides a user friendly, well documented, self validating user interface which is applicable to a wide range of scientific models and which minimises the developer input required to maintain and extend the model interface.

Journal article

Ham DA, Farrell PE, Gorman GJ, Maddison JR, Wilson CR, Kramer SC, Shipton J, Collins GS, Cotter CJ, Piggott MDet al., 2009, Spud 1.0: generalising and automating the user interfaces of scientific computer models, GEOSCIENTIFIC MODEL DEVELOPMENT, Vol: 2, Pages: 33-42, ISSN: 1991-959X

Journal article

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