A framework for mimetic discretization of the rotating shallow-water equations on arbitrary polygonal grids

Copyright © 2012 Society for Industrial and Applied MathematicsAccurate simulation of atmospheric flow in weather and climate prediction models requires the discretization of the governing equations to have a number of desirable properties. Although these properties can be achieved relatively straightforwardly on a latitude-longitude grid, they are much more challenging on the various quasi-uniform spherical grids that are now under consideration. A recently developed scheme—called TRiSK—has these desirable properties on grids that have an orthogonal dual. The present work extends the TRiSK scheme into a more general framework suitable for grids that have a nonorthogonal dual, such as the equiangular cubed sphere. We also show that this framework fits within the wider framework of mimetic discretizations and discrete exterior calculus. One key ingredient is the definition of certain mapping operators that are discrete analogues of the Hodge star operator, enabling the definition of a compatible inner product. Discrete Coriolis terms are also included within the mimetic framework, and in such a way as to conserve energy and ensure that discrete geostrophic balance can be maintained; this requires the definition of a further mapping operator, with special properties, that transfers the discrete velocity field from the primal to the dual grid

A primal-dual mimetic finite element scheme for the rotating shallow water equations on polygonal spherical meshes

Copyright © 2015 Elsevier. NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Computational Physics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Computational Physics Vol. 290 (2015), DOI: 10.1016/j.jcp.2015.02.045A new numerical method is presented for solving the shallow water equations on a rotating sphere using quasi-uniform polygonal meshes. The method uses special families of finite element function spaces to mimic key mathematical properties of the continuous equations and thereby capture several desirable physical properties related to balance and conservation. The method relies on two novel features. The first is the use of compound finite elements to provide suitable finite element spaces on general polygonal meshes. The second is the use of dual finite element spaces on the dual of the original mesh, along with suitably defined discrete Hodge star operators to map between the primal and dual meshes, enabling the use of a finite volume scheme on the dual mesh to compute potential vorticity fluxes. The resulting method has the same mimetic properties as a finite volume method presented previously, but is more accurate on a number of standard test cases.Natural Environment Research Council under the “GungHo” projec

Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere

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 illustrate our discretisation with some standard rotating sphere test problems.Comment: accepted versio

Compatible finite element methods for numerical weather prediction

This article takes the form of a tutorial on the use of a particular class of mixed finite element methods, which can be thought of as the finite element extension of the C-grid staggered finite difference method. The class is often referred to as compatible finite elements, mimetic finite elements, discrete differential forms or finite element exterior calculus. We provide an elementary introduction in the case of the one-dimensional wave equation, before summarising recent results in applications to the rotating shallow water equations on the sphere, before taking an outlook towards applications in three-dimensional compressible dynamical cores.Comment: To appear in ECMWF Seminar proceedings 201

Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere

This is the final version. Available from Elsevier via the DOI in this record.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.Natural Environment Research Council (NERC)Natural Environment Research Council (NERC)Engineering and Physical Sciences Research Council (EPSRC)Engineering and Physical Sciences Research Council (EPSRC

A mimetic, semi-implicit, forward-in-time, finite volume shallow water model: comparison of hexagonal–icosahedral and cubed-sphere grids

A new algorithm is presented for the solution of the shallow water equations on quasi-uniform spherical grids. It combines a mimetic finite volume spatial discretization with a Crank–Nicolson time discretization of fast waves and an accurate and conservative forward-in-time advection scheme for mass and potential vorticity (PV). The algorithm is implemented and tested on two families of grids: hexagonal–icosahedral Voronoi grids, and modified equiangular cubed-sphere grids. <br><br> Results of a variety of tests are presented, including convergence of the discrete scalar Laplacian and Coriolis operators, advection, solid body rotation, flow over an isolated mountain, and a barotropically unstable jet. The results confirm a number of desirable properties for which the scheme was designed: exact mass conservation, very good available energy and potential enstrophy conservation, consistent mass, PV and tracer transport, and good preservation of balance including vanishing &nabla; &times; &nabla;, steady geostrophic modes, and accurate PV advection. The scheme is stable for large wave Courant numbers and advective Courant numbers up to about 1. <br><br> In the most idealized tests the overall accuracy of the scheme appears to be limited by the accuracy of the Coriolis and other mimetic spatial operators, particularly on the cubed-sphere grid. On the hexagonal grid there is no evidence for damaging effects of computational Rossby modes, despite attempts to force them explicitly