Compatible finite element spaces for geophysical fluid dynamics

Abstract

Compatible finite elements provide a framework for preserving important structures in equations of geophysical fluid 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 fluid 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.

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

“…Although this has minimal effect on the results shown here, upwind‐based schemes such as that used by Natale et al . () are being investigated to improve on this aspect. The method of lines scheme used to transport scalars imposes a time‐step restriction on the model due to the CFL number constraint. Future work will investigate removing this constraint through using a flux‐form semi‐Lagrangian scheme for the scalars and possibly also for the velocity components.…”

“…Therefore the lowest-order version of the mixed finite-element scheme is used here but extended to the three-dimensional Euler equations (section 4). This is achieved by extending the hierarchy of finite-element spaces to include the particular temperature space proposed by Guerra and Ullrich (2016); Natale et al (2016) and Melvin et al (2018). At lowest order, this space resembles a finite-difference Charney-Phillips staggering of temperature.…”

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

“…The use of stabilized finite‐element methods in atmospheric simulations has been investigated by Marras et al (2013). Compatible finite‐element methods have been developed in order to extend conservation and stability properties from C‐grid staggered finite‐difference methods to the finite‐element setting (Cotter and Shipton, 2012; McRae and Cotter, 2014; Natale et al , 2016); these methods provide the context for this article.…”

Choice of function spaces for thermodynamic variables in mixed finite‐element methods

“…Mixed finite‐element methods (Cotter and Shipton, ) have a number of properties that make them appealing for modelling geophysical flows: they are inf‐sup stable and mimetic in that they carry over continuous properties, such as irrotationality of the gradient operator or solenoidality of the curl operator, at the discrete level (see also Natale et al () and references therein). Compatible finite‐element methods are useful for building discretizations for numerical weather prediction since they allow (a) flexibility in the type of grids that can be used, since they do not require orthogonal grids for their fundamental properties,(b) flexibility in the choice of finite element spaces leading to discretizations on triangular meshes that maintain the 2:1 ratio of velocity to pressure degrees of freedom that is necessary to avoid spurious modes (Staniforth and Thuburn, ), and(c) flexibility in the choice of the consistency order of the approximation through using higher‐order polynomials.The grid flexibility is particularly important since it allows the recovery of C‐grid wave dispersion properties on pseudo‐uniform grids such as the cubed sphere that avoid the parallel scalability issues associated with the poles of a latitude–longitude grid.…”

“…A third option of a space with continuous basis functions in both directions corresponds to the natural choice for scalars in the so‐called deRham complex of mixed finite‐element function spaces (Thuburn and Cotter, and references therein). Moreover, the fully continuous option eases integration by parts of the pressure gradient term in the momentum equation of the fully compressible Euler model (Natale et al ). Unlike in the first two cases, with the third option the buoyancy degrees of freedom are horizontally staggered with respect to those of the vertical velocity.…”