90 research outputs found
Compatible finite element methods for geophysical fluid dynamics
This article surveys research on the application of compatible finite element
methods to large scale atmosphere and ocean simulation. Compatible finite
element methods extend Arakawa's C-grid finite difference scheme to the finite
element world. They are constructed from a discrete de Rham complex, which is a
sequence of finite element spaces which are linked by the operators of
differential calculus. The use of discrete de Rham complexes to solve partial
differential equations is well established, but in this article we focus on the
specifics of dynamical cores for simulating weather, oceans and climate. The
most important consequence of the discrete de Rham complex is the
Hodge-Helmholtz decomposition, which has been used to exclude the possibility
of several types of spurious oscillations from linear equations of geophysical
flow. This means that compatible finite element spaces provide a useful
framework for building dynamical cores. In this article we introduce the main
concepts of compatible finite element spaces, and discuss their wave
propagation properties. We survey some methods for discretising the transport
terms that arise in dynamical core equation systems, and provide some example
discretisations, briefly discussing their iterative solution. Then we focus on
the recent use of compatible finite element spaces in designing structure
preserving methods, surveying variational discretisations, Poisson bracket
discretisations, and consistent vorticity transport.Comment: correction of some typo
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
Discrete conservation properties for shallow water flows using mixed mimetic spectral elements
A mixed mimetic spectral element method is applied to solve the rotating
shallow water equations. The mixed method uses the recently developed spectral
element histopolation functions, which exactly satisfy the fundamental theorem
of calculus with respect to the standard Lagrange basis functions in one
dimension. These are used to construct tensor product solution spaces which
satisfy the generalized Stokes theorem, as well as the annihilation of the
gradient operator by the curl and the curl by the divergence. This allows for
the exact conservation of first order moments (mass, vorticity), as well as
quadratic moments (energy, potential enstrophy), subject to the truncation
error of the time stepping scheme. The continuity equation is solved in the
strong form, such that mass conservation holds point wise, while the momentum
equation is solved in the weak form such that vorticity is globally conserved.
While mass, vorticity and energy conservation hold for any quadrature rule,
potential enstrophy conservation is dependent on exact spatial integration. The
method possesses a weak form statement of geostrophic balance due to the
compatible nature of the solution spaces and arbitrarily high order spatial
error convergence
A Quasi-Hamiltonian Discretization of the Thermal Shallow Water Equations
International audienceThe rotating shallow water (RSW) equations are the usual testbed for the development of numerical methods for three-dimensional atmospheric and oceanic models. However, an arguably more useful set of equations are the thermal shallow water equations (TSW), which introduce an additional thermodynamic scalar but retain the single layer, two-dimensional structure of the RSW. As a stepping stone towards a three-dimensional atmospheric dynamical core, this work presents a quasi-Hamiltonian discretization of the thermal shallow water equations using compatible Galerkin methods, building on previous work done for the shallow water equations. Structure-preserving or quasi-Hamiltonian discretizations methods, that discretize the Hamiltonian structure of the equations of motion rather than the equations of motion themselves, have proven to be a powerful tool for the development of models with discrete conservation properties. By combining these ideas with an energy-conserving Poisson time integrator and a careful choice of Galerkin spaces, a large set of desirable properties can be achieved. In particular, for the first time total mass, buoyancy and energy are conserved to machine precision in the fully discrete model
Compatible finite element spaces for geophysical fluid dynamics
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
Hybridizable compatible finite element discretizations for numerical weather prediction: implementation and analysis
There is a current explosion of interest in new numerical methods for atmospheric modeling. A driving force behind this is the need to be able to simulate, with high efficiency, large-scale geophysical flows on increasingly more parallel computer systems. Many current operational models, including that of the UK Met Office, depend on orthogonal meshes, such as the latitude-longitude grid. This facilitates the development of finite difference discretizations with favorable numerical properties. However, such methods suffer from the ``pole problem," which prohibits the model to make efficient use of a large number of computing processors due to excessive concentration of grid-points at the poles.
Recently developed finite element discretizations, known as ``compatible" finite elements, avoid this issue while maintaining the key numerical properties essential for accurate geophysical simulations. Moreover, these properties can be obtained on arbitrary, non-orthogonal meshes. However, the efficient solution of the resulting discrete systems depend on transforming the mixed velocity-pressure (or velocity-pressure-buoyancy) system into an elliptic problem for the pressure. This is not so straightforward within the compatible finite element framework due to inter-element coupling.
This thesis supports the proposition that systems arising from compatible finite element discretizations can be solved efficiently using a technique known as ``hybridization." Hybridization removes inter-element coupling while maintaining the desired numerical properties. This permits the construction of sparse, elliptic problems, for which fast solver algorithms are known, using localized algebra. We first introduce the technique for compatible finite element discretizations of simplified atmospheric models. We then develop a general software abstraction for the rapid implementation and composition of hybridization methods, with an emphasis on preconditioning.
Finally, we extend the technique for a new compatible method for the full, compressible atmospheric equations used in operational models.Open Acces
- …