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Image: D.D. Holm, T. Schmah & C. Stoica. Geometric Mechanics and Symmetry:

From Finite to Infinite Dimensions, Oxford University Press (2009).

### First AGM Meeting: Geometric and Structure-Preserving Numerics

In recent years, there has emerged an area broadly describable as Geometric and Structure-Preserving Numerics for modelling the atmosphere, geophysical fluid dynamics, and hydrodynamics problems. There has been significant recent work on applications of discrete differential geometric methods in the development of more accurate models in numeric weather prediction.

Gusto is a compatible finite element model used for prototyping atmosphere/ocean dynamical cores housing a variation of the next-generation Met Office dynamical core. In atmosphere models, it is desirable to capture certain key properties related to conservation and balance. These physical properties are related to geometrical properties of the governing equations such as vanishing of curl of gradient, and adjointness of the divergence and negative gradient operators. Certain classes of mixed finite elements, called compatible finite elements, give rise to a discrete de Rham complex, enabling a numerical model to obtain these desirable properties. For example, a mixed finite-element finite-volume shallow-water model can use a hexagonal mesh or a cubed sphere mesh. The model makes use of two families of compatible finite elements, one on the primal mesh and the other on a dual mesh, each with their own discrete de Rham complex. Mapping model fields between primal and dual meshes makes use of discrete Hodge star operators that arise in a very natural way.

GungHo is a new dynamical core that is being developed by the Met Office to enable efficient use of future supercomputer architectures. The spatial discretization uses compatible mixed finite-elements, which allow the use of an unstructured mesh, whilst the time-discretization uses an iterated-implicit scheme, which has good stability properties. This results in solving a mixed system of equations, which uses the associated Helmholtz equation as the precondition. Lateral boundary conditions are required to enable running the dynamical core over a local region rather than the whole globe.

Furthermore, recently significant progress has been made in the development of structure-preserving finite element integrators for compressible fluids and MHD. This approach combines the geometric formulation of fluid dynamics on groups of diffeomorphisms with finite element discretization techniques. A specific feature of the discrete geometric formulation is the occurrence of a nonholonomic right-invariant distribution on the discrete group of diffeomorphisms, that is shown to be isomorphic to a Raviart-Thomas finite element space. The resulting finite element discretizations correspond to weak forms of the compressible fluid equations that do not seem to have been used in the finite element literature.

Topics expected to be discussed include:

(1) Structure-preserving numerical methods used in the Met Office models:

Modelling the atmosphere on quasi-uniform spherical meshes and geometrical underpinnings. Numerical weather prediction with compound compatible finite elements. Implementation of lateral boundary conditions in the Met Office Dynamical Core GungHo. Mimetic finite element methods for geophysical fluids.

(2) Other hydrodynamics problems: Structure-preserving schemes in ionic transport. Structure-preserving finite element integrators for compressible fluids and MHD.

The date of the meeting, timetable, participants, speakers, and further details will be posted here.

For any enquiries, please contact Hamid Alemi Ardakani.