A primal-dual mixed finite element method for accurate and efficient atmospheric modelling on massively parallel computers
Efficient modelling of the atmosphere using massively parallel computers will require a quasi-uniform grid to avoid the communication bottleneck associated with the poles of the traditional latitude-longitude grid. However, achieving an accurate solution on a quasi-uniform grid is non-trivial. A mixed finite element method can provide the following desirable properties: mass conservation; a C-grid-like placement of variables for accurate wave dispersion and adjustment; vanishing curl of gradient; linear energy conservation; and steady geostrophic modes in the linear f-plane case. A further desirable property is that the potential vorticity (PV) should evolve as if advected by some chosen (accurate) advection scheme. This can be achieved by inserting the PV fluxes into the nonlinear Coriolis term that appears in the `vector invariant' form of the momentum equation, provided the PV fluxes themselves can be constructed. Introducing a dual family of function spaces, in which the PV lives in a piecewise constant function space, along with suitable maps between primal and dual spaces, provides a convenient framework in which the PV fluxes can be computed by a finite volume advection scheme in the dual space.
The scheme can be implemented in terms of a small number of sparse matrices that can be precomputed off-line, avoiding the need for numerical quadrature at run time. A mass matrix and two dual-primal mapping operators need to be inverted at each time step, but these are well conditioned and the inversion can be absorbed into the iterative solver used for implicit time stepping at only a modest increase in cost. Some sample shallow water model results on a hexagonal icosahedral grid and a cubed sphere grid will be presented.