55 research outputs found
A compatible finite element discretisation for the nonhydrostatic vertical slice equations
We present a compatible finite element discretisation for the vertical slice
compressible Euler equations, at next-to-lowest order (i.e., the pressure space
is bilinear discontinuous functions). The equations are numerically integrated
in time using a fully implicit timestepping scheme which is solved using
monolithic GMRES preconditioned by a linesmoother. The linesmoother only
involves local operations and is thus suitable for domain decomposition in
parallel. It allows for arbitrarily large timesteps but with iteration counts
scaling linearly with Courant number in the limit of large Courant number. This
solver approach is implemented using Firedrake, and the additive Schwarz
preconditioner framework of PETSc. We demonstrate the robustness of the scheme
using a standard set of testcases that may be compared with other approaches.Comment: Response to reviewers. Thanks to Golo Wimmer for pointing out the
wrong factor of h in the interior penalty for diffusion - this was also wrong
in the codes and we reran the dense bubble testcase
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
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
Conservation and stability in a discontinuous Galerkin method for the vector invariant spherical shallow water equations
We develop a novel and efficient discontinuous Galerkin spectral element
method (DG-SEM) for the spherical rotating shallow water equations in vector
invariant form. We prove that the DG-SEM is energy stable, and discretely
conserves mass, vorticity, and linear geostrophic balance on general curvlinear
meshes. These theoretical results are possible due to our novel entropy stable
numerical DG fluxes for the shallow water equations in vector invariant form.
We experimentally verify these results on a cubed sphere mesh. Additionally, we
show that our method is robust, that is can be run stably without any
dissipation. The entropy stable fluxes are sufficient to control the grid scale
noise generated by geostrophic turbulence without the need for artificial
stabilisation
Structure Preserving Model Order Reduction of Shallow Water Equations
In this paper, we present two different approaches for constructing
reduced-order models (ROMs) for the two-dimensional shallow water equation
(SWE). The first one is based on the noncanonical Hamiltonian/Poisson form of
the SWE. After integration in time by the fully implicit average vector field
method, ROMs are constructed with proper orthogonal decomposition/discrete
empirical interpolation method (POD/DEIM) that preserves the Hamiltonian
structure. In the second approach, the SWE as a partial differential equation
with quadratic nonlinearity is integrated in time by the linearly implicit
Kahan's method and ROMs are constructed with the tensorial POD that preserves
the linear-quadratic structure of the SWE. We show that in both approaches, the
invariants of the SWE such as the energy, enstrophy, mass, and circulation are
preserved over a long period of time, leading to stable solutions. We conclude
by demonstrating the accuracy and the computational efficiency of the reduced
solutions by a numerical test problem
Compatible finite element spaces for geophysical fluid dynamics
This is the final version. Available from Oxford University Press via the DOI in this record.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 non-linear rotating shallow water equations, and the three-dimensional compressible Euler equations. We summarize 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 compatible finite element discretisation for the nonhydrostatic vertical slice equations
We present a compatible finite element discretisation for the vertical slice compressible Euler equations, at next-to-lowest order (i.e., the pressure space is bilinear discontinuous functions). The equations are numerically integrated in time using a fully implicit timestepping scheme which is solved using monolithic GMRES preconditioned by a linesmoother. The linesmoother only involves local operations and is thus suitable for domain decomposition in parallel. It allows for arbitrarily large timesteps but with iteration counts scaling linearly with Courant number in the limit of large Courant number. This solver approach is implemented using Firedrake, and the additive Schwarz preconditioner framework of PETSc. We demonstrate the robustness of the scheme using a standard set of testcases that may be compared with other approaches
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