9 research outputs found
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
Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere
This is the final version. Available from Elsevier via the DOI in this record.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 TaylorâGalerkin 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 demonstrate that our discretisation achieves the expected second order convergence and provide results from some standard rotating sphere test problems.Natural Environment Research Council (NERC)Natural Environment Research Council (NERC)Engineering and Physical Sciences Research Council (EPSRC)Engineering and Physical Sciences Research Council (EPSRC
Energy conserving SUPG methods for compatible finite element schemes in numerical weather prediction
We present an energy conserving space discretisation based on a Poisson
bracket that can be used to derive the dry compressible Euler as well as
thermal shallow water equations. It is formulated using the compatible finite
element method, and extends the incorporation of upwinding for the shallow
water equations as described in Wimmer, Cotter, and Bauer (2019). While the
former is restricted to DG upwinding, an energy conserving SUPG scheme for the
(partially) continuous Galerkin thermal field space is newly introduced here.
The energy conserving property is validated by coupling the Poisson bracket
based spatial discretisation to an energy conserving time discretisation.
Further, the discretisation is demonstrated to lead to an improved temperature
field development with respect to stability when upwinding is included. An
approximately energy conserving full discretisation with a smaller
computational cost is also presented.Comment: 27 pages, 9 figures, first version: all comments welcom
A Mixed Mimetic Spectral Element Model of the Rotating Shallow Water Equations on the Cubed Sphere
In a previous article [J. Comp. Phys. (2018) 282-304], the
mixed mimetic spectral element method was used to solve the rotating shallow
water equations in an idealized geometry. Here the method is extended to a
smoothly varying, non-affine, cubed sphere geometry. The differential operators
are encoded topologically via incidence matrices due to the use of spectral
element edge functions to construct tensor product solution spaces in
, and . These incidence matrices
commute with respect to the metric terms in order to ensure that the mimetic
properties are preserved independent of the geometry. This ensures conservation
of mass, vorticity and energy for the rotating shallow water equations using
inexact quadrature on the cubed sphere. The spectral convergence of errors are
similarly preserved on the cubed sphere, with the generalized Piola
transformation used to construct the metric terms for the physical field
quantities
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
Finite Element Exterior Calculus with Applications to the Numerical Solution of the GreenâNaghdi Equations
The study of finite element methods for the numerical solution of differential equations is one of the gems of modern mathematics, boasting rigorous analytical foundations as well as unambiguously useful scientific applications. Over the past twenty years, several researchers in scientific computing have realized that concepts from homological algebra and differential topology play a vital role in the theory of finite element methods. Finite element exterior calculus is a theoretical framework created to clarify some of the relationships between finite elements, algebra, geometry, and topology. The goal of this thesis is to provide an introduction to the theory of finite element exterior calculus, and to illustrate some applications of this theory to the design of mixed finite element methods for problems in geophysical fluid dynamics. The presentation is divided into two parts. Part 1 is intended to serve as a selfâcontained introduction to finite element exterior calculus, with particular emphasis on its topological aspects. Starting from the basics of calculus on manifolds, I go on to describe Sobolev spaces of differential forms and the general theory of Hilbert complexes. Then, I explain how the notion of cohomology connects Hilbert complexes to topology. From there, I discuss the construction of finite element spaces and the proof that special choices of finite element spaces can be used to ensure that the cohomological properties of a particular problem are preserved during discretization. In Part 2, finite element exterior calculus is applied to derive mixed finite element methods for the GreenâNaghdi equations (GN). The GN extend the more wellâknown shallow water equations to the regime of nonâinfinitesimal aspect ratio, thus allowing for some nonâhydrostatic effects. I prove that, using the mixed formulation of the linearized GN, approximations of balanced flows remain steady. Additionally, one of the finite element methods presented for the fully nonlinear GN provably conserves mass, vorticity, and energy at the semiâdiscrete level. Several computational test cases are presented to assess the practical performance of the numerical methods, including a collision between solitary waves, the motion of solitary waves over variable bottom topography, and the breakdown of an unstable balanced state