31 research outputs found
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 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
Mimetic framework on curvilinear quadrilaterals of arbitrary order
In this paper higher order mimetic discretizations are introduced which are
firmly rooted in the geometry in which the variables are defined. The paper
shows how basic constructs in differential geometry have a discrete counterpart
in algebraic topology. Generic maps which switch between the continuous
differential forms and discrete cochains will be discussed and finally a
realization of these ideas in terms of mimetic spectral elements is presented,
based on projections for which operations at the finite dimensional level
commute with operations at the continuous level. The two types of orientation
(inner- and outer-orientation) will be introduced at the continuous level, the
discrete level and the preservation of orientation will be demonstrated for the
new mimetic operators. The one-to-one correspondence between the continuous
formulation and the discrete algebraic topological setting, provides a
characterization of the oriented discrete boundary of the domain. The Hodge
decomposition at the continuous, discrete and finite dimensional level will be
presented. It appears to be a main ingredient of the structure in this
framework.Comment: 69 page