Choose CG or DG

The horizontal spectral-element grid is continuous Galerkin (CG) or discontinuous Galerkin (DG). The choice is made once, when the grid is constructed, and a model written with Operators.tendency_completion runs on either. This page is the decision guide; Spectral elements: CG and DG has the theory.

Set the discretization

Pass discretization = Grids.DG() (or Grids.CG(), the default) to any constructor that builds a horizontal spectral-element grid:

space = Spaces.SpectralElementSpace2D(topology, quad; discretization = Grids.DG())
space = CubedSphereSpace(; radius, h_elem, n_quad_points, discretization = Grids.DG())

Read it back with Spaces.discretization(space); Spaces.is_continuous(space) is the Boolean form. An omitted keyword follows the quadrature: Gauss–Lobatto– Legendre nodes give CG(), Gauss–Legendre nodes give DG().

Write the tendency once

Build the completion from the tendency field at setup and pass the interface flux unconditionally; it is used on DG and ignored on CG:

numflux = Operators.RusanovNumericalFlux(physical_flux, wavespeed)
completion = Operators.tendency_completion(dydt; numflux)

function rhs!(dydt, y, (params, completion), t)
    wdiv = Operators.Divergence{Operators.WeakForm}()
    @. dydt = -wdiv(physical_flux(y, (params,)))
    Operators.complete_tendency!(completion, dydt, y, params)
    return dydt
end

The CG and DG tutorial runs this on both spaces.

Decide

Question	CG	DG
How is grid-scale energy removed?	Explicit fourth-order hyperdiffusion, tuned per resolution	The interface-flux penalty (Rusanov, Roe); no hyperdiffusion needed
What does inter-element coupling cost?	One DSS per completed tendency; the only horizontal communication	One numerical-flux evaluation per face per completed tendency, plus a halo exchange
Is the scheme conservative?	Yes, to round-off, through the inner-product-preserving DSS	Yes, to round-off, through antisymmetric interface fluxes
Which operators exist today?	All of them: strong and weak `Gradient`, `Divergence`, `Curl`; scalar and vector Laplacians; limiters; hypsography	The same element-local operators and the scalar Laplacian (with interior-penalty face terms), plus the flux-differencing divergence and face lifting; no vector Laplacian yet
Can the tendency be a `FieldVector`?	Yes, completed by one batched DSS	No; the state must be one field with a composite element type
Horizontal boundary conditions?	Periodic, or imposed by the model on the state; the operators take none	`PeriodicBC`, `ReflectingWallBC`, or a one-sided `boundary_numflux`

Check the coupling

On a periodic domain, the integral of a conservative tendency vanishes on both grids; a nonzero value means the completion was skipped or the interface flux lacks antisymmetry.

abs(sum(dydt.ρ)) < 1e-12 * sum(abs, dydt.ρ)