This paper presents a novel nodal finite-element method for either continuous
and discontinuous elements, as applied to the 2-D shallow-water equations on
the cubed sphere. The cornerstone of this method is the construction of a
robust derivative operator that can be applied to compute discrete
derivatives even over a discontinuous function space. A key advantage of the
robust derivative is that it can be applied to partial differential equations
in either a conservative or a non-conservative form. However, it is also
shown that discontinuous penalization is required to recover the correct
order of accuracy for discontinuous elements. Two versions with discontinuous
elements are examined, using either the

Modeling of the 2-D shallow-water equations is an important step in understanding the behavior of a numerical discretization for atmospheric modeling. In particular, the dynamical character of the global shallow-water equations is governed by features common with atmospheric motions including nonlinearity, barotropic Rossby waves and inertia-gravity waves, without the added complexity of a vertical dimension.

A comprehensive literature already exists on the development of numerical
methods for the global shallow-water equations spanning the past several
decades. Examples include the spectral transform method

This paper introduces a novel discrete derivative operator that is applied to
the shallow-water equations on a manifold using continuous and discontinuous
finite elements. This work is motivated by the flux correction methods of

Discontinuous elements are potentially more desirable than continuous
elements for several reasons: first, discontinuous elements only require
parallel communication along coordinate axes, whereas continuous elements
also require parallel communication along diagonals, a doubling of the total
number of communications in 2-D. Second, discontinuous elements provide a
natural mechanism to enforce stabilization via discontinuous penalization (or
Riemann solvers, for equations in conservation form). Third, discontinuous
elements can be used in conjunction with upwind methods, which are generally
better for tracer transport and associated problems. However, discontinuous
elements also have a number of disadvantages, including higher storage
requirements (for the same order of accuracy), a maximum time step size which
is typically smaller than that imposed for continuous elements

The outline of this paper is as follows.
Section

The 2-D shallow-water equations in on a Riemannian manifold with coordinate
indices

The prognostic variables are free-surface height

The mass equation, Eq. (

A 3-D view of the cubed-sphere grid shown here with

The Eqs. (

The contravariant 2-D metric on the equiangular cubed sphere of radius

Spherical coordinates

A nodal finite-element method is employed

The 2-D element

A depiction of the nodal grid for a reference element on GLL nodes
for

Figure

A robust differentiation operator is now constructed for both continuous and
discontinuous finite elements. Let

An analogous definition holds in the

With the definition of a robust discrete derivative
Eq. (

At element boundaries, the use of one-sided derivatives will cause the
discontinuity between neighboring elements to exhibit an error with magnitude

On the cubed-sphere grid, the discontinuous method has 6

The primary computational difference between the continuous and discontinuous
formulations is due to the evaluation of the penalty terms,
Eqs. (

Stabilization is typically needed for co-located (or unstaggered) finite-element methods, whether implicitly in the form of upwinding or explicitly in
the form of a diffusive operator, to avoid high-frequency dispersive errors
associated with spectral ringing. In general, it is preferred that this
operator is consistent with the underlying geometry of the Riemannian
manifold, which precludes, for example, the Boyd–Vandeven filter

Note that any viscosity operator will lead to a loss of energy conservation of the underlying numerical method. This loss is exhibited in two obvious ways: first, for geostrophically balanced flows the error will tend to grow over time. Second, energy conservation is lost leading to a decay in the total energy content of the system over time.

For stabilization of the method, diffusion is added in the form of either
viscosity or hyperviscosity, which corresponds to multiple applications of
the viscosity operator. A scalar viscosity operator is constructed to satisfy

Note that the contravariant derivatives

Vector viscosity is used for damping of the velocity field, and takes the form

Observe that if

For simplicity of calculation, we treat divergence damping and vorticity
damping separately. For divergence damping, the operator is constructed by
taking the inner product of

For vorticity damping an analogous procedure leads to

Note that for shallow-water flows, only the radial component of the vorticity
is relevant for this calculation. The discrete value of

Following another lengthy derivation (see
Appendix

Applying an analogous procedure for test function

The divergence and curl, which are needed for evaluation of the Laplacian,
are computed via

For stabilization of a high-order discretization, hyperviscosity is preferred
since it retains the order of accuracy of the underlying scheme. In practice,
hyperviscosity is implemented by repeated application of the viscosity
operator. For instance, for fourth-order hyperviscosity, the

Calculation of hyperviscosity in the form presented here requires one
parallel exchange per application of the Laplacian operator. For continuous
elements, this communication is manifested through the application of DSS,
which averages away any discontinuity that has been generated along element
edges. For discontinuous elements, scalar viscosity requires pointwise
updates along element edges computed from
Eq. (

In this section selected results are provided to evaluate the relative
performance of the methods described in this paper. Four test cases are
evaluated: from the

All simulations are performed with

When required, the standard

When applied, hyperviscosity uses a single coefficient for both scalar and vector hyperviscosity,

This choice of scaling for the hyperviscosity coefficient is based on

Test case 2 of

This height field also serves as the reference solution. The non-divergent
velocity field is specified in latitude–longitude

Height field with

Normalized total energy and potential enstrophy change for the zonal
flow over an isolated mountain test with

Height field with

Figure

Normalized total energy and potential enstrophy change for the
Rossby–Haurwitz wave test with

To understand the growth of error norms associated with each configuration,
additional simulations with

Relative vorticity field with

Normalized total energy and enstrophy change for the barotropic
instability test with

To verify that the model exhibits the correct convergence rate,
Fig.

Test case 5 in

Simulation results for this test case were computed at

To understand conservation of invariants over time, total energy

A time series of energy and potential enstrophy are plotted in
Fig.

Test case 6 in

Results for the Rossby–Haurwitz wave are given in
Figs.

The barotropic instability test case of

Simulation results for this test case were computed at

Normalized total energy and potential enstrophy are plotted for the
barotropic instability in Fig.

Following

From the

The non-conservative discontinuous element formulation has been shown to be a potential candidate for future atmospheric modeling. It has the advantage of requiring fewer parallel communications than continuous methods, and exhibits stability even when hyperviscosity is not used for explicit stabilization. However, with the reduced time step size it remains unclear whether the discontinuous formulation would be computationally competitive with continuous element methods.

The method discussed in this paper has been implemented in the Tempest
atmospheric model, available from

In this appendix equivalence of the variational formulation of the spectral
element method and the differential formulation using the robust derivative
is demonstrated. For continuous elements,

For simplicity consider a single quadrilateral spectral element with test
functions

Using the derivative operator (

Then using integration by parts,

For the last term, observe that on a manifold

On the other hand, by construction

Furthermore, in conjunction with Eq. (

Equivalence of this equation with Eq. (

In this appendix the derivation of the discrete viscosity operator is provided for scalar and vector hyperviscosity on a Riemannian manifold.

From the natural quadrature rule that arises from the nodal finite-element
formulation, the left-hand side of
Eq. (

The area integral term in Eq. (

From Eq. (

The infinitesimal length element along each edge is given by the covariant
metric,

Then along the right edge, using the nodal discretization of the boundary
integral,

The area integral that appears on the left-hand side of
Eqs. (

In nodal form, the divergence expands as

Further, the radial component of the vorticity expands as

Combining Eqs. (

Using Eqs. (

Repeating for all edges and using
Eq. (

For vorticity damping, along the right edge, Eq. (

Repeating for all edges and using Eq. (

The author would like to acknowledge Mark Taylor, Oksana Guba, David Hall, Hans Johansen and Jorge Guerra for many fruitful conversations and for their assistance in refining this manuscript. Edited by: H. Weller