A Latitude-Longitude Grid
|
In these studies, we present high-order finite element spatial discretization
method for the SWEs in spherical coordinates.
In the first study,
the SWEs are discretized in time with the semi-implicit leapfrog method,
and in space with the cubic spline collocation method on a skipped
latitude-longitude grid.
The simplest numerical grid covering the entire sphere is the familiar
latitude-longitude grid, but the convergence of the meridians toward
the poles necessitates special measures if such a grid is to be used for
a numerical simulation of the atmosphere.
When an Eulerian time integration method is used, the timestep is restricted
by the Courant-Friedrichs-Lewy (CFL) condition and is determined essentially
by the shortest resolvable scale divided by the largest phase speed of
modes treated explicitly.
Rather than allowing the atypically high resolution near the poles to dictate
the timestep, it is common practice to use a skipped grid in which
the longitudinal increment
is allowed to increase near the poles.
Boundary Conditions for Geopotential
|
A longstanding problem in the integration of numerical weather prediction
models is that, when an explicit Eulerian time discretization method is used,
the maximum permissible timestep is restricted by stability rather than
accuracy.
That is, in order for the integration to be stable, the timestep has to
be so small that the time truncation error is much smaller than the spatial
truncation error, thus giving rise to high computation cost.
Early models used an explicit leapfrog method, in which timestep is limited
by both the CFL condition as well as the
propagation of gravity waves.
Discretization schemes on the semi-Lagrangian treatment of advection offer the
promise of larger time steps, with no loss in accuracy, in
comparison with Eulerian-based advection schemes.
Since gravity terms may render the equations stiff and thus severely restrict
the timestep even with semi-Lagrangian advection approximations,
one needs to combine the semi-Lagrangian approximations
with semi-implicit approximations to obtain maximum benefit from the
semi-Lagrangian approach.
By combining a semi-Lagrangian treatment of advection, a semi-implicit
treatment of gravity terms, and the cubic spline collocation method,
we show in a second study that it is possible to increase
the timestep
substantially while maintaining numerical stability and with no loss
in accuracy.
|