The development of grids smoothly covering the sphere has had a long history in computational
meteorology that has led to two distinct approaches: (i) the stereographic approach in which the sphere is
covered by two overlapping patches obtained by stereographic projection about the North and South
poles [58]; and (ii) the cubed-sphere approach in which the sphere is covered by the 6 patches obtained by a
projection of the faces of a circumscribed cube [206
]. A discussion of the advantages of each of these
methods and a comparison of their performance in a standard fluid testbed are given in [58
].
In numerical relativity, the stereographic method has been reinvented in the context of the
characteristic evolution problem [180
]; and the cubed-sphere method has been reinvented in
building an apparent horizon finder [242
]. The cubed sphere module, including the interpatch
transformations, has been integrated into the Cactus toolkit and later applied to black hole excision [241
].
Perhaps the most ingenious treatment of the sphere, based upon a toroidal map, was devised by
the Canberra group in building their characteristic code [30
]. These methods are described
below.
Motivated by problems in meteorology, G. L. Browning, J. J. Hack and P. N. Swartztrauber [58]
developed the first finite difference scheme based upon a composite mesh with two overlapping stereographic
coordinate patches, each having a circular boundary centered about the North or South poles. Values for
quantities required at ghost points beyond the boundary of one of the patches were interpolated from values
in the other patch. Because a circular boundary does not fit regularly on a stereographic grid,
dissipation was found necessary to remove the short wavelength error resulting from the inter-patch
interpolations. They used the shallow water equations as a testbed to compare their approach to
existing spectral approaches in terms of computer time, execution rate and accuracy. Such
comparisons of different numerical methods can be difficult. Both the finite difference and spectral
approaches gave good results and were competitive in terms of overall operation count and
memory requirements For the particular initial data sets tested, the spectral approach had an
advantage but not enough to give clear indication of the suitability of one method over another. The
spectral method with M modes requires
operations per time step compared with
for a finite difference method on a
grid. However, assuming that the solution is
smooth, the accuracy of the spectral method is
compared to, say,
for a
sixth order finite difference method. Hence, for comparable accuracy,
which
implies that the operation count for the spectral and finite difference methods are
and
, respectively. Thus for sufficiently high accuracy, i.e. large
, the spectral
method requires fewer operations. The issue of spectral vs finite difference methods thus depends
on the nature of the smoothness of the physical problem being addressed and the accuracy
desired.
The Pitt null code was first developed using two stereographic patches with square boundaries, each overlapping the equator. This has recently been modified based upon the approach advocated in [58], which retains the original stereographic coordinates but shrinks the overlap region by masking a circular boundary near the equator. The original square boundaries aligned with the grid and did not require numerical dissipation. However, the corners of the square boundary, besides being a significant waste of economy, were a prime source of inaccuracy. The resolution at the corners is only 1/9th that at the poles due to the stretching of the stereographic map. Near the equator, the resolution is approximately 1/2 that at the poles. The use of a circular boundary requires an angular version of numerical dissipation to control the resulting high frequency error (see Section 4.2.1).
A crucial ingredient of the PITT code is the -module [118] which incorporates a computational
version of the Newman–Penrose eth-formalism [180]. The underlying method can be applied to any smooth
coordinatization
of the sphere based upon several patches. The unit sphere metric
, defined
by these coordinates, is decomposed in each patch in terms of a complex basis vector
,
C. Ronchi, R. Iacono and P. S. Paolucci [206], developed the “cubed-sphere” approach as a new gridding
method for solving global meteorological problems. The method decomposes the sphere into the 6 identical
regions obtained by projection of a cube circumscribed on its surface. This gives a variation of the
composite mesh method in which the 6 domains butt up against each other along shared grid boundaries.
As a result only 1-dimensional intergrid interpolations are necessary (as opposed to the 2-dimensional
interpolations of the stereographic grid), which results in enhanced accuracy. The symmetry of the scheme,
in which the six patches have the same geometric structure and grid, also allows efficient use
of parallel computer architectures. Their tests of the cubed sphere method based upon the
simulation of shallow water waves in spherical geometry show that the numerical solutions are as
accurate as those with spectral methods, with substantial saving in execution time. Recently, the
cubed-sphere method has also been developed for application to characteristic evolution in numerical
relativity [201, 111
]. The eth-calculus is used to treat tensor fields on the sphere in the same way as in the
stereographic method except the interpatch transformations now involve 6, rather than 2, sets of basis
vectors.
The Canberra group treats fields on the sphere by taking advantage of the existence of a smooth map from
the torus to the sphere [30]. The pullback of this map allows functions on the sphere to be expressed in
terms of toroidal coordinates. The intrinsic topology of these toroidal coordinates allow them to take
advantage of of fast-Fourier transforms to implement a highly efficient pseudo-spectral treatment. This
ingenious method has apparently not yet been adopted in other fields.
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