4.1 Coordinatization of the sphere

Any characteristic code extending to + ℐ requires the ability to handle tensor fields and their derivatives on the sphere. Spherical coordinates and spherical harmonics are natural analytic tools for the description of radiation, but their implementation in computational work requires dealing with the impossibility of smoothly covering the sphere with a single coordinate grid. Polar coordinate singularities in axisymmetric systems can be regularized by standard tricks. In the absence of symmetry, these techniques do not generalize and would be especially prohibitive to develop for tensor fields.

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 [58Jump To The Next Citation Point]; 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 [206Jump To The Next Citation Point]. A discussion of the advantages of each of these methods and a comparison of their performance in a standard fluid testbed are given in [58Jump To The Next Citation Point]. In numerical relativity, the stereographic method has been reinvented in the context of the characteristic evolution problem [180Jump To The Next Citation Point]; and the cubed-sphere method has been reinvented in building an apparent horizon finder [242Jump To The Next Citation Point]. The cubed sphere module, including the interpatch transformations, has been integrated into the Cactus toolkit and later applied to black hole excision [241Jump To The Next Citation Point]. Perhaps the most ingenious treatment of the sphere, based upon a toroidal map, was devised by the Canberra group in building their characteristic code [30Jump To The Next Citation Point]. These methods are described below.

4.1.1 Stereographic grids

Motivated by problems in meteorology, G. L. Browning, J. J. Hack and P. N. Swartztrauber [58Jump To The Next Citation Point] 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 O (M 3) operations per time step compared with 2 O (N ) for a finite difference method on a N × N grid. However, assuming that the solution is smooth, the accuracy of the spectral method is O (e−M ) compared to, say, O (N −6) for a sixth order finite difference method. Hence, for comparable accuracy, M = O (lnN ) which implies that the operation count for the spectral and finite difference methods are O[(lnN )3] and 2 O(N ), respectively. Thus for sufficiently high accuracy, i.e. large N, 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 xA of the sphere based upon several patches. The unit sphere metric qAB, defined by these coordinates, is decomposed in each patch in terms of a complex basis vector qA,

qAB = q(A¯qB ) (21 )
Vector and tensor fields on the sphere, and their covariant derivatives, are then represented by their basis components. For example, the vector field U A is represented by the complex spin-weight 1 field U = U AqA. The covariant derivative DA associated with qAB is then expressed in terms of the ∂ operator according to
A B A B qAqBD U = ∂U , ¯qAqBD U = ¯∂U. (22 )
The eth-calculus simplifies the underlying equations, avoids spurious coordinate singularities and allows accurate differentiation of tensor fields on the sphere in a computationally efficient and clean way. Its main weakness is the numerical noise introduced by interpolations between the patches.

4.1.2 Cubed sphere grids

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 [201Jump To The Next Citation Point111Jump To The Next Citation Point]. 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.

4.1.3 Toroidal grids

The Canberra group treats fields on the sphere by taking advantage of the existence of a smooth map from the torus to the sphere [30Jump To The Next Citation Point]. 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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