When the numerical grid does not extend to infinity, e.g., when solving for a hyperbolic PDE, the boundary defined by is a smooth surface, on which boundary conditions are much easier to impose. Finally, spherical harmonics, which are strongly linked with these coordinates, can simplify a lot the solution of Poisson-like or wave-like equations. On the other hand, there are some technical problems linked with this set of coordinates, as detailed hereafter, but spectral methods can handle them in a very efficient way.
The transformation from spherical to Cartesian coordinates is obtained byregular function, should give a well-defined result. The same should be true if one uses spherical coordinates: the operator (93) applied to a regular function should yield a regular result. This means that a regular function of spherical coordinates must have a particular behavior at the origin and on the axis, so that the divisions by or appearing in regular operators are always well defined. If one considers an analytic function in (regular) Cartesian coordinates , it can be expanded as a series of powers of and , near the origin
In addition, the -dependence translates into a Taylor series near the origin, with the same parity as . More details in the case of polar (2D) coordinates are given in Chapter 18 of Boyd .
If we go back to the evaluation of the Laplace operator (93), it is now clear that the result is always regular, at least for and . We detail the cases of and , using the fact that spherical harmonics are eigenfunctions of the angular part of the Laplace operator (see Equation (103)). For the scalar field is reduced to a Taylor series of only even powers of , therefore the first derivative contains only odd powers and can be safely divided by . Once decomposed on spherical harmonics, the angular part of the Laplace operator (93) acting on the component reads , which is a problem only for the first term of the Taylor expansion. On the other hand, this term cancels with the , providing a regular result. This is the general behavior of many differential operators in spherical coordinates: when applied to a regular field, the full operator gives a regular result, but single terms of this operator may give singular results when computed separately, the singularities canceling between two different terms.
As this may seem an argument against the use of spherical coordinates, let us stress that spectral methods are very powerful in evaluating such operators, keeping everything finite. As an example, we use Chebyshev polynomials in for the expansion of the field , being a positive constant. From the Chebyshev polynomial recurrence relation (46), one hasfinite part is always a regular and linear operation on the vector of coefficients. Thus, the singular terms of a regular operator are never computed, but the result is a good one, as if the cancellation of such terms had occurred. Moreover, from the parity conditions it is possible to use only even or odd Chebyshev polynomials, which simplifies the expressions and saves computer time and memory. Of course, relations similar to Equation (97) exist for other families of orthonormal polynomials, as well as relations that divide by a function developed on a Fourier basis. The combination of spectral methods and spherical coordinates is thus a powerful tool for accurately describing regular fields and differential operators inside a sphere . To our knowledge, this is the first reference showing that it is possible to solve PDEs with spectral methods inside a sphere, including the three-dimensional coordinate singularity at the origin.
Spherical harmonics are the pure angular functions
The first property makes the description of scalar fields on spheres very easy: spherical harmonics are used as a decomposition basis within spectral methods, for instance in geophysics or meteorology, and by some groups in numerical relativity [21, 109, 219]. However, they could be more broadly used in numerical relativity, for example for Cauchy-characteristic evolution or matching [228, 15], where a single coordinate chart on the sphere might help in matching quantities. They can also help to describe star-like surfaces being defined by as event or apparent horizons [152, 23, 1]. The search for apparent horizons is also made easier: since the function verifies a two-dimensional Poisson-like equation, the linear part can be solved directly, just by dividing by in the coefficient space.
The second property makes the Poisson equation,ordinary differential equations for the coefficients (see also ): [20, 21]). The same technique can be used to advance in time the wave equation with an implicit scheme and Chebyshev-tau method for the radial coordinate [44, 157].
The use of spherical-harmonics decomposition can be regarded as a basic spectral method, like Fourier decomposition. There are, therefore, publicly available “spherical harmonics transforms”, which consist of a Fourier transform in the -direction and a successive Fourier and Legendre transform in the -direction. A rather efficient one is the SpharmonicsKit/S2Kit , but writing one’s own functions is also possible .
All the discussion in Sections 3.2.1 – 3.2.2 has been restricted to scalar fields. For vector, or more generally tensor fields in three spatial dimensions, a vector basis (triad) must be specified to express the components. At this point, it is very important to stress that the choice of the basis is independent of the choice of coordinates. Therefore, the most straightforward and simple choice, even if one is using spherical coordinates, is the Cartesian triad . With this basis, from a numerical point of view, all tensor components can be regarded as scalars and therefore, a regular tensor can be defined as a tensor field, whose components with respect to this Cartesian frame are expandable in powers of and (as in Bardeen and Piran ). Manipulations and solutions of PDEs for such tensor fields in spherical coordinates are generalizations of the techniques for scalar fields. In particular, when using the multidomain approach with domains having different shapes and coordinates, it is much easier to match Cartesian components of tensor fields. Examples of use of Cartesian components of tensor fields in numerical relativity include the vector Poisson equation  or, more generally, the solution of elliptic systems arising in numerical relativity . In the case of the evolution of the unconstrained Einstein system, the use of Cartesian tensor components is the general option, as it is done by the Caltech/Cornell group [127, 188].
The use of an orthonormal spherical basis (see. Figure 19) requires more care. The interested reader can find more details in the work of Bonazzola et al. [44, 37]. Nevertheless, there are systems in general relativity in which spherical components of tensors can be useful:
Problems arise because of the singular nature of the basis itself, in addition to the spherical coordinate singularities. The consequences are first that each component is a multivalued function at the origin or on the -axis, and then that components of a given tensor are not independent from one another, meaning that one cannot, in general, specify each component independently or set it to zero, keeping the tensor field regular. As an example, we consider the gradient of the scalar field , where is the usual first Cartesian coordinate field. This gradient expressed in Cartesian components is a regular vector field . The spherical components of read
The other drawback of spherical coordinates is that the usual partial differential operators mix the components. This is due to the nonvanishing connection coefficients associated with the spherical flat metric . For example, the vector Laplace operator () reads of numerical relativity, technically more complicated, and the technique using scalar spherical harmonics (Section 3.2.2) is no longer valid. One possibility can be to use vector, and more generally tensor [146, 239, 218, 51], spherical harmonics as the decomposition basis. Another technique might be to build from the spherical components regular scalar fields, which can have a similar physical relevance to the problem. In the vector case, one can think of the following expressions
Living Rev. Relativity 12, (2009), 1
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