3.1 Spatial coordinate systems

Most of the interesting problems in numerical relativity involve asymmetries that require the use of a full set of three-dimensional coordinates. We briefly review several coordinate sets (all orthogonal) that have been used in numerical relativity with spectral methods. They are described through the line element ds2 of the flat metric in the coordinates we discuss.

3.1.1 Mappings

Choosing a smart set of coordinates is not the end of the story. As for finite elements, one would like to be able to cover some complicated geometries, like distorted stars, tori, etc…or even to be able to cover the whole space. The reason for this last point is that, in numerical relativity, one often deals with isolated systems for which boundary conditions are only known at spatial infinity. A quite simple choice is to perform a mapping from numerical coordinates to physical coordinates, generalizing the change of coordinates to [− 1,1 ], when using families of orthonormal polynomials or to [0,2π] for Fourier series.

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Figure 17: Regular deformation of the [− 1,1] × [− 1,1] square.

An example of how to map the [− 1,1] × [− 1,1] domain can be taken from Canuto et al. [56Jump To The Next Citation Point], and is illustrated in Figure 17View Image: once the mappings from the four sides (boundaries) of ˆΩ to the four sides of Ω are known, one can construct a two-dimensional regular mapping Π, which preserves orthogonality and simple operators (see Chapter 3.5 of [56Jump To The Next Citation Point]).

The case where the boundaries of the considered domain are not known at the beginning of the computation can also be treated in a spectral way. In the case where this surface corresponds to the surface of a neutron star, two approaches have been used. First, in Bonazzola et al. [38Jump To The Next Citation Point], the star (and therefore the domain) is supposed to be “star-like”, meaning that there exists a point from which it is possible to reach any point on the surface by straight lines that are all contained inside the star. To such a point is associated the origin of a spherical system of coordinates, so that it is a spherical domain, which is regularly deformed to coincide with the shape of the star. This is done within an iterative scheme, at every step, once the position of the surface has been determined. Then, another approach has been developed by Ansorg et al. [10Jump To The Next Citation Point] using cylindrical coordinates. It is a square in the plane (ρ,z), which is mapped onto the domain describing the interior of the star. This mapping involves an unknown function, which is itself decomposed in terms of a basis of Chebyshev polynomials, so that its coefficients are part of the global vector of unknowns (as the density and gravitational field coefficients).

In the case of black-hole–binary systems, Scheel et al. [188Jump To The Next Citation Point] have developed horizon-tracking coordinates using results from control theory. They define a control parameter as the relative drift of the black hole position, and they design a feedback control system with the requirement that the adjustment they make on the coordinates be sufficiently smooth that they do not spoil the overall Einstein solver. In addition, they use a dual-coordinate approach, so that they can construct a comoving coordinate map, which tracks both orbital and radial motion of the black holes and allows them to successfully evolve the binary. The evolutions simulated in [188Jump To The Next Citation Point] are found to be unstable, when using a single rotating-coordinate frame. We note here as well the work of Bonazzola et al. [42], where another option is explored: the stroboscopic technique of matching between an inner rotating domain and an outer inertial one.

3.1.2 Spatial compactification

As stated above, the mappings can also be used to include spatial infinity into the computational domain. Such a compactification technique is not tied to spectral methods and has already been used with finite-difference methods in numerical relativity by, e.g., Pretorius [176Jump To The Next Citation Point]. However, due to the relatively low number of degrees of freedom necessary to describe a spatial domain within spectral methods, it is easier within this framework to use some resources to describe spatial infinity and its neighborhood. Many choices are possible to do so, either directly choosing a family of well-behaved functions on an unbounded interval, for example the Hermite functions (see, e.g., Section 17.4 in Boyd [48Jump To The Next Citation Point]), or making use of standard polynomial families, but with an adapted mapping. A first example within numerical relativity was given by Bonazzola et al. [41Jump To The Next Citation Point] with the simple inverse mapping in spherical coordinates.

r = ---1----, x ∈ [− 1,1]. (89 ) α(x − 1)
This inverse mapping for spherical “shells” has also been used by Kidder and Finn [125Jump To The Next Citation Point], Pfeiffer et al. [171Jump To The Next Citation Point, 167Jump To The Next Citation Point], and Ansorg et al. in cylindrical [10Jump To The Next Citation Point] and spheroidal [8Jump To The Next Citation Point] coordinates. Many more elaborated techniques are discussed in Chapter 17 of Boyd [48Jump To The Next Citation Point], but to our knowledge, none have been used in numerical relativity yet. Finally, it is important to point out that, in general, the simple compactification of spatial infinity is not well adapted to solving hyperbolic PDEs and the above mentioned examples were solving only for elliptic equations (initial data, see Section 5). For instance, the simple wave equation (127View Equation) is not invariant under the mapping (89View Equation), as has been shown, e.g., by Sommerfeld (see [201Jump To The Next Citation Point], Section 23.E). Intuitively, it is easy to see that when compactifying only spatial coordinates for a wave-like equation, the distance between two neighboring grid points becomes larger than the wavelength, which makes the wave poorly resolved after a finite time of propagation on the numerical grid. For hyperbolic equations, is is therefore usually preferable to impose physically and mathematically well-motivated boundary conditions at a finite radius (see, e.g., Friedrich and Nagy [83], Rinne [179Jump To The Next Citation Point] or Buchman and Sarbach [53Jump To The Next Citation Point]).

3.1.3 Patching in more than one dimension

The multidomain (or multipatch) technique has been presented in Section 2.6 for one spatial dimension. In Bonazzola et al. [40Jump To The Next Citation Point] and Grandclément et al. [109Jump To The Next Citation Point], the three-dimensional spatial domains consist of spheres (or star-shaped regions) and spherical shells, across which the solution can be matched as in one-dimensional problems (only through the radial dependence). In general, when performing a matching in two or three spatial dimensions, the reconstruction of the global solution across all domains might need some more care to clearly write down the matching conditions (see, e.g., [167Jump To The Next Citation Point], where overlapping as well as nonoverlapping domains are used at the same time). For example in two dimensions, one of the problems that might arise is the counting of matching conditions for corners of rectangular domains, when such a corner is shared among more than three domains. In the case of a PDE where matching conditions must be imposed on the value of the solution, as well as on its normal derivative (Poisson or wave equation), it is sufficient to impose continuity of either normal derivative at the corner, the jump in the other normal derivative being spectrally small (see Chapter 13 of Canuto et al. [56Jump To The Next Citation Point]).

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Figure 18: Two sets of spherical domains describing a neutron star or black hole binary system. Each set is surrounded by a compactified domain of the type (89View Equation), which is not displayed

A now typical problem in numerical relativity is the study of binary systems (see also Sections 5.5 and 6.3) for which two sets of spherical shells have been used by Gourgoulhon et al. [100Jump To The Next Citation Point], as displayed in Figure 18View Image. Different approaches have been proposed by Kidder et al. [128Jump To The Next Citation Point], and used by Pfeiffer [167Jump To The Next Citation Point] and Scheel et al. [188Jump To The Next Citation Point] where spherical shells and rectangular boxes are combined together to form a grid adapted to black hole binary study. Even more sophisticated setups to model fluid flows in complicated tubes can be found in [144].

Multiple domains can thus be used to adapt the numerical grid to the interesting part (manifold) of the coordinate space; they can be seen as a technique close to the spectral element method [166Jump To The Next Citation Point]. Moreover, it is also a way to increase spatial resolution in some parts of the computational domain where one expects strong gradients to occur: adding a small domain with many degrees of freedom is the analog of fixed-mesh refinement for finite-differences.

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