3.3 Going further

The development of spectral methods linked with the problems arising in the field of numerical relativity has always been active and continues to be. Among the various directions of research one can foresee, quite interesting ones might be the beginning of higher-dimensional studies and the development of better-adapted mappings and domains, within the spirit of going from pure spectral methods to spectral elements [166, 29].

3.3.1 More than three spatial dimensions

There has been some interest in the numerical study of black holes in higher dimensions, as well as with compactified extra dimensions [202], as in brane world models [199Jump To The Next Citation Point, 132]; recently, some simulations of the head-on collision of two black holes have already been undertaken [230]. With the relatively low number of degrees of freedom per dimension needed, spectral methods should be very efficient in simulations involving four spatial dimensions or more. Here we give starting points to implement four-dimensional (as needed by, e.g., brane world models) spatial representation with spectral methods. The simplest approach is to take Cartesian coordinates (x, y,z,w ), but a generalization of spherical coordinates (r,πœƒ,φ, ξ) is also possible and necessitates less computational resources. The additional angle ξ is defined in [0,π ] with the following relations with Cartesian coordinates

x = rsinπœƒ cosφ sinξ, y = rsinπœƒ sin φ sin ξ, z = rcos πœƒsinξ, w = rcos ξ. (111 )
The four-dimensional flat Laplace operator appearing in constraint equations [199] reads
∂2Ο• 3 ∂Ο• 1 ( ∂2 Ο• 2 ∂ Ο• 1 ) Δ4 Ο• = --2-+ -----+ -2 ---2 + -------- + ---2-Δ πœƒφΟ• , (112 ) ∂r r ∂r r ∂ ξ tan ξ∂ ξ sin ξ
where Δ πœƒφ is the two-dimensional angular Laplace operator (103View Equation). As in the three-dimensional case, it is convenient to use the eigenfunctions of the angular part, which are here
β„“ m im φ G k(cosξ)P β„“ (cos πœƒ)e , (113 )
with k, β„“, m integers such that |m | ≤ β„“ ≤ k. Pmβ„“ (x) are the associated Legendre functions defined by Equation (100View Equation). G β„“(x ) k are the associated Gegenbauer functions
β„“ β„“ (β„“) (β„“) dβ„“Gk-(x-) G k(cosξ) = (sin ξ)G k (cosξ) with G k (x ) = dx β„“ , (114 )
where Gk(x ) is the k-th Gegenbauer polynomial (λ) C k with λ = 1, as the Gk are also a particular case of Jacobi polynomials with α = β = 1βˆ•2 (see, for example, [131]). Jacobi polynomials are also solutions of a singular Sturm–Liouville problem, which ensures fast convergence properties (see Section 2.4.1). The Gk (x) fulfill recurrence relations that make them easy to implement as a spectral decomposition basis, like the Legendre polynomials. These eigenfunctions are associated with the eigenvalues − k(k + 2):
Δ4 (G β„“(cosξ)P m(cos πœƒ)eim φ) = − k(k + 2)G β„“(cos ξ)Pm (cosπœƒ)eimφ. (115 ) k β„“ k β„“
So, as in 3+1 dimensions, after decomposing in such a basis, the Poisson equation turns into a collection of ODEs in the coordinate r. This type of construction might be generalized to even higher dimensions, with a choice of the appropriate type of Jacobi polynomials for every new introduced angular coordinate.

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