### 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 [199, 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 , but a generalization of spherical
coordinates is also possible and necessitates less computational resources. The
additional angle is defined in with the following relations with Cartesian coordinates

The four-dimensional flat Laplace operator appearing in constraint equations [199] reads
where is the two-dimensional angular Laplace operator (103). As in the three-dimensional case, it is
convenient to use the eigenfunctions of the angular part, which are here
with integers such that . are the associated Legendre functions defined by
Equation (100). are the associated Gegenbauer functions
where is the -th Gegenbauer polynomial with , as the are also a particular
case of Jacobi polynomials with (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 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 :
So, as in 3+1 dimensions, after decomposing in such a basis, the Poisson equation turns into a collection of
ODEs in the coordinate . 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.