Going further

The development of spectral methods linked with the problems arising in the field of numerical relativity has always been active and still is now. Among the various directions of research one can foresee, the most interesting might be the improvement of time-integration techniques and the beginning of higher-dimensional studies. In addition to these, it might also be relevant to consider the development of better-adapted mappings and domains, within the spirit of going from pure spectral methods to spectral elements [115, 22].

3.4.1 High-order time schemes

When using spectral methods in time-dependent problems, it is sometimes frustrating to have so accurate numerical techniques for the evaluation of spatial derivatives, and the integration

of elliptic PDEs, whereas the time derivatives, and hyperbolic PDEs, do not benefit from the exponential convergence. Some tentative studies are being undertaken in order to represent also the time interval by spectral methods [6] and, if these techniques can be applied in general three- dimensional simulations, it would really be a great improvement.

Nevertheless, there are other, also more sophisticated and accurate, time-integration techniques that are currently investigated for several stiff PDEs [88], among which Korteweg-de Vries and nonlinear Schr¨odinger equations [93]. Many such PDEs share the properties of being stiff (very different time-scales/ characteristic frequencies) and combining low-order non-linear terms with higher-order linear terms. Einstein evolution equations can also be written in such a way [29]. Let us consider a PDE

∂u

∂t = Lu +N u, (119)

with the notations of Sec. 3.1.1 and N being a nonlinear spatial operator. Following the same notations and within spectral approximation, one recovers

∂UN

∂t = LNUN +NNUN. (120)

We detail hereafter five methods to solve this type of ODEs (see also [88]):

• Implicit-explicit techniques use some explicit multi-step scheme to advance the nonlinear part_NN, and an implicit one for the linear one.

• Split-step are effective when the equation splits into two equation which can be directly integrated (see [93] for examples with the nonlinear Schr¨odinger and Korteweg-de Vries equa- tions).

• Integrating factor is a change of variable that allows for the exact solution of the linear part

VN = e−LNtUN, (121)

and to use an explicit multi-step method for the integration of the new nonlinear part ∂VN

∂t = e

−LNt

NNeLNtVN. (122)

• Sliders can be seen as an extension of the implicit-explicit method described above. In addition to splitting to problem into a linear and nonlinear part, the linear part itself is split into two or three regions (in Fourier space), depending on the wavenumber. Then, different numerical schemes are used for different groups of wavenumbers: implicit schemes for high wavenumbers and explicit high-order methods for the low wavenumbers. This method is restricted to Fourier spectral techniques in space.

• Exponential time-differencing have been known for some time in computational electro- dynamics. These methods are similar to the integrating factor technique, but one considers the exact equation over one time-step

U_NJ+1= eLN∆t_UJ N+ eLN∆t Z ∆t 0 e−LNτ NN(UN(N ∆t + τ ), N ∆t + τ )dτ. (123)

Various orders for these schemes come from the approximation order of the integral. For example Kassam and Trefethen [88] consider a fourth-order Runge-Kutta type approximation to this integral, where the difficulty comes from the accurate computation of functions which suffer from cancellation errors.

3.4.2 More than three spatial dimensions

There have been some interest for the numerical study of black holes in higher dimensions: as well with compactified extra-dimensions [138], as in brane world models [136, 96]; recently, some simulations on the head-on collision of two black holes have already been undertaken [156]. 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. We give here starting points to implement 4-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, θ, 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 = r sin θ cos ϕ sin ξ, y = r sin θ sin ϕ sin ξ, z = r cos θ sin ξ,

w = r cos ξ.

The four-dimensional flat Laplace operator appearing in constraint equations [136] reads ∆4φ =∂ 2_φ ∂r2 + 3 r ∂φ ∂r+ 1 r2  ∂2_φ ∂ξ2 + 2 tan ξ ∂φ ∂ξ + 1 sin2ξ∆θϕφ  , (124)

where ∆θϕ is the two-dimensional angular Laplace operator (111). As in the three-dimensional

case, it is convenient to use the eigenfunctions of the angular part, which are here

Gℓk(cos ξ)Pℓm(cos θ)eimϕ, (125)

with k, ℓ, m integers such that|m| ≤ ℓ ≤ k. Pm

ℓ (x) are the associated Legendre functions defined

by Eq. (108). Gℓ

k(x) are the associated Gegenbauer functions

Gℓk(cos ξ) = (sinℓξ)G (ℓ) k (cos ξ) with G (ℓ) k (x) = dℓ_G k(x) dxℓ , (126)

and Gk(x) being the k-th Gegenbauer polynomial Ck(λ)with λ = 1. Since the Gkare also particular

case of Jacobi polynomials with α = β = 1/2 (see, for example [95]), they fulfill recurrence relations that make them easy to implement as spectral decomposition basis, like the Legendre polynomials. These eigenfunctions are associated with the eigenvalues_{−k(k + 2):}

∆4 Gℓk(cos ξ)Pℓm(cos θ)eimϕ
= −k(k + 2)Gℓk(cos ξ)Pℓm(cos θ)eimϕ. (127)

So as in 3+1 dimensions, after decomposing on 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 the choice of appropriate type of Jacobi polynomials for every new introduced angular coordinate.