arXiv:0706.2286v2 [gr-qc] 23 Oct 2008 - CiteSeerXwww.researchgate.net › publication › fulltext › Spectral-

Spectral Methods for Numerical Relativity Philippe Grandcl´ement Laboratoire Univers et Th´eories UMR 8102 du C.N.R.S., Observatoire de Paris F-92195 Meudon Cedex, France email: [email protected] http://www.luth.obspm.fr/minisite.php?nom=Grandclement J´erˆome Novak Laboratoire Univers et Th´eories UMR 8102 du C.N.R.S., Observatoire de Paris F-92195 Meudon Cedex, France email: [email protected] http://www.luth.obspm.fr/minisite.php?nom=Novak Abstract Equations arising in General Relativity are usually too complicated to be solved analytically and one has to rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses on a class called spectral methods where, typically, the various functions are expanded onto sets of orthogonal polynomials or functions. A theoretical introduction on spectral expansion is first given and a particular emphasis is put on the fast convergence of the spectral approximation. We present then different approaches to solve partial differential equations, first limiting ourselves to the one-dimensional case, with one or several domains. Generalization to more dimensions is then discussed. In particular, the case of time evolutions is carefully studied and the stability of such evolutions investigated. One then turns to results obtained by various groups in the field of General Relativity by means of spectral methods. First, works which do not involve explicit time-evolutions are discussed, going from rapidly rotating strange stars to the computation of binary black holes initial data. Finally, the evolutions of various systems of astrophysical interest are presented, from supernovae core collapse to binary black hole mergers.

1

1

Introduction

Einstein’s equations represent a complicated set of nonlinear partial differential equations for which some exact [30] or approximate [31] analytical solutions are known. But these solutions are not always suitable for physically or astrophysically interesting systems, which require an accurate description of their relativistic gravitational field without any assumption on the symmetry or with the presence of matter fields, for instance. Therefore, many efforts have been undertaken to solve Einstein’s equations with the help of computers in order to model relativistic astrophysical objects. Within this field of numerical relativity, several numerical methods have been experimented and a large variety of them are currently being used. Among them, spectral methods are now increasingly popular and the goal of this review is to give an overview (at the moment it is written or updated) of the methods themselves, the groups using them and the obtained results. Although some theoretical framework of spectral methods is given in Sections 2 to 4, more details about spectral methods can be found in the books by Gottlieb and Orszag [94], Canuto et al. [56, 57, 58], Fornberg [79], Boyd [48] and Hesthaven et al. [117]. While these references have of course been used for writing this review, they can also help the interested reader to get deeper understanding of the subject. This review is organized as follows: hereafter in the introduction, we briefly introduce the spectral methods, their usage in computational physics and give a simple example. Section 2 gives important notions concerning polynomial interpolation and the solution of ordinary differential equations (ODE) with spectral methods. Multi-domain approach is also introduced there, whereas some of the multi-dimensional techniques are described in Section 3. The cases of timedependent partial differential equations (PDE), are treated in Section 4. The last two sections are then reviewing results obtained using spectral methods: on stationary configurations and initial data (Section 5), and on the time-evolution (Section 6) of stars, gravitational waves and black holes.

1.1

About spectral methods

When doing simulations and solving PDEs, one faces the problem of representing and deriving functions on a computer, which deals only with (finite) integers. Let us take a simple example of a function f : [−1, 1] → R. The most straightforward way to approximate its derivative is through finite-differences methods: first one must setup a grid {xi }i=0...N ⊂ [−1, 1] of N + 1 points in the interval, and represent f by its N + 1 values on these grid points {fi = f (xi )}i=0...N .

Then, the (approximate) representation of the derivative f ′ shall be, for instance ∀i < N, fi′ = f ′ (xi ) ≃

fi+1 − fi . xi+1 − xi

(1)

If we suppose an equidistant grid, so that ∀i < N, xi+1 − xi = ∆x = 1/N , the error in the approximation (1) will decay as ∆x (first-order scheme). One can imagine higher-order schemes, with more points involved for the computation of each derivative and, for a scheme of order n, the n accuracy can vary as (∆x) = 1/N n . Spectral methods represent an alternate way: the function f is no longer represented through its values on a finite number of grid points, but using its coefficients (coordinates) {ci }i=0...N in a finite bas