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 with and a large
variety 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 results obtained. Although some of
the theoretical framework of spectral methods is given in Sections 2 to 4, more details 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 may also help the interested reader to get a deeper understanding
of the subject. This review is organized as follows: hereafter in the introduction, we briefly
introduce 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. Multidomain approach is also
introduced there, whereas some multidimensional techniques are described in Section 3. The cases of
time-dependent partial differential equations (PDE) are treated in Section 4. The last two sections
then review results obtained using spectral methods: for stationary configurations and initial
data (Section 5), and for the time evolution (Section 6) of stars, gravitational waves and black
holes.
