From pure fluid-dynamics simulations, spectral methods have rapidly been used in connected fields of research: geophysics , meteorology and climate modeling . In this last research category, global circulation models are used as boundary conditions to more specific (lower-scale) models, with improved micro-physics. In this way, spectral methods are only a part of the global numerical model, combined with other techniques to bring the highest accuracy, for a given computational power. A solution to the Maxwell equations can, of course, also be obtained with spectral methods and therefore, magneto-hydrodynamics (MHD) have been studied with these techniques (see, e.g., Hollerbach ). This has been the case in astrophysics too, where, for example, spectral three-dimensional numerical models of solar magnetic dynamo action realized by turbulent convection have been computed . And Kompaneet’s equation, describing the evolution of photon distribution function in a plasma bath at thermal equilibrium within the Fokker-Planck approximation, has been solved using spectral methods to model the X-ray emission of Her X-1 [33, 40]. In simulations of cosmological structure formation or galaxy evolution, many N-body codes rely on a spectral solver for the computation of the gravitational force by the particle-mesh algorithm. The mass corresponding to each particle is decomposed onto neighboring grid points, thus defining a density field. The Poisson equation giving the Newtonian gravitational potential is then usually solved in Fourier space for both fields .
To our knowledge, the first published result of the numerical solution of Einstein’s equations, using spectral methods, is the spherically-symmetric collapse of a neutron star to a black hole by Gourgoulhon in 1991 . He used spectral methods as they were developed in the Meudon group by Bonazzola and Marck . Later studies of quickly-rotating neutron stars  (stationary axisymmetric models), the collapse of a neutron star in tensor-scalar theory of gravity  (spherically-symmetric dynamic spacetime), and quasiequilibrium configurations of neutron star binaries  and of black holes  (three-dimensional and stationary spacetimes) have grown in complexity, up to the three-dimensional time-dependent numerical solution of Einstein’s equations . On the other hand, the first fully three-dimensional evolution of the whole Einstein system was achieved in 2001 by Kidder et al. , where a single black hole was evolved to using excision techniques. They used spectral methods as developed in the Cornell/Caltech group by Kidder et al.  and Pfeiffer et al. . Since then, they have focused on the evolution of black-hole–binary systems, which has recently been simulated up to merger and ring down by Scheel et al. . Other groups (for instance Ansorg et al. , Bartnik and Norton , Frauendiener  and Tichy ) have also used spectral methods to solve Einstein’s equations; Sections 5 and 6 are devoted to a more detailed review of these works.
Living Rev. Relativity 12, (2009), 1
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