There are relatively few rigorous results on convergence and stability of Chebyshev collocation methods for IBVPs; some of them are [211] and [210].

Even though this review is concerned with time-dependent problems, we note in passing that there are a significant number of efforts in relativity using spectral methods for the constraint equations; see [215]. The use of spectral methods in relativistic evolutions can be traced back to pioneering work in the mid -1980s [66] (see also [67, 68, 213]). Over the last decade they have gained popularity, with applications in scenarios as diverse as relativistic hydrodynamics [313, 427, 428], characteristic evolutions [43], absorbing and/or constraint-preserving boundary conditions [314, 369, 365, 363], constraint projection [244], late time “tail” behavior of black-hole perturbations [382, 420], cosmological studies [19, 49, 50], extreme–mass-ratio inspirals within perturbation theory and self-forces [112, 162, 111, 425, 114, 113, 123] and, prominently, binary black-hole simulations (see, for example, [384, 329, 71, 381, 132, 288, 402, 131, 90, 289]) and black-hole–neutron-star ones [150, 168]. The method of lines (Section 7.3) is typically used with a small enough timestep so that the time integration error is smaller than the one due to the spatial approximation and spectral convergence is observed. Spectral collocation methods were first used in spherically-symmetric black-hole evolutions of the Einstein equations in [255] and in three dimensions in [254]. The latter work showed that some constraint violations in the Einstein–Christoffel [22] type of formulations do not go away with resolution but are a feature of the continuum evolution equations (though the point – namely, that time instabilities are in some cases not a product of lack of resolution – applies to many other scenarios).

Most of these references use explicit symmetric hyperbolic first-order formulations. More recently, progress has been made towards using spectral methods for the BSSN formulation of the Einstein equations directly in second-order form in space [419, 163], and, generally, on multi-domain interpatch boundary conditions for second-order systems [413] (numerical boundary conditions are discussed in the next Section 10). A spectral spacetime approach (as opposed to spectral approximation in space and marching in time) for the 1+1 wave equation in compactified Minkowski was proposed in [233]; in higher dimensions and dynamical spacetimes the cost of such approach might be prohibitive though.

[83] presents an implementation of the harmonic formulation of the Einstein equations on a spherical domain using a double Fourier expansion and, in addition, significant speed-ups using Graphics Processing Units (GPUs).

[215] presents a detailed review of spectral methods in numerical relativity.

A promising approach, which, until recently, has been largely unexplored within numerical relativity is the use of discontinuous Galerkin methods [238, 457, 162, 163, 339].

Living Rev. Relativity 15, (2012), 9
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