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].
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