9.8 Going further, applications in numerical relativity

Based on the minimum gridspacing between spectral collocation points, one would naively expect the CFL limit to scale as 2 1∕N, where N is the number of points. The expectation indeed holds, but the reason is related to the 2 𝒪 (N ) scaling of the eigenvalues of Jacobi polynomials as solutions to Sturm-Liouville problems (in fact, the result holds for non-collocation spectral methods as well) [212].

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 [215Jump To The Next Citation Point]. 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 [314Jump To The Next Citation Point, 369Jump To The Next Citation Point, 365Jump To The Next Citation Point, 363Jump To The Next Citation Point], 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, 162Jump To The Next Citation Point, 111, 425Jump To The Next Citation Point, 114, 113, 123] and, prominently, binary black-hole simulations (see, for example, [384Jump To The Next Citation Point, 329, 71, 381Jump To The Next Citation Point, 132, 288, 402Jump To The Next Citation Point, 131Jump To The Next Citation Point, 90Jump To The Next Citation Point, 289Jump To The Next Citation Point]) and black-hole–neutron-star ones [150Jump To The Next Citation Point, 168Jump To The Next Citation Point]. 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, 163Jump To The Next Citation Point], and, generally, on multi-domain interpatch boundary conditions for second-order systems [413Jump To The Next Citation Point] (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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