Spectral Methods for Numerical Relativity
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Abstract
Equations arising in general relativity are usually too complicated to be solved analytically and one must rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses on a class called spectral methods in which, typically, the various functions are expanded in sets of orthogonal polynomials or functions. First, a theoretical introduction of spectral expansion is given with a particular emphasis on the fast convergence of the spectral approximation. We then present different approaches to solving partial differential equations, first limiting ourselves to the onedimensional case, with one or more domains. Generalization to more dimensions is then discussed. In particular, the case of time evolutions is carefully studied and the stability of such evolutions investigated. We then present results obtained by various groups in the field of general relativity by means of spectral methods. Work, which does not involve explicit timeevolutions, is discussed, going from rapidlyrotating strange stars to the computation of blackholebinary initial data. Finally, the evolution of various systems of astrophysical interest are presented, from supernovae core collapse to blackholebinary mergers.
Keywords
Black Hole Neutron Star Gravitational Wave Spherical Harmonic Spectral Method1 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 timedependent 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.
1.1 About spectral methods

Finite differences: the trial functions are overlapping local polynomials of fixed order (lower than N).

Finite elements: the trial functions are local smooth functions, which are nonzero, only on subdomains of U.

Spectral methods: the trial functions are global smooth functions on U.
Various choices of the test functions define different types of spectral methods, as detailed in Section 2.5. Usual choices for the trial functions are (truncated) Fourier series, spherical harmonics or orthogonal families of polynomials.
1.2 Spectral methods in physics
We do not give here all the fields of physics in which spectral methods are employed, but sketching the variety of equations and physical models that have been simulated with such techniques. Spectral methods originally appeared in numerical fluid dynamics, where large spectral hydrodynamic codes have been regularly used to study turbulence and transition to the turbulence since the seventies. For fully resolved, direct numerical calculations of NavierStokes equations, spectral methods were often preferred for their high accuracy. Historically, they also allowed for two or threedimensional simulations of fluid flows, because of their reasonable computer memory requirements. Many applications of spectral methods in fluid dynamics have been discussed by Canuto et al. [56, 58], and the techniques developed in that field are of some interest to numerical relativity.
From pure fluiddynamics simulations, spectral methods have rapidly been used in connected fields of research: geophysics [189], meteorology and climate modeling [216]. In this last research category, global circulation models are used as boundary conditions to more specific (lowerscale) models, with improved microphysics. 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, magnetohydrodynamics (MHD) have been studied with these techniques (see, e.g., Hollerbach [119]). This has been the case in astrophysics too, where, for example, spectral threedimensional numerical models of solar magnetic dynamo action realized by turbulent convection have been computed [52]. And Kompaneet’s equation, describing the evolution of photon distribution function in a plasma bath at thermal equilibrium within the FokkerPlanck approximation, has been solved using spectral methods to model the Xray emission of Her X1 [33, 40]. In simulations of cosmological structure formation or galaxy evolution, many Nbody codes rely on a spectral solver for the computation of the gravitational force by the particlemesh 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 [118].
To our knowledge, the first published result of the numerical solution of Einstein’s equations, using spectral methods, is the sphericallysymmetric collapse of a neutron star to a black hole by Gourgoulhon in 1991 [95]. He used spectral methods as they were developed in the Meudon group by Bonazzola and Marck [44]. Later studies of quicklyrotating neutron stars [41] (stationary axisymmetric models), the collapse of a neutron star in tensorscalar theory of gravity [156] (sphericallysymmetric dynamic spacetime), and quasiequilibrium configurations of neutron star binaries [39] and of black holes [110] (threedimensional and stationary spacetimes) have grown in complexity, up to the threedimensional timedependent numerical solution of Einstein’s equations [37]. On the other hand, the first fully threedimensional evolution of the whole Einstein system was achieved in 2001 by Kidder et al. [127], where a single black hole was evolved to t ≃ 600 M − 1300 M using excision techniques. They used spectral methods as developed in the Cornell/Caltech group by Kidder et al. [125] and Pfeiffer et al. [171]. Since then, they have focused on the evolution of blackholebinary systems, which has recently been simulated up to merger and ring down by Scheel et al. [185]. Other groups (for instance Ansorg et al. [10], Bartnik and Norton [21], Frauendiener [81] and Tichy [219]) have also used spectral methods to solve Einstein’s equations; Sections 5 and 6 are devoted to a more detailed review of these works.
1.3 A simple example
 1.
Setup an adapted grid for the computation of spectral coefficients (e.g., equidistant in the angular directions and ChebyshevGaussLobatto collocation points; see Section 2.4.3).
 2.
Get the values of the source σ on these grid points.
 3.
Perform a sphericalharmonics transform (for example, using some available library [151]), followed by the Chebyshev transform (using a Fast Fourier Transform (FFT), or a GaussLobatto quadrature) of the source σ.
 4.
For each couple of values (ℓ, m), build the corresponding matrix \({\bar L}\) with the boundary conditions, and invert the system (using any available linearalgebra package) with the coefficients of as the righthand side.
 5.
Perform the inverse spectral transform to get the values of ϕ on the grid points from its coefficients.
A numerical implementation of this algorithm has been reported by Grandclément et al. [109], who have observed that the error decayed as \({e^{ {\ell _{\max}}}} \cdot {e^{ N}}\), provided that the source σ was smooth. Machine roundoff accuracy can be reached with ℓ_{max} ∼ N ∼ 30, which makes the matrix inversions of step 4 very cheap in terms of CPU and the whole method affordable in terms of memory usage. These are the main advantages of using spectral methods, as shall be shown in the following sections.
2 Concepts in One Dimension
In this section the basic concept of spectral methods in one spatial dimension is presented. Some general properties of the approximation of functions by polynomials are introduced. The main formulae of the spectral expansion are then given and two sets of polynomials are discussed (Legendre and Chebyshev polynomials). A particular emphasis is put on convergence properties (i.e., the way the spectral approximation converges to the real function).
In Section 2.5, three different methods of solving an ordinary differential equation (ODE) are exhibited and applied to a simple problem. Section 2.6 is concerned with multidomain techniques. After giving some motivations for the use of multidomain decomposition, four different implementations are discussed, as well as their respective merits. One simple example is given, which uses only two domains.
For problems in more than one dimension see Section 3.
2.1 Best polynomial approximation
Polynomials are the only functions that a computer can exactly evaluate and so it is natural to try to approximate any function by a polynomial. When considering spectral methods, we use global polynomials on a few domains. This is to be contrasted with finite difference schemes, for instance, where only local polynomials are considered.
2.2 Interpolation on a grid
A grid X on the interval [−1, 1] is a set of N + 1 points x_{i} ∈ [−1, 1], 0 ≤ i ≤ N. These points are called the nodes of the grid X.
2.3 Polynomial interpolation
2.3.1 Orthogonal polynomials
At first sight, the projection seems to be an interesting means of numerically representing a function. However, in practice this is not the case. Indeed, to determine the projection of a function, one needs to compute the integrals (30), which requires the evaluation of at a great number of points, making the whole numerical scheme impractical.
2.3.2 Gaussian quadratures

Gauss quadrature: δ = 1.

GaussRadau: δ = 0 and x_{0} = −1.

GaussLobatto: δ = −1 and x_{0} = −1, x_{N} = 1.
Gauss quadrature is the best choice because it applies to polynomials of higher degree but GaussLobatto quadrature is often more useful for numerical purposes because the outermost collocation points coincide with the boundaries of the interval, making it easier to impose matching or boundary conditions. More detailed results and demonstrations about those quadratures can be found for instance in [57].
2.3.3 Spectral interpolation
2.3.4 Two equivalent descriptions

in the configuration space, if the function is described by its value at the N + 1 collocation points f(x_{i});

in the coefficient space, if one works with the N + 1 coefficients \({{\tilde f}_i}\).
Depending on the operation one has to perform on a given function, it may be more clever to work in one space or the other. For instance, the square root of a function is very easily given in the collocation space by \(\sqrt {f({x_i})}\), whereas the derivative can be computed in the coefficient space if, and this is generally the case, the derivatives of the basis polynomials are known, by \({f^{\prime}}(x) = \sum\limits_{n = 0}^N {{{\tilde f}_n}{p^{\prime}}_n(x)}\).
2.4 Usual polynomials
2.4.1 SturmLiouville problems and convergence

p(x) is continuously differentiable, strictly positive and continuous at x = ±1.

q(x) is continuous, nonnegative and bounded.

w(x) is continuous, nonnegative and integrable.
Singular SturmLiouville problems are particularly important for spectral methods. A SturmLiouville problem is singular if and only if the function p vanishes at the boundaries x = ±1. One can show, that if the functions of the spectral basis are chosen to be the solutions of a singular SturmLiouville problem, then the convergence of the function to its interpolant is faster than any power law of N, N being the order of the expansion (see Section 5.2 of [57]). One talks about spectral convergence. Let us be precise in saying that this does not necessarily imply that the convergence is exponential. Convergence properties are discussed in more detail for Legendre and Chebyshev polynomials in Section 2.4.4.
Conversely, it can be shown that spectral convergence is not ensured when considering solutions of regular SturmLiouville problems [57].
In what follows, two usual types of solutions of singular SturmLiouville problems are considered: Legendre and Chebyshev polynomials.
2.4.2 Legendre polynomials

LegendreGauss: x_{i} are the nodes of P_{N+1} and \({w_i} = {2 \over {(1  x_i^2){{[P_{N  1}^{\prime}({x_i})]}^2}}}\).

LegendreGaussRadau: x_{0} = −1 and x_{i} are the nodes of P_{N} + P_{N+1}. The weights are given by \({w_0} = {2 \over {{{(N + 1)}^2}}}\) and \({w_i} = {1 \over {{{(N + 1)}^2}}}\).

LegendreGaussLobatto: x_{0} = −1, x_{N} = 1 and x_{i} are the nodes of P′_{N}. The weights are \({w_i} = {2 \over {N(N + 1)}}{1 \over {{{[{P_N}({x_i})]}^2}}}\).
These values have no analytic expression, but they can be computed numerically in an efficient way.
 if is multiplication by x then:$${b_n} = {n \over {2n  1}}{a_{n  1}} + {{n + 1} \over {2n + 3}}{a_{n + 1}}\quad (n \geq 1);$$(41)
 if is the derivative:$${b_n} = (2n + 1)\sum\limits_{p = n + 1,p + n\;{\rm{odd}}}^N {{a_p}};$$(42)
 if is the second derivative:$${b_n} = (n + 1/2)\sum\limits_{p = n + 2,p + n\;{\rm{even}}}^N {[p(p + 1)  n(n + 1)]{a_p}}.$$(43)
These kind of relations enable one to represent the action of H as a matrix acting on the vector of a_{n}, the product being the coefficients of Hf, i.e. the b_{n}.
2.4.3 Chebyshev polynomials

ChebyshevGauss: \({x_i} = \cos {{(2i + 1)\pi} \over {2N + 2}}\) and \({w_i} = {\pi \over {N + 1}}\).

ChebyshevGaussRadau: \({x_i} = \cos {{2\pi i} \over {2N + 1}}\). The weights are \({w_0} = {\pi \over {2N + 1}}\) and \({w_i} = {{2\pi} \over {2N + 1}}\).

ChebyshevGaussLobatto: \({x_i} = \cos {{\pi i} \over N}\). The weights are \({w_0} = {w_N} = {\pi \over {2N}}\) and \({w_i} = {\pi \over N}\).
 if is multiplication by x:$${b_n} = {1 \over 2}[(1 + {\delta _{0\;n  1}}){a_{n  1}} + {a_{n + 1}}]\quad (n \geq 1);$$(47)
 if H is the derivative:$${b_n} = {2 \over {(1 + {\delta _{0\;n}})}}\sum\limits_{p = n + 1,p + n\;{\rm{odd}}}^N {p{a_p}};$$(48)
 if H is the second derivative:$${b_n} = {1 \over {(1 + {\delta _{0\;n}})}}\sum\limits_{p = n + 2,p + n\;{\rm{even}}}^N {p({p^2}  {n^2}){a_p}}.$$(49)
2.4.4 Convergence properties
 For Legendre:$${\left\Vert {{I_N}f  f} \right\Vert _{{L^2}}} \leq {{{C_1}} \over {{N^{m  1/2}}}}\sum\limits_{k = 0}^m {{{\left\Vert {{f^{(k)}}} \right\Vert}_{{L^2}}}}.$$(50)
 For Chebyshev:$${\left\Vert {{I_N}f  f} \right\Vert _{L_w^2}} \leq {{{C_2}} \over {{N^m}}}\sum\limits_{k = 0}^m {{{\left\Vert {{f^{(k)}}} \right\Vert}_{L_w^2}}}.$$(51)$${\left\Vert {{I_N}f  f} \right\Vert _\infty} \leq {{{C_3}} \over {{N^{m  1/2}}}}\sum\limits_{k = 0}^m {{{\left\Vert {{f^{(k)}}} \right\Vert}_{L_w^2}}}.$$(52)
The C_{i} are positive constants. An interesting limit of the above estimates concerns a \({{\mathcal C}^\infty}\) function. One can then see that the difference between f and I_{N}f decays faster than any power of N. This is spectral convergence. Let us be precise in saying that this does not necessarily imply that the error decays exponentially (think about exp \(\left({ \sqrt N} \right)\) for instance). Exponential convergence is achieved only for analytic functions, i.e. functions that are locally given by a convergent power series.
Finally, Equation (52) shows that, if m > 0, the interpolant converges uniformly to the function. Continuous functions that do not converge uniformly to their interpolant, whose existence has been shown by Faber [73] (see Section 2.2), must belong to the \({{\mathcal C}^0}\) functions. Indeed, for the case m = 0, Equation (52) does not prove convergence (neither do Equations (50) or (51)).
2.4.5 Trigonometric functions
A detailed presentation of the theory of Fourier transforms is beyond the scope of this work. However, there is a close link between discrete Fourier transforms and their spectral interpolation, which is briefly outlined here. More detail can be found, for instance, in [57].
The approximation made by using discrete coefficients in place of real ones is of the same nature as the one made when computing coefficients of projection (30) by means of the Gaussian quadratures. Let us mention that, in the case of a discrete Fourier transform, the first and last collocation points lie on the boundaries of the interval, as for a GaussLobatto quadrature. As for the polynomial interpolation, the convergence of \({I_N}f\) to f is spectral for all periodic and \({{\mathcal C}^\infty}\) functions.
2.4.6 Choice of basis
For periodic functions of [0, 2π], the discrete Fourier transform is the natural choice of basis. If the considered function has also some symmetries, one can use a subset of the trigonometric polynomials. For instance, if the function is i) periodic on [0, 2π] and is also odd with respect to x = π, then it can be expanded in terms of sines only. If the function is not periodic, then it is natural to expand it either in Chebyshev or Legendre polynomials. Using Legendre polynomials can be motivated by the fact that the associated measure is very simple: w(x) = 1. The multidomain technique presented in Section 2.6.5 is one particular example in which such a property is required. In practice, Legendre and Chebyshev polynomials usually give very similar results.
2.5 Spectral methods for ODEs
2.5.1 The weighted residual method
2.5.2 The tau method
In this particular method, the test functions coincide with the basis used for the spectral expansion, for instance the Chebyshev polynomials. Let us denote ũ_{i} and \({{\tilde {\mathcal S}}_i}\) the coefficients of the solution u and of the source \(S\), respectively.
The tau method thus ensures that L_{u} and S have the same coefficients, excepting the last ones. If the functions are smooth, then their coefficients should decrease in a spectral manner and so the “forgotten” conditions are less and less stringent as N increases, ensuring that the computed solution converges rapidly to the real one.
2.5.3 The collocation method
Let us note that, even if the collocation method imposes that Lu and S coincide at each collocation point, the unknowns of the system written in the form (66) are the coefficients ũ_{n} and not u(x_{n}). As for the tau method, system (66) is not invertible and boundary conditions must be enforced by additional equations. In this case, the relaxed conditions are the two associated with the outermost points, i.e. n = 0 and n = N, which are replaced by appropriate boundary conditions to get an invertible system.
2.5.4 Galerkin method
The basic idea of the Galerkin method is to seek the solution u as a sum of polynomials G_{i} that individually verify the boundary conditions. Doing so, u automatically fulfills those conditions and they do not have to be imposed by additional equations. Such polynomials constitute a Galerkin basis of the problem. For practical reasons, it is better to choose a Galerkin basis that can easily be expressed in terms of the original orthogonal polynomials.
2.5.5 Optimal methods
A spectral method is said to be optimal if it does not introduce an additional error to the error that would be introduced by interpolating the exact solution of a given equation.
Let us call u_{exact} such an exact solution, unknown in general. Its interpolant is I_{N}u_{exact} and the numerical solution of the equation is u_{num}. The numerical method is then optimal if and only if \(\Vert {I_N}{u_{{\rm{exact}}}}  {u_{{\rm{exact}}}}{\Vert _\infty}\) and \(\Vert {u_{{\rm{num}}}}  {u_{{\rm{exact}}}}{\Vert _\infty}\) behave in the same manner when N → ∞.
2.6 Multidomain techniques for ODEs
2.6.1 Motivations and setting
As seen in Section 2.4.4, spectral methods are very efficient when dealing with \({{\mathcal C}^\infty}\) functions. However, they lose some of their appeal when dealing with less regular functions, the convergence to the exact functions being substantially slower. Nevertheless, the physicist has sometimes to deal with such functions. This is the case for the density jump at the surface of strange stars or the formation of shocks, to mention only two examples. In order to maintain spectral convergence, one then needs to introduce several computational domains such that the various discontinuities of the functions lie at the interface between the domains. Doing so in each domain means that one only deals with \({{\mathcal C}^\infty}\) functions.
Multidomain techniques can also be valuable when dealing with a physical space either too complicated or too large to be described by a single domain. Related to that, one can also use several domains to increase the resolution in some parts of the space where more precision is required. This can easily be done by using a different number of basis functions in different domains. One then talks about fixedmesh refinement.
Efficient parallel processing may also require that several domains be used. Indeed, one could set a solver, dealing with each domain on a given processor, and interprocessor communication would then only be used for matching the solution across the various domains. The algorithm of Section 2.6.4 is well adapted to such purposes.
In the following, four different multidomain methods are presented to solve an equation of the type Lu = S on [−1, 1]. L is a secondorder linear operator and S is a given source function. Appropriate boundary conditions are given at the boundaries x = −1 and x = 1.

first domain: x ≤ 0 described by x_{1} = 2x + 1, x_{1} ∈ [−1, 1],

second domain: x ≥ 0 described by x_{2} = 2x − 1, x_{2} ∈ [−1, 1].
If x ≤ 0, a function u is described by its interpolant in terms of \({x_1}:{I_N}u(x) = \sum\limits_{i = 0}^N {\tilde u_i^1{T_i}({x_1}(x))}\). The same is true for x ≥ 0 with respect to the variable x_{2}. Such a setup is obviously appropriate to deal with problems where discontinuities occur at x = 0, that is x_{1} = 1 and x_{2} = −1.
2.6.2 The multidomain tau method
Two supplementary equations are enforced to ensure that the boundary conditions are fulfilled. Finally, the operator L being of second order, one needs to ensure that the solution and its first derivative are continuous at the interface x = 0. This translates to a set of two additional equations involving both domains.

N − 1 residual equations in the first domain,

N − 1 residual equations in the second domain,

2 boundary conditions,

2 matching conditions,
2.6.3 Multidomain collocation method
Two supplementary equations are enforced to ensure that the boundary conditions are fulfilled. Finally, the operator L being second order, one needs to ensure that the solution and its first derivative are continuous at the interface x = 0. This translates to a set of two additional equations involving the coefficients in both domains.

N − 1 residual equations in the first domain,

N − 1 residual equations in the second domain,

2 boundary conditions,

2 matching conditions,
2.6.4 Method based on homogeneous solutions
The method described here proceeds in two steps. First, particular solutions are computed in each domain. Then, appropriate linear combinations with the homogeneous solutions of the operator L are performed to ensure continuity and impose boundary conditions.
In order to compute particular solutions, one can rely on any of the methods described in Section 2.5. The boundary conditions at the boundary of each domain can be chosen (almost) arbitrarily. For instance, one can use in each domain a collocation method to solve Lu = S, demanding that the particular solution u_{part} is zero at both ends of each interval.
Then, in order to have a solution over the whole space, one needs to add homogeneous solutions to the particular ones. In general, the operator L is second order and admits two independent homogeneous solutions g and h in each domain. Let us note that, in some cases, additional regularity conditions can reduce the number of available homogeneous solutions. The homogeneous solutions can either be computed analytically if the operator L is simple enough or numerically, but one must then have a method for solving Lu = 0.
2.6.5 Variational method
Contrary to previously presented methods, the variational one is only applicable with Legendre polynomials. Indeed, the method requires that the measure be w(x) = 1. It is also useful to extract the secondorder term of the operator L and to rewrite it as Lu = u″ + H, H being first order only.
As usual, two additional equations are provided by appropriate boundary conditions at both ends of the global domain. One also gets an additional condition by matching the solution across the boundary between the two domains.
2.6.6 Merits of the various methods
From a numerical point of view, the method based on an explicit matching using the homogeneous solutions is somewhat different from the two others. Indeed, one must solve several systems in a row and each one is of the same size as the number of points in one domain. This splitting of the different domains can also be useful for designing parallel codes. On the contrary, for both the variational and the tau method one must solve only one system, but its size is the same as the number of points in a whole space, which can be quite large for many domains settings. However, those two methods do not require one to compute the homogeneous solutions, computation that could be tricky depending on the operators involved and the number of dimensions.
The variational method may seem more difficult to implement and is only applicable with Legendre polynomials. However, on mathematical grounds, it is the only method that is demonstrated to be optimal. Moreover, some examples have been found in which the others methods are not optimal. It remains true that the variational method is very dependent on both the shape of the domains and the type of equation that needs to be solved.
3 Multidimensional Cases
3.1 Spatial coordinate systems

Cartesian (rectangular) coordinates are of course the simplest and most straightforward to implement; the line element reads ds^{2} = dx^{2} + dy^{2} + dz^{2}. These coordinates are regular in all space, with vanishing connection, which makes them easy to use, since all differential operators have simple expressions and the associated triad is also perfectly regular. They are particularly well adapted to cubelike domains, see for instance [167, 171] and [81] in the case of toroidal topology.

Circular cylindrical coordinates have a line element ds^{2} = dρ^{2} + ρ^{2} dϕ^{2} + dz^{2} and exhibit a coordinate singularity on the zaxis (ρ = 0). The associated triad being also singular for ρ = 0, regular vector or tensor fields have components that are multivalued (depending on ϕ) at any point of the zaxis. As for the spherical coordinates, this can be handled quite easily with spectral methods. This coordinate system can be useful for axisymmetric or rotating systems, see [10].

Spherical (polar) coordinates will be discussed in more detail in Section 3.2. Their line element reads ds^{2} = dr^{2} + r^{2} dθ^{2} + r^{2} sin^{2} θ dφ^{2}, showing a coordinate singularity at the origin (r = 0) and on the axis for which θ = 0, π. They are very useful in numerical relativity for the numerous spherelike objects under study (stars, black hole horizons) and have mostly been implemented for shelllike domains [40, 109, 167, 219] and for spheres including the origin [44, 109].

Prolate spheroidal coordinates consist of a system of confocal ellipses and hyperbolae, describing an (x, z)plane, and an angle φ giving the position as a rotation with respect to the focal axis [131]. The line element is ds^{2} = a^{2}(sinh^{2} μ + sin^{2} ν) (dμ^{2} + dν^{2}) + a^{2} sinh^{2} μ sin^{2} ν dφ^{2}. The foci are situated at z = ±a and represent coordinate singularities for μ = 0 and ν = 0, π. These coordinates have been used in [8] with blackholepuncture data at the foci.

Bispherical coordinates are obtained by the rotation of bipolar coordinates around the focal axis, with a line element ds^{2} = a^{2} (cosh η − cos χ)^{−2} (dη^{2} + dχ^{2} + sin^{2} χdφ^{2}). As with prolate spheroidal coordinates, the foci situated at z = ±a (η → ±∞, χ = 0, π) and more generally, the focal axis, exhibit coordinate singularities. Still, the surfaces of constant are spheres situated in the z > 0(< 0) region for η > 0(< 0), respectively. Thus, these coordinates are very well adapted for the study of binary systems and in particular for excision treatment of black hole binaries [6].
3.1.1 Mappings
Choosing a smart set of coordinates is not the end of the story. As for finite elements, one would like to be able to cover some complicated geometries, like distorted stars, tori, etc… or even to be able to cover the whole space. The reason for this last point is that, in numerical relativity, one often deals with isolated systems for which boundary conditions are only known at spatial infinity. A quite simple choice is to perform a mapping from numerical coordinates to physical coordinates, generalizing the change of coordinates to [−1, 1], when using families of orthonormal polynomials or to [0, 2π] for Fourier series.
The case where the boundaries of the considered domain are not known at the beginning of the computation can also be treated in a spectral way. In the case where this surface corresponds to the surface of a neutron star, two approaches have been used. First, in Bonazzola et al. [38], the star (and therefore the domain) is supposed to be “starlike”, meaning that there exists a point from which it is possible to reach any point on the surface by straight lines that are all contained inside the star. To such a point is associated the origin of a spherical system of coordinates, so that it is a spherical domain, which is regularly deformed to coincide with the shape of the star. This is done within an iterative scheme, at every step, once the position of the surface has been determined. Then, another approach has been developed by Ansorg et al. [10] using cylindrical coordinates. It is a square in the plane (ρ, z), which is mapped onto the domain describing the interior of the star. This mapping involves an unknown function, which is itself decomposed in terms of a basis of Chebyshev polynomials, so that its coefficients are part of the global vector of unknowns (as the density and gravitational field coefficients).
In the case of blackholebinary systems, Scheel et al. [188] have developed horizontracking coordinates using results from control theory. They define a control parameter as the relative drift of the black hole position, and they design a feedback control system with the requirement that the adjustment they make on the coordinates be sufficiently smooth that they do not spoil the overall Einstein solver. In addition, they use a dualcoordinate approach, so that they can construct a comoving coordinate map, which tracks both orbital and radial motion of the black holes and allows them to successfully evolve the binary. The evolutions simulated in [188] are found to be unstable, when using a single rotatingcoordinate frame. We note here as well the work of Bonazzola et al. [42], where another option is explored: the stroboscopic technique of matching between an inner rotating domain and an outer inertial one.
3.1.2 Spatial compactification
3.1.3 Patching in more than one dimension
The multidomain (or multipatch) technique has been presented in Section 2.6 for one spatial dimension. In Bonazzola et al. [40] and Grandclément et al. [109], the threedimensional spatial domains consist of spheres (or starshaped regions) and spherical shells, across which the solution can be matched as in onedimensional problems (only through the radial dependence). In general, when performing a matching in two or three spatial dimensions, the reconstruction of the global solution across all domains might need some more care to clearly write down the matching conditions (see, e.g., [167], where overlapping as well as nonoverlapping domains are used at the same time). For example in two dimensions, one of the problems that might arise is the counting of matching conditions for corners of rectangular domains, when such a corner is shared among more than three domains. In the case of a PDE where matching conditions must be imposed on the value of the solution, as well as on its normal derivative (Poisson or wave equation), it is sufficient to impose continuity of either normal derivative at the corner, the jump in the other normal derivative being spectrally small (see Chapter 13 of Canuto et al. [56]).
Multiple domains can thus be used to adapt the numerical grid to the interesting part (manifold) of the coordinate space; they can be seen as a technique close to the spectral element method [166]. Moreover, it is also a way to increase spatial resolution in some parts of the computational domain where one expects strong gradients to occur: adding a small domain with many degrees of freedom is the analog of fixedmesh refinement for finitedifferences.
3.2 Spherical coordinates and harmonics
When the numerical grid does not extend to infinity, e.g., when solving for a hyperbolic PDE, the boundary defined by r = const is a smooth surface, on which boundary conditions are much easier to impose. Finally, spherical harmonics, which are strongly linked with these coordinates, can simplify a lot the solution of Poissonlike or wavelike equations. On the other hand, there are some technical problems linked with this set of coordinates, as detailed hereafter, but spectral methods can handle them in a very efficient way.
3.2.1 Coordinate singularities

near θ = 0, a regular scalar field is equivalent to f(θ) ∼ sin^{∣m∣} θ,

near r = 0, a regular scalar field is equivalent to f(r) ∼ r^{ℓ}.
In addition, the rdependence translates into a Taylor series near the origin, with the same parity as ℓ. More details in the case of polar (2D) coordinates are given in Chapter 18 of Boyd [48].
If we go back to the evaluation of the Laplace operator (93), it is now clear that the result is always regular, at least for ℓ ≥ 2 and m ≥ 2. We detail the cases of ℓ = 0 and ℓ =1, using the fact that spherical harmonics are eigenfunctions of the angular part of the Laplace operator (see Equation (103)). For ℓ = 0 the scalar field f is reduced to a Taylor series of only even powers of r, therefore the first derivative contains only odd powers and can be safely divided by r. Once decomposed on spherical harmonics, the angular part of the Laplace operator (93) acting on the ℓ = 1 component reads −2/r^{2}, which is a problem only for the first term of the Taylor expansion. On the other hand, this term cancels with the \({2 \over r}{\partial \over {\partial r}}\), providing a regular result. This is the general behavior of many differential operators in spherical coordinates: when applied to a regular field, the full operator gives a regular result, but single terms of this operator may give singular results when computed separately, the singularities canceling between two different terms.
3.2.2 Spherical harmonics
The first property makes the description of scalar fields on spheres very easy: spherical harmonics are used as a decomposition basis within spectral methods, for instance in geophysics or meteorology, and by some groups in numerical relativity [21, 109, 219]. However, they could be more broadly used in numerical relativity, for example for Cauchycharacteristic evolution or matching [228, 15], where a single coordinate chart on the sphere might help in matching quantities. They can also help to describe starlike surfaces being defined by r = h(θ, φ) as event or apparent horizons [152, 23, 1]. The search for apparent horizons is also made easier: since the function h verifies a twodimensional Poissonlike equation, the linear part can be solved directly, just by dividing by −ℓ(ℓ + 1) in the coefficient space.
The use of sphericalharmonics decomposition can be regarded as a basic spectral method, like Fourier decomposition. There are, therefore, publicly available “spherical harmonics transforms”, which consist of a Fourier transform in the φdirection and a successive Fourier and Legendre transform in the θdirection. A rather efficient one is the SpharmonicsKit/S2Kit [151], but writing one’s own functions is also possible [99].
3.2.3 Tensor components
All the discussion in Sections 3.2.1–3.2.2 has been restricted to scalar fields. For vector, or more generally tensor fields in three spatial dimensions, a vector basis (triad) must be specified to express the components. At this point, it is very important to stress that the choice of the basis is independent of the choice of coordinates. Therefore, the most straightforward and simple choice, even if one is using spherical coordinates, is the Cartesian triad \(\left({{{\rm{e}}_x} = {\partial \over {\partial x}},{{\bf{e}}_y} = {\partial \over {\partial y}},{{\bf{e}}_z} = {\partial \over {\partial z}}} \right)\). With this basis, from a numerical point of view, all tensor components can be regarded as scalars and therefore, a regular tensor can be defined as a tensor field, whose components with respect to this Cartesian frame are expandable in powers of x, y and z (as in Bardeen and Piran [19]). Manipulations and solutions of PDEs for such tensor fields in spherical coordinates are generalizations of the techniques for scalar fields. In particular, when using the multidomain approach with domains having different shapes and coordinates, it is much easier to match Cartesian components of tensor fields. Examples of use of Cartesian components of tensor fields in numerical relativity include the vector Poisson equation [109] or, more generally, the solution of elliptic systems arising in numerical relativity [171]. In the case of the evolution of the unconstrained Einstein system, the use of Cartesian tensor components is the general option, as it is done by the Caltech/Cornell group [127, 188].

When doing excision for the simulation of black holes, the boundary conditions on the excised sphere for elliptic equations (initial data) may be better formulated in terms of spherical components for the shift or the threemetric [62, 104, 123]. In particular, the component that is normal to the excised surface is easily identified with the radial component.

Still, in the 3+1 approach, the extraction of gravitational radiation in the wave zone is made easier if the perturbation to the metric is expressed in spherical components, because the transverse part is then straightforward to obtain [218].
3.3 Going further
The development of spectral methods linked with the problems arising in the field of numerical relativity has always been active and continues to be. Among the various directions of research one can foresee, quite interesting ones might be the beginning of higherdimensional studies and the development of betteradapted mappings and domains, within the spirit of going from pure spectral methods to spectral elements [166, 29].
3.3.1 More than three spatial dimensions
4 TimeDependent Problems
From a relativistic point of view, the time coordinate could be treated in the same way as spatial coordinates and one should be able to achieve spectral accuracy for the time representation of a spacetime function f(t, x, y, z) and its derivatives. Unfortunately, this does not seem to be the case and we are neither aware of any efficient algorithm dealing with the time coordinate, nor of any published successful code solving any of the PDEs coming from Einstein’s equations, with the recent exception of the 1+1 dimensional study by Hennig and Ansorg [113]. Why is time playing such a special role? It is not easy to find in the literature on spectral methods a complete and comprehensive study. A first standard explanation is the difficulty, in general, of predicting the exact time interval on which one wants to study the time evolution. In addition, time discretization errors in both finite difference techniques and spectral methods are typically much smaller than spatial ones. Finally, one must keep in mind that, contrary to finite difference techniques, spectral methods store all global information about a function over the whole time interval. Therefore, one reason may be that there are strong memory and CPU limitations to fully threedimensional simulations; it is already very CPU and memory consuming to describe a complete field depending on 3+1 coordinates, even with fewer degrees of freedom, as is the case for spectral methods. But the strongest limitation is the fact that, in the full 3+1 dimensional case, the matrix representing a differential operator would be very big; it would therefore be very time consuming to invert it in a general case, even with iterative methods.
More details on the standard, finitedifference techniques for time discretization are given in Section 4.1. Due to the technical complexity of a general stability analysis, we restrict the discussion of this section to the eigenvalue stability (Section 4.1) with the following approach: the eigenvalues of spatial operator matrices must fall within the stability region of the timemarching scheme. Although this condition is only a necessary one and, in general, is not sufficient, it provides very useful guidelines for selecting timeintegration schemes. A discussion of the imposition of boundary conditions in timedependent problems is given in Section 4.2. Section 4.3 then details the stability analysis of spatial discretization schemes, with the examples of heat and advection equations, before the details of a fullydiscrete analysis are given for a simple case (Section 4.4).
4.1 Time discretization
4.1.1 Method of lines
At every instant t, one can represent the function u_{N}(x, t) by a finite set U_{N}(t), composed of its timedependent spectral coefficients or its values at the collocation points. We denote L_{N} the spectral approximation to the operator L, together with the boundary conditions, if the tau or collocation method is used. L_{N} is, therefore, represented as an N × N matrix. This is the method of lines, which allows one to reduce a PDE to an ODE, after discretization in all but one dimensions. The advantage is that many ODE integration schemes are known (RungeKutta, symplectic integrators, …) and can be used here. We shall suppose an equallyspaced grid in time, with the timestep noted Δt and \(U_N^J = {U_N}(J \times \Delta t)\).
4.1.2 Stability
The basic definition of stability for an ODE integration scheme is that, if the timestep is lower than some threshold, then \(\Vert U_N^{J + 1}\Vert \leq A{e^{KJ\Delta t}}\), with constants A and K independent of the timestep. This is perhaps not the most appropriate definition, since in practice one often deals with bounded functions and an exponential growth in time would not be acceptable. Therefore, an integration scheme is said to be absolutely stable (or asymptotically stable), if \(\Vert U_N^J\Vert\) remains bounded, ∀J ≥ 0. This property depends on a particular value of the product λ_{i} × Δt. For each time integration scheme, the region of absolute stability is the set of the complex plane containing all the λ_{i}Δt for which the scheme is absolutely stable.
4.1.3 Spectrum of simple spatial operators
4.1.4 Semiimplicit schemes
4.2 Imposition of boundary conditions
The timedependent PDE (116) can be written as a system of ODEs in time either for the timedependent spectral coefficients {c_{i}(t)}_{i=0…N} of the unknown function u(x, t) (Galerkin or tau methods), or for the timedependent values at collocation points {u(xi, t)}_{i=0…N} (collocation method). Implicit timemarching schemes (like the backward Euler scheme (119)) are technically very similar to a succession of boundaryvalue problems, as for elliptic equations or Equation (62) described in Section 2.5. The coefficients (or the values at collocation points) are determined at each new timestep by inversion of the matrix of type I + ΔtL or its higherorder generalization. To represent a wellposed problem, this matrix needs, in general, the incorporation of boundary conditions, for tau and collocation methods. Galerkin methods are not so useful if the boundary conditions are time dependent: this would require the construction of a new Galerkin basis at each new timestep, which is too complicated and/or time consuming. We shall therefore discuss in the following sections the imposition of boundary conditions for explicit time schemes, with the tau or collocation methods.
4.2.1 Strong enforcement
4.2.2 Penalty approach
4.3 Discretization in space: stability and convergence
After dealing with temporal discretization, we now turn to another fundamental question of numerical analysis of initial value problems, which is to find conditions under which the discrete (spatial) approximation u_{N}(x, t) converges to the right solution u(x, t) of the PDE (116) as N → ∞ and t ∈ [0, T]. The time derivative term is treated formally, as one would treat a source term on the righthand side, that we do not consider here, for better clarity.
 consistency: an approximation to the PDE (116) is consistent if ∀ν ∈ ℌ both$$\left. {\begin{array}{*{20}c} {{{\left\Vert {{P_N}(Lv  {L_N}v)} \right\Vert}_{L_w^2}} \rightarrow 0}\\ {{{\left\Vert {{P_N}v  {v_N}} \right\Vert}_{L_w^2}} \rightarrow 0\quad \;\;\;}\\ \end{array}} \right\}\;\;{\rm{as}}\;N \rightarrow \infty;$$(142)
 stability: with the formal notations of Equation (117), an approximation to the PDE (116) is stable ifwhere C(t) is independent of N and bounded for t ∈ [0, T].$$\forall N,\quad \left\Vert {{e^{{L_N}t}}} \right\Vert = \underset{v}{\sup} {{{{\left\Vert {{e^{{L_N}t}}v} \right\Vert}_{L_w^2}}} \over {{{\left\Vert v \right\Vert}_{L_w^2}}}} \leq C(t),$$(143)
4.3.1 LaxRichtmyer theorem
The direct proof of convergence of a given scheme is usually very difficult to obtain. Therefore, a natural approach is to use the LaxRichtmyer equivalence theorem: “a consistent approximation to a wellposed linear problem is stable if and only if it is convergent”. Thus, the study of convergence of discrete approximations can be reduced to the study of their stability, assuming they are consistent. Hereafter, we sketch out the proof of this equivalence theorem.
4.3.2 Energy estimates for stability
The most straightforward technique for establishing the stability of spectral schemes is the energy method: it is based on choosing the approximate solution itself as a test function in the evaluation of residual (60). However, this technique only provides a sufficient condition and, in particular, crude energy estimates indicating that a spectral scheme might be unstable can be very misleading for nonnormal evolution operators (see the example in Section 8 of Gottlieb and Orszag [94]).
 If the operator L is semibounded:where I is the identity operator.$$\exists \gamma, \quad L + {L^{\ast}} \leq \gamma I,$$(148)
 In the parabolic case, if L satisfies the coercivity condition (see also Chapter 6.5 of Canuto et al. [57]^{1}):and the continuity condition:$$\exists A > 0,\forall (u,v),\quad \vert(Lu,v)\vert\; \leq A\vert\vert u\vert\vert\vert\vert v\vert\vert,$$(149)$$\exists \alpha > 0,\forall u,\quad (Lu,u)\; \leq  \alpha \vert\vert u\vert\vert^{2}.$$(150)
 In the hyperbolic case, if there exists a constant C > 0 such thatand if the operator verifies the negativity condition:$$\forall u,\quad \vert\vert Lu\vert\vert \leq C\vert\vert u\vert\vert,$$(151)$$\forall u,\quad (Lu,u) \leq 0.$$(152)
4.3.3 Examples: heat equation and advection equation
4.3.3.1 Heat equation
4.3.3.2 Advection equation
4.4 Fullydiscrete analysis
4.4.1 Strong stabilitypreserving methods
The above fullydiscrete analysis must, in principle, be performed for every timemarching scheme. Therefore, it is very convenient to have a way of extending the results from the firstorder Euler method to higherorder methods. Strong stabilitypreserving RungeKutta and multistep methods preserve these kinds of stability properties, including those following from nonlinear stability analysis. A general review of the subject has been done by Shu [200], and we list some results here.
4.5 Going further: Highorder time schemes
When using spectral methods in timedependent problems, it is sometimes frustrating to have such accurate numerical techniques for the evaluation of spatial derivatives and the integration of elliptic PDEs, while the time derivatives and hyperbolic PDEs do not benefit from spectral convergence. Some tentative studies are being undertaken in order to represent the time interval by spectral methods as well [113]. In their sphericallysymmetric study of the wave equation in Minkowski spacetime, Hennig and Ansorg have applied spectral methods to both spatial and time coordinates. Moreover, they have used a conformal compactification of Minkowski spacetime, making the wave equation singular at null infinity. They have obtained nicely accurate and spectrally convergent solutions, even to a nonlinear wave equation. If these techniques can be applied in general threedimensional simulations, it would really be a great improvement.

Implicitexplicit techniques use an explicit multistep scheme to advance the nonlinear part \({{\mathcal N}_N}\), and an implicit one for the linear part.

Splitstep techniques are effective when the equation splits into two equations, which can be directly integrated (see [129] for examples with the nonlinear Schrödinger and Kortewegde Vries equations).
 Integrating factor is a change of variable that allows for the exact solution of the linear partand explicit multistep method for the integration of the new nonlinear part$${V_N} = {e^{ {L_N}t}}{U_N},$$(183)$${{\partial {V_N}} \over {\partial t}} = {e^{ {L_N}t}}{{\mathcal N}_N}{e^{{L_N}t}}{V_N}.$$(184)

Sliders can be seen as an extension of the implicitexplicit method described above. In addition to splitting the problem into a linear and nonlinear part, the linear part itself is split into two or three regions (in Fourier space), depending on the wavenumber. Then, different numerical schemes are used for different groups of wavenumbers: implicit schemes for high wavenumbers and explicit highorder methods for low wavenumbers. This method is restricted to Fourier spectral techniques in space.
 Exponential timedifferencing techniques have been known for some time in computational electrodynamics. These methods are similar to the integrating factor technique, but one considers the exact equation over one timestepVarious orders for these schemes come from the approximation order of the integral. For example Kassam and Trefethen [124] consider a fourthorder RungeKutta type approximation to this integral, where the difficulty comes from the accurate computation of functions, that suffer from cancellation errors.$$U_N^{J + 1} = {e^{{L_N}\Delta t}}U_N^J + {e^{{L_N}\Delta t}}\int\nolimits_0^{\Delta t} {{e^{ {L_N}\tau}}{{\mathcal N}_N}({U_N}(N\Delta t + \tau),N\Delta t + \tau)d\tau.}$$(185)
5 Stationary Computations and Initial Data
5.1 Introduction
In this section, we restrict ourselves to problems in which time does not appear explicitly. This is especially the case for systems, that are stationary, like neutron stars in rotation or binary systems in circular orbits. The computation of initial data also falls into this class, given that it consists in finding a particular solution of Einstein’s equations at a given time only. Indeed, when using the standard 3+1 decomposition of spacetime, the initial data that are passed to the evolution equations cannot be totally arbitrary and must satisfy a set of equations called Einstein’s constraint equations. For more details of the initial data problem we refer to the review by Cook [61]. So, in treating the problems considered here, one may forget about the issues specific to time presented in Section 4.
It must be said that spectral methods are not the only technique that has been successfully used to generate stationary spacetimes. [24, 222, 60, 147] give some examples of this, especially in the case of binary systems, for neutron stars or black holes. More references can be found in [61].
5.2 Single compact stars
The computation of the structure of stationary compact stars dates back to 1939 with the famous solution of TolmanOppenheimerVolkoff. During the last few years, the need for accurate models has become more pressing, especially with the coming online of gravitational wave detectors, which could help to probe the interior of such compact stars. Isolated stars in rotation are essentially axisymmetric, but some physical effects can induce a symmetry breaking that could lead to the emission of gravitational waves. In the following, we will review some computations that aim at including some of these effects, such as spontaneous symmetry breaking, the inclusion of magnetic fields, the effects of exotic dense matter (mainly with strange quarks) and the influence of an interior composed of two different superfluids.
5.2.1 Formalisms
The first computation of models of relativistic rotating stars in general relativity, by means of spectral methods, is presented in [41]. The equations are solved in spherical coordinates (see Section 3.2). Doing so, the fields only depend on the azimuthal angle θ and on the radius r. The fields are expanded in terms of spherical harmonics with respect to the angle and in Chebyshev polynomials with respect to r. The use of spherical harmonics is a natural way of dealing with the coordinate singularity on the zaxis. In [41] the whole space is divided into two spherical domains, the outer one extending up to infinity by making use of the compactification in 1/r seen in Section 3.1.2. With such a setting, Einstein’s equations reduce to a set of four elliptic equations with sources extending up to infinity that are solved using a version of the algorithm based on matching with the homogeneous solutions (presented in Section 2.6.4) for each spherical harmonic. The system is complete once a description of the matter is given. The simplest choice is to consider a polytropic fluid, with or without a magnetic field. The system is solved by iteration.
In [41], a particular emphasis is put on various methods to measure the accuracy of the code. For nonrotating stars, the error is found to decrease exponentially, as the number of coefficients increases (see Figures 5 and 6 of [41]). However, for fastrotating configurations, the error only decays as a power law (see Figure 7 of [41]). This comes from the fact that quantities like the energy density are no longer \({{\mathcal C}^\infty}\) across the star’s surface. Nevertheless, the results are in good agreement (to 0.1%) with those obtained by other numerical methods, as can be seen in [160].
Spectral convergence can be recovered by using surfaceadapted coordinates as first done in [36]. A regular mapping of the numerical coordinates to the physical ones is introduced, so that the surface of the star lies at the boundary between two domains (see Section 3.1.1). For polytropes with γ < 2, this is sufficient to recover spectral convergence (see Figures 5 and 6 of [38]). However, for γ > 2, some quantities are still diverging at the surface but the convergence can be made closer and closer to the spectral one by analytically regularizing the density (see Section IV of [38]). Doing so, the error decreases as a power law, but the decrease can be made arbitrary fast at the cost of increasing the number of operations and so the computational time.
Up until 2006, neutron stars were computed using quasiisotropic coordinates. However, in order to use these configurations as initial data for evolutionary codes, it may be useful to allow for other choices. Among the possible gauges, Dirac’s is one of the most promising [37]. In [134] models of rotating neutron stars in the Dirac gauge are computed for both polytropic and realistic equations of state. Contrary to the quasiisotropic coordinates, the use of this new gauge implies that one must solve one tensorlike Poisson equation. Configurations obtained with the two different formalisms are shown to be in very good agreement.
5.2.2 Rotating neutron star models
Even before adapted mappings were available, interesting results could be obtained. In [183, 184], models of rotating neutron stars with various equations of state have been computed. Among the most surprising findings is the existence of supramassive stars. These stars do not connect to the nonrotating limit. Indeed, their high mass can only be supported by the presence of a centrifugal force. One of the remarkable features of such stars is the fact that they actually spin up when they lose angular momentum, contrary to what is observed for normal stars. This effect can also be seen for neutron stars containing hyperons and, thus, a softer equation of state [238]. Let us mention that, in this case, the stability analysis of the configurations required the great precision that spectral methods with adapted coordinates could provide.
It is known that isolated pulsars spin down due to magnetic braking. As the rotational frequency decreases, it is possible that the star will encounter a transition from one state of matter to another. Stationary rotating models have been used to determine the properties of such transitions [231, 232, 233]. A puzzling result is that the amount of energy released in a firstorder phase transition does not depend on the orbital velocity of the star and is the same as for nonrotating ones. This is shown to be the case for both weak [232] and strong [233] firstorder transitions.
5.2.3 Spontaneous symmetry breaking
It is known that stars can undergo spontaneous symmetry breaking when rotating fast enough. When such a phenomenon occurs, triaxial configurations are formed that are potential emitters of gravitational waves. The departure from axisymmetry is studied in two papers by the Meudon group [34, 35]. The idea of the method is to start from an axisymmetric neutron star configuration and to follow the growth or decay of triaxial instabilities. This work reaffirms wellestablished results from the Newtonian regime and presents the first results in general relativity for various equations of states. For a few of them, the frequency at which symmetry breaking occurs lies in the frequency band of the LIGO and Virgo detectors.
In 2002, this work was extended [89] by making use of surfacefitting coordinates. This enabled the authors to obtain results in the incompressible case by properly dealing with discontinuities lying at the surface of the star. Classical results in the incompressible case are, thus, recovered and it is found that the inclusion of relativity has only a moderate effect. Indeed, the critical ratio between the kinetic energy and the absolute gravitational energy T/W at which the triaxial instability occurs is only 30% larger for relativistic stars, with respect to their classical counterparts.
If relativistic effects only slightly stabilize stars, the same is not true for differential rotation. Indeed, in [182], the authors study various rotation profiles and equations of state using the same technique as in [34, 35] to determine the onset of instability. It appears that the critical value of T/W can be almost twice as high as for uniformly rotating stars.
5.2.4 Configurations with magnetic fields
Even if magnetic fields are strong in neutron stars, the structure of the object is not affected until it reaches huge values, on the order of at least 10^{10} T. In [32], models of rapidlyrotating stars with poloidal fields are constructed for various equations of state. The magnetic effects are taken into account consistently by solving the appropriate Maxwell equations, as well as by means of spectral methods. The maximum mass of highlymagnetized neutrons stars is found to be higher by 13–29% than for nonmagnetized stars. The magnetic field induces an additional pressure, which can help to support more massive stars, thus explaining this increase.
The presence of a magnetic field can also lead to the deformation of a neutron star. Such deformation could lead to the formation of a triaxial configuration, which would then emit gravitational waves. In [36] the emitted signal is computed. Typically the system radiates at two frequencies: Ω and 2Ω where Ω is the angular velocity of the star.
In more recent work by the Meudon group [159], magnetized configurations have been computed using coordinates matched to the surface of the star, thus making the computation much more accurate. Gyromagnetic ratios of rapidlyrotating neutron stars of various equations of state are obtained. The limit of a ratio of g = 2 value for a charged black hole is never reached.
5.2.5 Strange stars
It is possible that the fundamental state of nuclear matter is not the ordinary matter but rather a plasma of deconfined quarks u, d and s, called strange matter. If this is the case, neutron stars should rather be called strange stars. The main difference between these two types of compact stars is that strange ones are somewhat smaller and thus more compact. In particular, they support higher rotation rates. There is a strong density jump at the surface of a strange star and surfacefitting coordinates are required in order to deal with it.
Fast rotating models of strange stars are computed in [103, 88]. Due to higher compactness, it is found that strange stars can rotate significantly faster than their neutron star counterparts. The attained T/W can be twice as large. As in the neutron star case, supermassive configurations that spin up with angular momentum loss are found. The influence of strange matter on the emission of gravitational waves is studied in [90] where viscosity effects and triaxial instabilities are carefully taken into account.
It is believed that millisecond pulsars have been spun up by accreting matter from a companion. However, the details of this mechanism depend on the nature of the compact object. In [237], the differences between accretion onto a neutron star and onto a strange star are investigated, using 2D stationary models computed by spectral methods.
5.2.6 Quasiperiodic oscillations
Quasiperiodic oscillations (QPOs) are observed in the kHz regime and are believed to be the signature of matter falling into a compact object. In the standard picture, the frequency of the QPOs is that of the last stable orbit around the compact object. Let us mention that the presence of a last stable orbit around an extended body is not an effect of relativity but can also be seen in the Newtonian regime, as shown in [234].
The precise structure of the accreting object has a great influence on the QPO. In a series of papers [235, 92, 3, 27], comparisons are made between observations and various compact star models that could account for QPOs.
Using a multidomain approach, strange stars with a crust can also be computed [236], one domain describing the interior of the star and another the crust. It is shown that the presence of the crust could change the value of the QPO by up to 20%.
5.2.7 More complex configurations
In this section, objects in more exotic configurations are presented. This is an illustration of both the complexity of neutron stars physics and the ability of spectral methods to deal with complicated systems.
The observation of glitches in isolated pulsars is consistent with the presence of a superfluid interior. The simplest model considers two fluids, one composed of neutrons and the other of protons and electrons, both components being superfluids. However, these two components could have different velocities, in particular different rotation rates. Such configurations are computed in [177]. A multidomain setting is crucial to be able to follow the two different fluids because the components do not have the same shape. Among the various results obtained, let us mention the confirmation of the existence of prolateoblate configurations.
Neutron stars are usually assumed to be at zero temperature. However, this approximation is known to no longer be true for newborn neutron stars just after the supernova. Models of newborn neutron stars in uniform rotations are constructed in [106] using an extension of the code developed in [41]. Various hypothesis about the interior (different lepton numbers, isothermal versus isentropic) are considered. Sequences of fixed baryon mass and angular momentum are constructed. Such sequences approximate the evolution of the protoneutron star into a cold neutron star. The results have been extended to differentiallyrotating protoneutron stars in [107].
The effect of finite temperature is also investigated in [226]. The authors found that newborn neutron stars are unlikely to undergo bar mode instability, but that the secular ones could take place and result in a significant emission of gravitational waves. Another interesting result of [226] is the existence of toroidallike configurations, which appear for a broad range of parameters and before the massshedding limit. In such cases, the surface of the star is highly deformed and surfacefitting coordinates are required.
Axisymmetric rotating neutron stars have also been computed by a code developed by Ansorg et al. [9, 10]. This code is based on LewisPapapetrou coordinates (ρ,ξ), which are closely related to the usual cylindrical coordinates. Typically, space is decomposed into two domains: one for the interior of the star and another for the exterior, which extends up to spatial infinity. Compactification of space and surfacefitting mappings are used. Both variables are expanded in terms of Chebyshev polynomials. Instead of solving the equations harmonic by harmonic and then iterating, as is done by the Meudon group, the equations are written with a collocation method (see Section 2.5.3) and solved as one single system. The price to pay is that the size of the system is somewhat larger (i.e. in m^{2}, m being the total number of coefficients for each coordinate). The system is solved by means of the NewtonRaphson method. At each step, the linear system is solved using iterative techniques with preconditioning. With this method, impressive accuracy is reached.
The coordinates used in [9, 10] are more general than the ones used by the Meudon group, especially with respect to their surfacefitting capabilities. They can account for more complicated configurations and, in particular, highlydistorted matter distribution can be obtained. This is shown in [12, 11], where relativistic axisymmetric toroidal configurations of matter, known as the Dyson rings, are computed. Such rings are obtained up to the massshedding limit. Transition to the limit of an extreme Kerr black hole is also discussed.
5.3 Single black holes
Compared to the compact star case, single black holes have not been studied very much. This is probably because the structure of a stationary black hole is somewhat simpler than the one of a compact star. However, as will be seen, there are still properties that must be investigated.
Spacetimes containing a single black hole constitute a good benchmark for numerical methods, a lot of results being known analytically. In [125], the authors have implemented a spectral solver and applied it to various test problems. The solver itself is two dimensional and thus applicable only to axisymmetric systems. There is a single domain that consists of the whole space outside a sphere of given radius (i.e. the black hole). Space is compactified by using the standard variable 1/r. The two physical variables (r, θ) are mapped onto squares in ℝ^{2} and then expanded in terms of Chebyshev polynomials. The equations are written using a 2D collocation method (see Section 2.5.3) and the resulting system is solved by an iterative algorithm (here Richardson’s method with preconditioning). This solver is applied to solve Einstein’s constraint equations for three different systems: i) a single black hole ii) a single black hole with angular momentum iii) a black hole plus Brill waves. In all three cases, spectral convergence is achieved and accuracy on the order of 10^{−10} is reached with 30 points in each dimension.
A black hole is somewhat simpler than a neutron star, in the sense that there is no need for a description of matter (no equation of state, for instance). However, in some formalisms, the presence of a black hole is enforced by imposing a nontrivial solution on a surface (typically a sphere). The basic idea is to demand that the surface be a trapped surface. Such surfaces are known to lie inside event horizons and so are consistent with the presence of a black hole. Discussions about such boundary conditions can be found in [62]. However, in nonstationary cases, the set of equations to be used is not easy to derive. The authors of [123] implement various sets of boundary conditions to investigate their properties. Two different and independent spectral codes are used. Both codes are very close to those used in the case of neutron stars, one of them based on Lorene library [99] (see Section 5.2.1) and the other one developed by Ansorg and sharing a lot a features with [9, 10]. Such numerical tools have proved useful in clarifying the properties of some sets of boundary conditions that could be imposed on black hole horizons.
The reverse problem is also important in the context of numerical relativity. In some cases one needs to know if a given configuration contains a trapped surface and if it can be located at each timestep. Several algorithms have been proposed in the past to find the locus at which the expansion of the outgoing light rays vanishes (thus defining the trapped surface). Even if the term is not used explicitly, the first application of spectral expansions to this problem is detailed in [23]. The various fields are expanded in a basis of symmetric tracefree tensors. The algorithm is applied to spacetimes containing one or two black holes. However, results seem to indicate that high order expansions are required to locate horizons with a sufficient precision.
More recently, another code [135] using spectral methods has been used to locate apparent horizons. It is based on the Lorene library with its standard setting, i.e. a multidomain decomposition of space and spherical coordinates (see Section 5.2.1 for more details). The horizon finder has been successfully tested on known configurations, like KerrSchild black holes. The use of spectral methods makes it both fast and accurate. Even if the code uses only one set of spherical coordinates (hence its presentation in this section), it can be applied to situations with more than one black hole, like the wellknown BrillLindquist data [50].
5.4 Rings around black holes
The problem of uniformly rotating rings surrounding a black hole can be viewed as an intermediate step between onebody axisymmetric configurations and the two body problem. Indeed, even if one has to deal with two components, the problem is still axisymmetric. In [13], configurations of a black hole surrounded by a uniformly rotating ring of matter are computed in general relativity. The matter is assumed to be a perfect fluid. To solve the equations, space is divided into five computational domains. One of them describes the ring itself, another one the region around the black hole and another extends up to infinity. The two other domains are used to connect these regions. One of the difficulties is that the surface of the ring is not know a priori and so the domains must be dynamically adapted to its surface. Cylindricaltype coordinates are used and, in each domain, are mapped onto squares of numerical coordinates. The actual mappings depend on the domain and can be found in Section IV of [13].
Numerical coordinates are expanded in terms of Chebyshev polynomials. The system to be solved is obtained by writing Einstein’s equations in collocation space including regularity conditions on the axis and appropriate boundary conditions on both the horizon of the black hole and at spatial infinity. As in [9, 10], the system is solved iteratively, using the NewtonRaphson method.
Both the Newtonian and relativistic configurations are computed. The ratio between the mass of the black hole and the mass of the ring is varied from zero (no black hole) to 144. The inner mass shedding of the ring can be obtained. One of the most interesting results is the existence of configurations for which the ratio \({J_c}/M_c^2\) of the black hole angular momentum and the square of its mass exceeds one, contrary to what can be achieved for an isolated black hole.
5.5 Compact star binaries
5.5.1 Formalism
Systems consisting of two compact objects are known to emit gravitational waves. Due to this emission, no closed orbits can exist and the objects follow a spirallike trajectory. This implies that such systems have no symmetries that can be taken into account and full time evolutions should be made. However, when the objects are relatively far apart, the emission of gravitational waves is small and the inspiral slow. In this regime, one can hope to approximate the real trajectory with a sequence of closed orbits. Moreover, the emission of gravitational waves is known to efficiently circularize eccentric orbits so that only circular orbits are usually considered.
So, a lot of effort has been devoted to the computation of circular orbits in general relativity. This can be done by demanding that the system admit a helical Killing vector ∂_{t} + Ω∂_{φ}, Ω being the orbital angular velocity of the system. Roughly speaking, this means that advancing in time is equivalent to turning the system around its axis. Working in the corotating frame, one is left with a timeindependent problem.
Additional approximations must be made in order to avoid any diverging quantities. Indeed, when using helical symmetry, the system has an infinite lifetime and can fill the whole space with gravitational waves, thus causing quantities like the total mass to be infinite. To deal with that, various techniques can be used, the simplest one consisting of preventing the appearance of any gravitational waves. This is usually done by demanding that the spatial metric be conformally flat. This is not a choice of coordinates but a true approximation that has a priori no reason to be verified. Indeed, even for a single rotating black hole, one can not find coordinates in which the spatial threemetric is conformally flat, so that we do not expect it to be the case for binary systems. However, comparisons with postNewtonian results or nonconformallyflat results tend to indicate that this approximation is relatively good.
Under these assumptions, Einstein’s equations reduce to a set of five elliptic equations for the lapse, the conformal factor and the shift vector. These equations encompass both the Hamiltonian and momentum constraint equations and the trace of the evolution equations. To close the system, one must provide a description of the matter. It is commonly admitted that the fluid is irrotational, the viscosity in neutron stars being too small to synchronize the motion of the fluid with the orbital motion. It follows that the motion of the fluid is described by an additional elliptic equation for the potential of the flow. The matter terms entering the equations via the stressenergy tensor can then be computed, once an equation of state is given. An evolutionary sequence can be obtained by varying the separation between the stars.
5.5.2 Numerical procedure
Up to now, only the Meudon group has solved these equations by means of spectral methods in the case of two neutron stars. Two sets of domains are used, one centered on each star. Each set consists of spherelike domains that match the surface of the star and extend up to infinity. Functions are expanded in terms of spherical harmonics with respect to the angles (θ, φ) and Chebyshev polynomials with respect to the radial coordinates. Each Poisson equation ΔN = S_{N} is split into two parts \(\Delta {N_1} = {S_{{N_1}}}\) and \(\Delta {N_2} = {S_{{N_2}}}\), such that \({S_N} = {S_{{N_1}}} + {S_{{N_2}}}\) and N = N_{1} + N_{2}. The splitting is, of course, not unique and only requires that \({S_{{N_i}}}\) be mainly centered around star i so that it is well described by spherical coordinates around it. The equation labeled i is then solved using domains centered on the appropriate star. The splittings used for the various equations can be found explicitly in Section IVC of [100].
The elliptic equations are solved using the standard approach by the Meudon group found in [109]. For each spherical harmonic, the equation is solved using the tau method and the matching between the various domains is made using the homogeneous solutions method (see Section 2.6.4). The whole system of equations is solved by iteration and most of the computational time is spent when quantities are passed from one set of domains to the other by means of a spectral summation (this requires N^{6} operations, N being the number of collocation points in one dimension). A complete and precise description of the overall procedure can be found in [100].
5.5.3 Neutron star binaries
The first sequence of irrotational neutron star binaries computed by spectral means is shown in [40]. Both stars are assumed to be polytropes with an index γ = 2. The results are in good agreement with those obtained simultaneously by other groups with other numerical techniques (see, for instance, [24, 222]). One of the important points that has been clarified by [40] concerns the evolution of the central density of the stars. Indeed, at the end of the 1990s, there was a claim that both stars could individually collapse to black holes before coalescence, due to the increase of central density as the two objects spiral towards each other. Should that have been true, this would have had a great impact on the emitted gravitational wave signal. However, it turned out that this came from a mistake in the computation of one of the matter terms. The correct behavior, confirmed by various groups and in particular by [40], is a decrease in central density as the stars get closer and closer (see Figure I of [40]).
If the first sequence computed by spectral methods is shown in [40], the complete description and validation of the method are given in [100]. Convergence of the results with respect to the number of collocation points is exhibited. Global quantities like the total energy or angular momentum are plotted as a function of the separation and show remarkable agreement with results coming from analytical methods (see Figures 8–15 of [100]). Relativistic configurations are also shown to converge to their Newtonian counterparts when the compactness of the stars is small (see Figures 16–20 of [100]).
Newtonian configurations of compact stars with various equations of state are computed for both equal masses [215] and various mass ratios [212]. One of the main results of the computations concerns the nature of the end point of the sequence. For equal masses, the sequence ends at contact for synchronized binaries and at mass shedding for irrotational configurations. This is to be contrasted with the nonequalmass case, in which the sequence always ends at the mass shedding limit of the smallest object.
Properties of the sequences in the relativistic regime are discussed in [213, 214]. In [213] sequences with γ = 2 are computed for various compactness and mass ratios for both synchronized and irrotational binaries. The nature of the end point of the sequences is discussed and similar behavior to the Newtonian regime is observed. The existence of a configuration of maximum binding energy is also discussed. Such existence could have observational implications because it could be an indication of the onset of a dynamic instability. Sequences of polytropes with various indexes ranging from 1.8 to 2.2 are discussed in [214]. In particular, the authors are lead to conjecture that, if a configuration of maximum binding energy is observed in the Newtonian regime, it will also be observed in conformal relativity for the same set of parameters.
In [76] the authors derive from the sequences computed in [213] a method to constrain the compactness of the stars from the observations. Indeed, from the results in [213] one can easily determine the energy emitted in gravitational waves per interval of frequency (i.e. the power spectrum of the signal). For large separation, that is, for small frequencies, the curves follow the Newtonian. However, there is a break frequency at the higher end (see Figure 2 of [76]). The location of this frequency depends mainly on the compactness of the stars. More precisely, the more compact the stars are, the higher the break frequency is. Should such frequency be observed by the gravitational wave detectors, this could help to put constraints on the compactness of the neutron stars and, thus, on the equation of state of such objects.
5.5.4 Extensions
The framework of [100] is applied to more realistic neutron stars in [26]. In this work, the equations of state are more realistic than simple polytropes. Indeed, three different equations are considered for the interior, all based on different microscopic models. The crust is also modeled. For all the models, the end point of the evolution seems to be given by the mass shedding limit. However, the frequency at which the shedding occurs depends rather strongly on the EOS. The results are in good agreement with postNewtonian ones, until hydrodynamic effects begin to be dominant. This occurs at frequencies in the range of 500–1000 Hz, depending on the EOS.
Sequences of strange star binaries have also been computed [133]. Contrary to the neutron star case, matter density does not vanish at the surface of the stars and one really needs to use surfacefitting domains to avoid any Gibbs phenomenon that would spoil the convergence of the overall procedure. Sequences are computed for both synchronized and irrotational binaries and a configuration of maximum binding energy is attained in both cases. This is not surprising: strange stars are more compact than neutron stars and are less likely to be tidally destroyed before reaching the extremum of energy, making it easier to attain dynamic instability. More detailed results on both neutron star and strange star binaries are discussed in [87, 91].
All the work presented above was done in the conformal flatness approximation. As already stated in Section 5.5.1, this is only an approximation and one expects that the true conformal threemetric will depart from flatness. However, in order to maintain asymptotic flatness of spacetime, one needs to get rid of the gravitational wave content. One such waveless approximation is presented in [195] and implemented in [223]. Two independent codes are used, one of them being an extension of the work described in [100]. The number of equations to be solved is then greater than in conformal flatness (one has to solve for the conformal metric), but the algorithms are essentially the same. It turns out that the deviation from conformal flatness is rather small. The new configurations are slightly further from postNewtonian results than the conformallyflat ones, which is rather counterintuitive and might be linked to a difference in the definition of the waveless approximations.
5.6 Blackholebinary systems
5.6.1 Digging the holes
Though the computation of black hole binaries in circular orbit has a lot of common features with the neutron star case, there are also some differences that need to be addressed. In at least one aspect, black holes are much simpler objects because they are a solution of Einstein’s equations without matter. So the whole issue of investigating various equations of state is irrelevant and there is no need to solve any equation for the matter. However, there is a price to pay and one must find a way to impose the presence of black holes in the spacetime. Two ideas have been proposed.
In the puncture method, the full spacetime contains three asymptoticallyflat regions. One is located at r = ∞ and the other two at two points, M1 an M2, which are called the punctures. The presence of flat regions near the punctures is enforced by demanding that some quantities, like the conformal factor, diverge at those points (typically as 1/r). The discontinuities are taken out analytically and the equations are solved numerically for the regular parts in the whole space. This idea dates back to the work of Brill and Lindquist [50], at least in the case of black holes initially at rest. The puncture approach has been successfully applied to the computation of quasicircular orbits by means of spectral methods in [8].
The apparent horizon method relies on initial works by Misner [150] and Lindquist [140]. In this case, the space has only two asymptoticallyflat regions. One can show that this is equivalent to solving Einstein’s equations outside two spheres on which boundary conditions must be imposed. The boundary conditions are based on the concept of trapped surfaces and apparent horizons. The physical state of the black holes are precisely encoded in the boundary conditions.
5.6.2 First configurations
The first configurations of black hole binaries computed by means of spectral methods can be found in [110]. The formalism and various hypotheses are given in the companion paper [98]. The assumptions are very similar to those used for neutron star binaries (see Section 5.5.1). Helical symmetry is enforced and conformal flatness assumed. The holes are described by the apparent horizon technique. However, the boundary conditions used have been shown to be only approximately valid, up to a rather good accuracy. This effect is discussed in the original paper [110] and further explored by Cook in [62]. The numerical techniques are very similar to the ones employed for neutronstarbinary configurations (see Section 5.5.2). Two sets of spherical domains are used, one for each black hole. Boundary conditions are imposed on the surface between the nucleus and the first shell. Both sets extend up to infinity using a compactification in 1/r.
For the first time, good agreement was found between numerical results and postNewtonian ones. A detailed comparison can be found in [67]. In particular, the location of the energy minimum is shown to coincide to within a few percent. This improvement with respect to previous numerical work is mainly due to a difference in the physical hypothesis (i.e. the use of helical symmetry). One important product of [110] is the use of a new criterion to determine the appropriate value of the orbital angular velocity Ω. Indeed, for neutron stars, this is done by demanding that the fluid of both stars be in equilibrium [100]. This, of course, is not applicable for black holes. Instead, in [98, 110] it is proposed that one find Ω by demanding that the ADM mass and the Komarlike mass coincide. Without going into too much detail, this amounts to demanding that, far from the binary and at first order in 1/r, the metric behave like the Schwarzschild. It is shown in [98] that it can be linked to a relativistic virial theorem. Since then it has been shown that this criterion can also be used for neutron stars [214] and that it is equivalent to the use of a variational principle called the effective potential method [60, 22, 172, 59], where the binding energy is minimized with respect to Ω.
5.6.3 Further investigation
More recently, two other spectral codes have been developed in the context of black hole binaries and successfully applied to address some of the issues raised by the work of [98, 110].
One of these codes comes from the Caltech/Cornell group of Pfeiffer et al. and is described extensively in [171, 167]. The code is multidomain and two main types of domains are used i) square domains in which each Cartesianlike coordinate is expanded in terms of Chebyshev polynomials and ii) spherical domains in which spherical harmonics are used for the angles (θ, φ) and Chebyshev polynomials for the radial coordinate. Space can be compactified by standard use of the variable 1/r. The two types of domains can be set up in various manners to accommodate the desired geometry. When using both square and spherical domains, there are regions of space that belong to more than one domain. This is to be contrasted with work by the Meudon group in which different domains are only touching but not overlapping. The algorithm of [171] solves differential equations by using a multidimensional collocation method. The size of the resulting system is roughly equal to the number of collocation points. It is then solved iteratively via a NewtonRaphson algorithm with a line search. At each step of the NewtonRaphson method, the linear system is solved by means of an iterative scheme (typically GMRES). This inner iterative solver requires careful preconditioning to work properly. Various tests are passed by the code in [171], in which elliptic equations and systems are solved in either spherical or bispherical topologies. In the cases presented, the error decays spectrally.
In [169] the code is used to investigate different ways of solving the constraint equations. Three different decompositions are used: the conformal TT one, the physical TT one and the thinsandwich decomposition. When solving for the constraint equations only, one must also choose some freely specifiable variables, which describe the physical state of the system. In [169], these specifiable variables are fixed using a superposition of two KerrSchild black holes. The net result of [169] is that global quantities, like the total energy, are very sensitive to the choice of decomposition. The variation of total energy can be as large as 5%, which is the order of the energy released by gravitational waves. It is also shown that the choice of extrinsic curvature tensor is more crucial than the one of conformal metric, in accordance with an underlying result of [110]. Let us point out that the equations derived form the helical Killing vector approach in [98, 110] are equivalent to the ones obtained by making use of the thinsandwich decomposition of the constraints. The freely specifiable variables are obtained by both the imposition of the helical Killing symmetry and by solving an additional equation for the lapse function (resulting in the extended thinsandwich formalism).
In [63] the boundary conditions based on the apparent horizon formalism [62] are implemented and tested numerically in the case of one and two black holes. In the latter case, the main difference from [110] lies in the use of more elaborate and better boundary conditions on the horizons of the black holes. By allowing for a nonvanishing lapse on the horizons, the authors of [63] solve the constraint equations exactly. This is to be contrasted with [110], where the momentum constraint equation is only solved up to a small correction. Both results show rather good agreement. This is not surprising as the correction used by the Meudon group was known to be small (see Figures 10 and 11 of [110]). More results are presented in [59], for both corotating and irrotational black holes. An important result of [59] is the comparison of the two criteria for determining the orbital angular velocity Ω. They indeed show that the effective potential method first introduced in [60] and the method based on the virial theorem proposed in [98] are in very good agreement.
By slightly extending the boundary conditions used in [59], the authors of [168] propose to reduce the eccentricity of the blackholebinary configurations. This is done by giving the black holes a small radial velocity by modifying the boundary condition on the shift vector. The code and other equations are the same as in [59]. Time evolution of the obtained initial data does show that this technique can reduce the eccentricity of the binary. However, the effect on the emitted gravitational wave is small and probably unimportant.
Another application of the Caltech/Cornell solver can be found in [143], where the emphasis is put on nearly maximum spinning black holes. Initial data are constructed for both single black holes and binaries. Three families of initial data are investigated. Using a formalism based on the KerrSchild spacetime, the authors are able to reach spins as large as a = 0.9998. Such nearlymaximum spinning black holes may be relevant from the astrophysical point of view. Evolutions of these data are also discussed there.
The other spectral code used to compute configurations of black hole binaries comes from Ansorg [6]. It shares a lot of features with the code developed by the same author in the context of rotating stars [9, 10] already discussed in Section 5.2.7. Space is decomposed into two domains. One of them lies just outside the horizons of the black holes and bisphericallike coordinates are used. The other domain extends up to infinity and an appropriate mapping is used in order to avoid the singularity of the bispherical coordinates at spatial infinity (see Section IV of [6]). The angle of the bispherical coordinates (i.e. the angle around the xaxis joining the two black holes) is expanded in terms of a Fourier series and the two other coordinates in terms of Chebyshev polynomials. As in [13, 171], the partial differential equations are solved using a collocation method and the resulting system is solved by the NewtonRaphson method. Once again the linear system coming from the Jacobian is solved by an iterative scheme with preconditioning. The code is used to compute essentially the same configuration as those shown in [59]. An interesting point of [6] is the detailed investigation of the convergence of the results with increased resolution. As can be seen in Figure 4 of [6], the error initially decreases exponentially, but, for high number of points, it seems that the error follows only a power law. This is an indication that some non\({{\mathcal C}^\infty}\) fields must be present. It is conjectured in [6] that this comes from logarithm terms that cannot be dealt with properly with a compactification in 1/r. The same kind of effect is investigated in some detail in [109], where some criteria for the appearance of such terms are discussed.
A code very similar to the one used in [6] has also been used to compute spacetimes with black holes using the puncture approach [8]. Given that the black holes are no longer described by their horizons, one does not need to impose inner boundary conditions. The absence of this requirement enables the author of [8] to use a single domain to describe the whole space, from the puncture up to infinity. The other features of the spectral solver are the same as in [6]. This method has been successfully applied to the computation of blackholebinary configurations in the puncture framework. The authors have, in particular, investigated high mass ratios between the bodies and compared their results with the ones given in the testmass limit around a Schwarzschild black hole. The discrepancy is found to be on the order of 50% for the total energy. It is believed that this comes from the fact that the mass of each puncture cannot be directly related to the local black hole mass (see discussion in Section VII of [8]).
Finally, let us mention that the algorithms developed by Ansorg in [9, 10, 8, 6] have all been unified in [7] to accommodate any type of binaries. Various domain decompositions are exhibited that can be used to represent neutron stars, excised black holes or puncture black holes, with the compactification of space. The algorithms are shown to be applicable to limiting cases such as large mass ratios.
5.7 Blackholeneutronstar binaries
Until recently binaries consisting of a neutron star and a black hole received fewer attention than other types of systems. It was believed, and this was partially true, that this case could easily be handled once the cases of neutron star and black hole binaries were understood. However, such binaries are of evident observational interest and could be the most promising source of gravitational waves for groundbased detectors [28].
The first application of spectral methods to blackholeneutronstar binaries can be found in [208]. The main approximation is to consider that the black hole is not influenced by the neutron star. Technically, this means that Einstein’s equations are split into two parts (i.e. as for neutron star binaries 5.5.2). However, the part of the fields associated with the black hole is fixed to its analytical value. As the fields are not solved for the blackhole part, the results should depend on the actual splitting, the equations being nonlinear. The part of the fields associated with the neutron star are solved using the standard setting for the Meudon group. Of course, this whole procedure is only valid if the black hole is much more massive than the neutron star and this is why [208] is limited to mass ratios of 10. In this particular case, it is shown that the results depend, to the level of a few percent, on the choice of splitting, which is a measure of the reached accuracy. It is also shown that the state of rotation of the star (i.e. synchronized or irrotational) has little influence on the results.
In [209] the equations of the extended thinsandwich formulation are solved consistently. As for the neutronstarbinary case, two sets of spherical coordinates are used, one centered around each object. The freely specifiable variables are derived from the KerrSchild approach. Configurations are obtained with a moderate mass ratio of five. However, the agreement with postNewtonian results is not very good and the data seem to be rather noisy (especially the deformation of the star).
Quasiequilibrium configurations based on a helical Killing vector and conformal flatness have been obtained independently by [108] and [210]. Both codes are based on the Lorene library [99] and use two sets of spherical coordinates. They differ mainly in their choice of boundary conditions for the black hole. However, it is shown in the erratum of [108] that the results match pretty well and are in very good agreement with postNewtonian results. Mass ratios ranging from 1 to 10 are obtained in [210] and the emitted energy spectra are estimated. The work of [210] has been extended in [211], where the parameter space of the binary is extensively explored. In particular, the authors determine whether the end point of the sequences is due to an instability or to the massshedding limit. It turns out that the star is more likely to reach the massshedding limit if it is less compact and if the mass ratio between the black hole and the star is important, as expected.
More recently, the Caltech/Cornell group has applied the spectral solver of [171, 167] in order to compute blackholeneutronstar configurations [80]. Some extensions have been made to enable the code to deal with matter by making use of surfacefitting coordinates. Thanks to the domain decomposition used (analogous to the one of [171, 167]), the authors of [80] can reach an estimated accuracy 5 × 10^{−5}, which is better than the precision of previous works (by roughly an order of magnitude). Configurations with one spinning black hole and configurations with reduced eccentricity are also presented, along the lines of [168].
5.8 Spacetimes with waves
[170] presents a method to produce initial data configurations containing waves. Given a fixed background metric, it shows how to superimpose a given gravitational wave content. The equations are solved numerically using a multidomain spectral code based on [171, 167]. Space is covered by various spherelike shells and is described up to infinity. When no black hole is present, the origin is covered by a square domain because regularity conditions at the origin, in spherical coordinates, are not handled by [171, 167]. Such a setting is used to generate spacetimes containing i) pure quadrupolar waves ii) flat space with an ingoing pulse and iii) a single black hole superimposed with an ingoing quadrupolar wave.
5.9 Hyperboloidal initial data
If the 3+1 decomposition is the most widely used for numerical relativity, some other possibilities have been proposed, with possibly better features. In particular, one can vary the foliation of spacetime to get hyperboloidal data. With such a setting, at infinity spacetime is foliated by light cones instead of spatial hypersurfaces, which makes the extraction of gravitational waves, in principle, easier.
In [81], Frauendiener is interested in generating hyperboloidal initialdata sets from data in physical space. The technique proceeds in two steps. First a nonlinear partial differential equation (the Yamabe equation) must be solved to determine the appropriate conformal factor ω. Then, the data are constructed by dividing some quantities by this ω. This second step involves an additional difficulty: ω vanishes at infinity but the ratios are finite and smooth. It is demonstrated in [81] that spectral methods can deal with these two steps. Some symmetry is assumed so that the problem reduces to a twodimensional one. The first variable is periodic and expanded in terms of a Fourier series, whereas Chebyshev polynomials are used for the other. The Yamabe equation is solved using an iterative scheme based on Richardson’s iteration procedure. The construction of the fields, and hence the division by a field vanishing at infinity, is then handled by making use of the nonlocal nature of the spectral expansion (i.e. by working in the coefficient space; see Section 4 of [81] for more details).
6 Dynamic Evolution of Relativistic Systems
The modeling of timedependent physical systems is traditionally the ultimate goal in numerical simulations. Within the field of numerical relativity, the need for studies of dynamic systems is even more pronounced because of the search for gravitational wave patterns. Unfortunately, as presented in Section 4.1, there is no efficient spectral time discretization yet and one normally uses finiteorder timedifferentiation schemes. Therefore, in order to get high temporal accuracy, one must use highorder explicit timemarching schemes (e.g., fourth or sixthorder RungeKutta [49]). This requires quite a lot of computational power and might explain why, except for gravitational collapse [95, 156], very few studies using spectral methods have dealt with dynamic situations until the Caltech/Cornell group began to use spectral methods in numerical relativity in the early part of this century [128, 127]. This group now has a very welldeveloped pseudospectral collocation code, “Spectral Einstein Code” (SpEC), for the solution of full threedimensional dynamic Einstein equations.
In this section, we review the status of numerical simulations that use spectral methods in some fields of general relativity and relativistic astrophysics. Although we may give at the beginning of each section a very short introduction to the context of the relevant numerical simulations, our point is not to give detailed descriptions of them, as dedicated reviews exist for most of the themes presented here and the interested reader should consult them for details of the physics and comparisons with other numerical and analytic techniques. Among the systems that have been studied, one can find gravitational collapse [84] (supernova core collapse or collapse of a neutron star to a black hole), oscillations of relativistic stars [205, 130] and evolution of “vacuum” spacetimes. These include the cases of pure gravitational waves or scalar fields, evolving in the vicinity of a black hole or as (selfgravitating) perturbations of Minkowski flat spacetime. Finally, we will discuss the situation of compact binary [174, 31] spectral numerical simulations.
6.1 Single Stars
The numerical study of the evolution of stars in general relativity involves two parts: first, one must solve for the evolution of matter (relativistic hydrodynamics, see [77]), and second, one must compute the new configuration of the gravitational field. Whereas spectralmethods based codes are now able to study quite well the second part (see Section 6.2), the first has not benefited from the efforts of groups using spectral methods in the past decade. Thus, one faces a paradox: spectral methods have been primarily developed for the simulation of hydrodynamic systems (see Section 1.2), but they are not often used for relativistic hydrodynamics. This might be understood as a consequence of the general problem of spectral methods to deal with discontinuous fields and supersonic flows: the Gibbs phenomenon (see Section 2.4.4). Relativistic flows in astrophysics are often supersonic and therefore contain shocks. Although some techniques have been devised to deal with them in onedimensional studies (see, e.g., [45]), there has been no convincing multidimensional convincing work. Another problem of multidimensional relativistic hydrodynamics that can spoil rapid convergence properties is sharp density profiles near neutron star surfaces. These can imply a diverging or discontinuous radial derivative of the density, thus slowing down the convergence of the spectral series.
6.1.1 Supernova core collapse
The physical scenario studied here is the formation of a neutron star from the gravitational collapse of a degenerate stellar core. This core can be thought of as the iron core of a massive star at the end of its evolution (standard mechanism of type II supernova). The degeneracy pressure of the electrons can no longer support the gravity and the collapse occurs. When the central density reaches nuclear values, the strong interaction stiffens the equation of state, stopping the collapse in the central region and generating a strong shock. This mechanism has long been thought to be a powerful source of gravitational radiation, but recent simulations show that the efficiency is much lower than previously estimated [70, 195]. The first numerical study of this problem was the sphericallysymmetric approach by May and White [148] using artificial viscosity to damp the spurious numerical oscillations caused by the presence of shock waves in the flow solution. Currently, stateoftheart codes use more sophisticated HighResolution ShockCapturing (HRSC) schemes or HighResolution Central (HRC) schemes (for details on these techniques, see the review by Font [77]). The first axisymmetric fully (general) relativistic simulations of the core collapse scenario were done by Shibata [191] and Shibata and Sekiguchi [195], which used HRSC schemes and a parametric equation of state. More recently, magnetohydrodynamic effects have been taken into account in the axisymmetric core collapse by Shibata et al. [193] using HRC schemes. Threedimensional core collapse simulations, including a more realistic equation of state and deleptonization scheme have been performed within the cactuscarpetwhisky [217, 17] framework by Ott et al. [164, 163]. These simulations have been compared with those of the CoCoNuT code (see [68, 69] and later in this section). A more detailed historical presentation can be found in the Living Review by Fryer and New [84].
The appearance of a strong hydrodynamic shock is, in principle, a serious problem to numerical models using spectral methods. Nevertheless, a first preliminary study in spherical symmetry and the Newtonian theory of gravity was undertaken in 1986 by Bonazzola and Marck [43], with the use of “natural” viscosity. The authors show a mass conservation to a level better than 10^{−4} using one domain with only 33 Chebyshev polynomials. In 1993, the same authors performed the first threedimensional simulation (still in Newtonian theory) of the prebounce phase [46], giving a computation of the gravitational wave amplitude, which was shown to be lower than standard estimates. Moreover, they showed that for a given mass, the gravitational wave amplitude depends only on the deformation of the core. These threedimensional simulations were made possible thanks to the use of spectral methods, particularly for the solution of the Poisson equation for the gravitational potential.
Thus, shock waves pose a problem to spectral codes and have either been smoothed with spectral vanishing viscosity [112] or ignored by the code stopping before their appearance. Another idea developed first between the Meudon and Valencia groups was to use more appropriate techniques for the simulation of shock waves: namely the HighResolution ShockCapturing techniques, also known as Godunov methods (see the Living Reviews by Martí and Müller [145] and by Font [77]). On the other hand, one wants to keep the fewest degrees of freedom required by spectral methods for an accurateenough description of functions, in particular for the solution of elliptic equations or for the representation of more regular fields, such as gravitational ones. Indeed, even in the case where a hydrodynamic shock is present, since it only appears as a source for the metric in Einstein’s equations, the resulting gravitational field is at least \({{\mathcal C}^1}\) and the spectral series do converge, although slower than in the smooth case. Moreover, in a multidomain approach, if the shock is contained within only one such domain, it is then necessary to increase resolution in only this particular domain and it is still possible to keep the overall number of coefficients lower than the number of points for the HRSC scheme. The combination of both types of methods (HRSC and spectral) was first achieved in 2000 by Novak and Ibáñez [158]. They studied a sphericallysymmetric core collapse in tensorscalar theory of gravity, which is a more general theory than general relativity and allows a priori for monopolar gravitational waves. The system of PDEs to be solved resembles that of general relativity, with the addition of a scalar nonlinear wave equation for the monopolar dynamic degree of freedom. It was solved by spectral methods, whereas the relativistic hydrodynamics equations were solved by Godunov techniques. Two grids were used, associated to each numerical technique, and interpolations between the two were done at every timestep. Although strong shocks were present in this simulation, they were sharply resolved with HRSC techniques and the gravitational field represented through spectral methods did not exhibit any Gibbslike oscillations. Monopolar gravitational waveforms could, thus, be given. In collaboration with the Garching relativistic hydrodynamics group, this numerical technique was extended in 2005 to three dimensions, but in the conformallyflat approximation of general relativity (see Sections 5.5 and 5.6) by Dimmelmeier et al. [69]. This approach using spectral methods for the gravitational field computation is now sometimes referred to as the “Marriage des Maillages” (French for “grid wedding”) and is currently employed by the corecollapse code CoCoNuT of Dimmelmeier et al. [68, 69] to study general relativistic simulations of protoneutron stars, with a microphysical equation of state as well as an approximate description of deleptonization [70].
6.1.2 Collapse to a black hole
Stellar collapse to a black hole has been widely studied, starting with sphericallysymmetric computations; in the case of dust (matter with no pressure), an analytical solution by Oppenheimer and Snyder [162] was found in 1939. Pioneering numerical works by Nakamura and Sato [153, 154] studied the axisymmetric general relativistic collapse to a black hole; Stark and Piran [203] gave the gravitational wave emission from such a collapse in the formalism of Bardeen and Piran [19]. Fully general relativistic collapse simulations in axisymmetry have also been performed by Shibata [190], and the first threedimensional calculations of gravitationalwave emission from the collapse of rotating stars to black holes was done by Baiotti et al. [17]. Recently, Stephens et al. [204] developed an evolution code for the coupled EinsteinMaxwellMHD equations, with application to the collapse to a black hole of a magnetized, differentiallyrotating neutron star.
To our knowledge, all studies of the collapse to a black hole, which use spectral methods are currently restricted to spherical symmetry. However, in this case and contrary to the corecollapse scenario, there is a priori no shock wave appearing in the evolution of the system and spectral methods are highly accurate at modeling the hydrodynamics as well. Thus, assuming spherical symmetry, the equations giving the gravitational field are very simple, first because of Birkhoff’s theorem, which gives the gravitational field outside the star, and then from the absence of any dynamic degree of freedom in the gravitational field. For example, when choosing the radial (Schwarzschild) gauge and polar slicing, Einstein’s equations, expressed within 3+1 formalism, turn into two firstorder constraints, which are simply solved by integrating with respect to the radial coordinate (see [95]).
In the work of Gourgoulhon [95] a Chebyshevtau method is used. The evolution equations for the relativistic fluid variables are integrated with a semiimplicit time scheme and a quasiLagrangian grid: the boundary of the grid is comoving with the surface of the star, but the grid points remain the usual GaussLobatto collocation points (Section 2.3.2). Due to the singularityavoiding gauge choice, the collapsing star ends in the “frozenstar” state, with the collapse of the lapse. This induces strong gradients on the metric potentials, but the code is able to follow the collapse down to very small values of the lapse, at less than 10^{−6}. The code is very accurate at determining whether a star at equilibrium is unstable, by triggering the physical instability from numerical noise at very low level. This property was later used by Gourgoulhon et al. [102] to study the stability of equilibrium configurations of neutron stars near the maximal mass, taking into account the effect of weak interaction processes. The addition of an inward velocity field to the initial equilibrium configurations enabled Gourgoulhon [96] to partially answer the question of the minimal mass of black holes: can the effective massenergy potential barrier associated with stable equilibrium states be penetrated by stars with substantial inward radial kinetic energy? In [96], Gourgoulhon was able to form a black hole with a starting neutron star that was 10% less massive than the usual maximal mass.
The spectral numerical code developed by Gourgoulhon [95] has been extended to also simulate the propagation of neutrinos, coming from the thermal effect and nonequilibrium weak interaction processes. With this tool, Gourgoulhon and Haensel [101] have simulated the neutrino bursts coming from the collapse of neutron stars, with different equations of state. Another modification of this spectral code has been done by Novak [156], extending the theory of gravity to tensorscalar theories. This allowed for the simulation of monopolar gravitational waves coming from the sphericallysymmetric collapse of a neutron star to a black hole [156]. From a technical point of view, the solution of a nonlinear wave equation on curved spacetime has been added to the numerical model. It uses an implicit secondorder CrankNicolson scheme for the linear terms and an explicit scheme for the nonlinear part. In addition, as for the hydrodynamics, the wave equation is solved on a grid, partly comoving with the fluid. The evolution of the scalar field shows that the collapsing neutron star has “expelled” all of its scalar charge before the appearance of the black hole.
6.1.3 Relativistic stellar pulsations
Oscillations of relativistic stars are often studied as a timeindependent, linear eigenvalue problem [130]. Nevertheless, numerical approaches via time evolutions have proved to bring interesting results, as obtained by Font et al. [78] for the first quasiradial mode frequencies of rapidlyrotating stars in full general relativity. Nonlinear evolution of the gravitationalradiationdriven instability in the rmodes of neutron stars has been studied by many authors (for a presentation of the physical problem, see Section 13 of [5]). In particular, the first study of nonlinear rmodes in rapidlyrotating relativistic stars, via threedimensional generalrelativistic hydrodynamic evolutions has been done by Stergioulas and Font [206]. Different approaches to numerical hydrodynamic simulations in Newtonian gravity have been attempted by Lindblom et al. [139], with an additional braking term, as by Villain and Bonazzola [224] (see the following).
Because of their very high accuracy, spectral methods are able to track dynamic instabilities in the evolution of equilibrium neutron star configurations, as shown in section 6.1.2 by the work of Gourgoulhon et al. [95, 102]. In this work, when the initial data represents a stable neutron star, some oscillations appear, which corresponds to the first fundamental mode of the star. As another illustration of the accuracy, let us mention the work of Novak [155], who followed the dynamic evolution of unstable neutron stars in the tensorscalar theory of gravity. The instability is linked with the possibility for these stars of undergoing some “spontaneous scalarization”, meaning that they could gain a very high scalar charge, whereas the scalar field would be very weak (or even null) outside the star. Thus, for a given number of baryons, there would be three equilibria for a star: two stable ones with high scalar charges (opposite in sign) and an unstable one with a weak scalar charge. Starting from this last one, the evolution code described in [156] was able to follow the transition to a stable equilibrium, with several hundreds of damped oscillations for the star. This damping is due to the emission of monopolar gravitational waves, which carry away the star’s kinetic energy. The final state corresponds to the equilibrium configuration, independently computed by a simple code solving the generalized TolmanOppenheimerVolkoff system with a scalar field, up to 1% error, after more than 50,000 timesteps. These studies could be undertaken with spectral methods because in these scenarios the flow remains subsonic and one does not expect any shock to be formed.
It is therefore quite natural to try and simulate stellar pulsations using spectral methods. Unfortunately, there have been only a few such studies, which are detailed in the following. Lockitch et al. [142] have studied the inertial modes of slowlyrotating stars in full general relativity. They wrote down perturbation equations in the form of a system of ordinary differential equations, thanks to a decomposition into vector and tensor spherical harmonics. This system is then a nonlinear eigenvalue problem for the dimensionless mode frequency in the rotating frame. Equilibrium and perturbation variables are then expanded in terms of a basis of Chebyshev polynomials, taking into account the coordinate singularity at the origin and parity requirements. The authors were therefore able to determine the values of the mode frequency, making the whole system singular and looked for eigenfunctions, through their spectral decomposition. They found that inertial modes were slightly stabilized by relativistic effects.
A different and maybe more natural approach, namely the time integration of the evolution equations, has been undertaken by Villain et al. [224, 225] with a spectral magnetohydrodynamics code in spherical coordinates. The code solves the linearized Euler or NavierStokes equations, with the anelastic approximation. This approximation, which is widely used in other fields of astrophysics and atmospheric physics, consists in neglecting acoustic waves by assuming that time derivatives of the pressure and the density perturbations are negligible. It allows for a characteristic time, which is not set by acoustic propagation time, but is much longer and the timestep can be chosen so as to follow the inertial modes themselves. In their 2002 paper [224], Villain et al. study inertial modes (i.e. modes whose restoring force is the Coriolis force, among which the rmodes [5]) in slowly rotating polytropes with γ = 2 in the linear regime. First, this is done in the framework of Newtonian gravity, where the anelastic approximation implies that the Eulerian perturbations of the gravitational potential do not play any role in the velocity perturbations. Second, they study the relativistic case, but with the Cowling approximation, meaning again that the metric perturbations are discarded. In both regimes and trying different boundary conditions for the velocity field at the surface of the star, they note the appearance of a polar part of the mode and the “concentration of the motion” near the surface, showing up in less than 15 periods of the linear rmode. A more recent work [225] deals with the study of gravity modes, in addition to inertial modes, in neutron stars. The interesting point of this work is the use of a quite realistic equation of state for nuclear matter, which is valid even when beta equilibrium is broken. The authors were, thus, able to show that the coupling between polar and axial modes is increasing with the rotation of the star, and that the coupling of inertial modes with gravity modes in nonbarotropic stars can produce fast energy exchanges between polar and axial parts of the fluid motion. From a numerical point of view, one of the key ingredients is the solution of the vector heat equation, coming from the Euler or NavierStokes equations. This is done by a poloidaltoroidal [47] decomposition of the velocity field into two scalar potentials, which is very natural in spectral methods. Moreover, to ensure correct analytical behavior at the origin, all scalar quantities are projected at each timestep to a modified Legendre function basis.
More recently, a complete nonlinear study of rotating star pulsations has been set by Dimmelmeier et al. [71]. They used the general relativistic code CoCoNuT (see above, Section 6.1.1) in axial symmetry, with an HRSC hydrodynamic solver, and spectral methods for the simplified Einstein equations (conformallyflat three metric). They noted that the conformal flatness condition did not have much effect on the dynamics when comparing with the Cowling approximation. Nevertheless, they found that differential rotation was shifting the modes to lower frequencies and they confirmed the existence of the masssheddinginduced damping of pulsations.
6.2 Vacuum and black hole evolutions
If one wants to simulate the most interesting astrophysical sources of gravitational radiation, one must have a code able to follow, in a stable manner, gravitational waves themselves on a background spacetime. It has been observed by all numerical relativity groups that the stability of a numerical code, which solves Einstein’s field equations, not only depends on the numerical algorithm, but also on the particular formulation of the equations. Successes in the simulations of binary systems of compact objects in general relativity (see Section 6.3) are also due to numerous studies and advances in the formulations of Einstein’s equations. The methods known at present that work for the numerical evolution of binaries are the generalized harmonic coordinate method [82, 86, 176] and the BSSN method (for BaumgarteShapiroShibataNakamura [25, 194]). In particular, these two formulations of the field equations have the important property that constraint violating modes (see discussion following) stay at a reasonable level during the evolution. Use of the generalized harmonic gauge requires constraint damping terms; and a particular method suited for harmonic evolution, which was proposed by Gundlach et al. [111], enabled Pretorius to evolve black hole spacetimes [176, 175].
It is, therefore, a crucial step to devise such a stable formulation, and more particularly with spectral methods, because they add very little numerical dissipation and thus, even the smallest instability is not dissipated away and can grow to an unacceptable level. The situation becomes even more complicated with the setup of an artificial numerical boundary at a finite distance from the source, needing appropriate boundary conditions to control the physical wave content, and possibly to limit the growth of unstable modes. All these points have been extensively studied since 2000 by the Caltech/Cornell groups and their pseudospectral collocation code SpEC [125, 127, 187, 186, 138, 120, 126, 137, 49]; they were followed in 2004 by the Meudon group [37] and in 2006 by Tichy [219].
Next, it is necessary to be able to evolve black holes. Successful simulations of black hole binaries have been performed using the blackhole puncture technique [55, 18]. Unfortunately, the dynamic part of Einstein fields are not regular at the puncture points and it seems difficult to regularize them so as to avoid any Gibbslike phenomenon using spectral methods. Therefore, punctures are not generally used for spectral implementations; instead the excision technique is employed, removing part of the coordinate space inside the apparent horizon. There is no need for boundary conditions on this new artificial boundary, provided that one uses a freeevolution scheme [187], solving only hyperbolic equations. In the considered scheme, and for hydrodynamic equations as well, one does not need to impose any boundary condition, nor do any special treatment on the excision boundary, contrary to finite difference techniques, where one must construct special onesided differencing stencils. On the other hand, with a constrained scheme, elliptictype equations are to be solved [37] and, as for initial data (see Sections 5.3 and 5.6), boundary conditions must be provided, e.g., on the apparent horizon, from the dynamic horizon formalism [105].
Finally, good outer boundary conditions, which are at the same time mathematically well posed, consistent with the constraints and prevent as much as possible reflections of outgoing waves, must be devised. In that respect, quite complete boundary conditions have been obtained by Buchman and Sarbach [53].
6.2.1 Formulation and boundary conditions
Several formulations have been proposed in the literature for the numerical solution of Einstein’s equations using spectral methods. The standard one is the 3+1 (a.k.a. ArnowittDeserMisnerADM) formalism of general relativity [14, 229] (for a comprehensive introduction, see the lecture notes by Gourgoulhon [97]), which has been reformulated into the BSSN [25, 194] for better stability. But first, let us mention an alternative characteristic approach based on expanding null hypersurfaces foliated by metric twospheres developed by Bartnik [20]. This formalism allows for a simple analysis of the characteristic structure of the equations and uses the standard “edth” (ð) operator on S^{2} to express angular derivatives. Therefore, Bartnik and Norton [21] use spinweighted spherical harmonics (see Section 3.2.2) to numerically describe metric fields.
Coming back to the 3+1 formalism, Einstein’s equations split into two subsets of equations. First, the dynamic equations specifying the way the gravitational field evolves from one timeslice to the next; then, the constraint equations, which must be fulfilled on each timeslice. Still, it is well known that for the Einstein system, as well as for Maxwell’s equations of electromagnetism, if the constraints are verified on the initial timeslice, then the dynamic equations guarantee that they shall be verified in the future of that timeslice. Unfortunately, when numerically doing such free evolution, i.e. solving only for the dynamic equations, small violations of the constraints due to roundoff errors appear to grow exponentially (for an illustration with spectral methods, see, e.g., [187, 219]). The opposite strategy is to discard some of the evolution equations, keeping the equations for the two physical dynamic degrees of freedom of the gravitational field, and to solve for the four constraint equations: this is a constrained evolution [37].
The advantages of the free evolution schemes are that they usually allow one to write Einstein’s equations in the form of a strongly or symmetrichyperbolic system, for which there are many mathematical theorems of existence and wellposedness. In addition, it is possible to analyze such systems in terms of characteristics, which can give very clear and easytoimplement boundary conditions [126]. Using finitedifference numerical methods, it is also very CPUtime consuming to solve for constraint equations, which are elliptic in type, but this is not the case with spectral methods. On the other hand, constrained evolution schemes have, by definition, the advantage of not being subject to constraintviolation modes. Besides, the equations describing stationary spacetimes are usually elliptic and are naturally recovered when taking the steadystate limit of such schemes. Finally, elliptic PDEs usually do not exhibit instabilities and are known to be well posed. To be more precise, constrained evolution using spectral methods has been implemented by the Meudon group [37], within the framework of the BSSN formulation. Freeevolution schemes have been used by Tichy [219] (with the BSSN formulation) and by the Caltech/Cornell group, which has developed their KidderScheelTeukolsky (KST) scheme [127] and have later used the GeneralizedHarmonic (GH) scheme [137]. The KST scheme is, in fact, a 12parameter family of hyperbolic formulations of Einstein’s equations, which can be fine tuned in order to stabilize the evolution of, e.g., black hole spacetimes [187].
Even when doing so, constraintviolating modes grow exponentially and three ways of controlling their growth have been studied by the Caltech/Cornell group. First, the addition of multiples of the constraints to the evolution system in order to minimize this growth. The parameters linked with these additional terms are then adjusted to control the evolution of the constraint norm. This generalized version of the dynamic constraint control method used by Tiglio et al. [221] has been presented by Lindblom et al. [138] and tested on a particular representation of the Maxwell equations. Second, Lindblom et al. [138] devised constraint preserving boundary conditions from those of Calabrese et al. [54], where the idea was to get maximally dissipative boundary conditions on the constraint evolution equations [138, 126]. This second option appeared to be more efficient, but still did not completely eliminate the instabilities. Finally, bulk constraint violations cannot be controlled by constraintpreserving boundary conditions alone, so Holst et al. [120] derived techniques to project at each timestep the solution of the dynamic equations onto the constraint submanifold of solutions. This method necessitates the solution of a covariant inhomogeneous Helmholtz equation to determine the optimal projection. Nevertheless, the most efficient technique seems to be the use of the GH formulation, which also incorporates multiples of the constraints, thus exponentially suppressing bulk constraint violation, together with constraintpreserving boundary conditions [137].
Boundary conditions are not only important for the control of the constraintviolation modes in free evolutions. Because they cannot be imposed at spatial infinity (see Section 3.1.2), they must be completely transparent to gravitational waves and prevent any physical wave from entering the computational domain. A first study of interest for numerical relativity was done by Novak and Bonazzola [157], in which gravitational waves are considered in the wave zone, as perturbations of flat spacetime. The specificity of gravitational waves is that they start at the quadrupole level (ℓ = 2) in terms of spherical harmonics expansion. Standard radiative boundary conditions (known as Sommerfeld boundary conditions [201]) being accurate only for the ℓ = 0 component, a generalization of these boundary conditions has been done to include quadrupolar terms [157]. They strongly rely on the spectral decomposition of the radiative field in terms of spherical harmonics and on spherical coordinates. More specific boundary conditions for the Einstein system, in order to control the influx of the radiative part of the Weyl tensor, have been devised by Kidder et al. [126] for the KST formulation, generalizing earlier work by Stewart [207] and Calabrese et al. [54]. They were then adapted to the GH formulation by Lindblom et al. [137]. Rinne [179] has studied the wellposedness of the initialboundaryvalue problem of the GH formulation of Einstein’s equations. He has considered firstorder boundary conditions, which essentially control the incoming gravitational radiation through the incoming fields of the Weyl tensor. He has also tested the stability of the whole system with a collocation pseudospectral code simulating a Minkowski or Schwarzschild perturbed spacetime. Generalizing previous works, a hierarchy of absorbing boundary conditions has been introduced by Buchman and Sarbach [53], which have then been implemented in the Caltech/Cornell SpEC code by Ruiz et al. [181], together with new sets of absorbing and constraintpreserving conditions in the generalized harmonic gauge. Ruiz et al. have shown that their secondorder conditions can control the incoming gravitational radiation, up to a certain point. In addition, they have analyzed the wellposedness of the initialboundaryvalue problems arising from these boundary conditions. Rinne et al. [180] have compared various methods for treating outer boundary conditions. They have used the SpEC code to estimate the reflections caused by the artificial boundary, as well as the constraint violation it can generate.
6.2.2 Gauges and wave evolution
6.2.3 Black hole spacetimes
As stated at the beginning of Section 6.2, the detailed strategy to perform numerical simulations of black hole spacetimes depends on the chosen formulation. With the characteristic approach, Bartnik and Norton [21] modeled gravitational waves propagating on a black hole spacetime in spherical coordinates, but with a null coordinate z = t − r. Interestingly, they combined a spectral decomposition on spinweighted spherical harmonics for the angular coordinates and an eighthorder scheme using spline convolution to calculate derivatives in the r or z direction. Integration in these directions was done with a fourth or eighthorder RungeKutta method. For the spectral part, they had to use Orszag’s 2/3 rule [56] for antialiasing. This code achieved a global accuracy of 10^{−5} and was able to evolve the black hole spacetime up to z = 55 M. More recently, Tichy has evolved a Schwarzschild black hole in KerrSchild coordinates in the BSSN formulation, up to t ≃ 100 M [219]. He used spherical coordinates in a shelllike domain, excising the interior of the black hole. The expansion functions are Chebyshev polynomials for the radial direction, and Fourier series for the angular ones.

GH formalism, exponentially suppressing all small shortwavelength constraint violations,

constraintpreserving boundary conditions,

the filtering of spherical harmonics spectral coefficients.
6.3 Binary systems
As seen in Section 6.2, not many groups using spectral methods are able to solve all the threedimensional Einstein equations in a stable way. When dealing with black holes, the situation is even worse. Only very recently, the Caltech/Cornell group succeeded in following 16 orbits, merger and ringdown of an equalmass nonspinning blackholebinary system [185]. Moreover, we can report on three recent partial results in the field using spectral methods, dealing with each type of binary system (neutron stars and/or black holes) and leaving space for future study in this rapidly evolving field. We note, of course, that successful numerical evolutions of such systems have been performed with other numerical methods, which we very briefly summarize here. The first successful fullyrelativistic simulation of neutron star binaries was obtained by Shibata et al. [196, 192] and now, more groups are also able to study such systems: the Louisiana State University (LSU) group [4] and the Albert Einstein Institute (AEI, Golm) group [16]. We should also mention here the more microphysicallydetailed simulations by Oechslin and Janka [161], although with the conformallyflat approximation, and those of Liu et al. [141] evolving magnetized neutron star binaries. Shibata and Uryū [197, 198] have successfully evolved blackholeneutronstar binaries using the puncture technique for the modeling of the black hole. As far as black hole binary systems are concerned, after many years of hard work and codes evolving the binary system for a restricted time, a first stable simulation up to the merger phase has been performed by Pretorius [175], who used general harmonic coordinates together with constraintdamping terms and a compactification of spatial infinity. He also used the excision technique for a region surrounding the singularity inside the horizon. This first success was followed a few moths later by the Texas/Brownsville group [55] and the NASA/Goddard group [18], using very different techniques, namely BSSN with moving punctures and “1+log” slicing together with a “Γdriver” shift condition. These techniques have rapidly become standards for many other groups, which are now able to stably evolve black hole binaries, such as the AEI/LSU collaboration [173], the group in Jena [93], that at Pennsylvania State University [114] and at Florida Atlantic University [220]. The results have reached a high level of confidence and it is possible to compare gravitational waveforms obtained from numerical evolution with postNewtonian template families [165].
6.3.1 Neutron star binaries
Numerical simulations of the final stage of inspiral and merger of neutron star binaries have been performed by Faber et al. [75], who used spectral methods in spherical coordinates (based on Lorene library [99]) to solve Einstein’s equations in the conformallyflat approximation (see Sections 5 and 6.1.1). The hydrodynamic evolution has been computed using a Lagrangian smoothed particle hydrodynamics (SPH) code. As for the initial conditions described in Section 5.5, the equations for the gravitational field reduce, in the case of the conformallyflat approximation, to a set of five nonlinear coupled elliptic (Poissontype) PDEs. The considered fields (lapse, shift and conformal factor) are “split” into two parts, each component being associated with one of the stars in the binary. Although this splitting is not unique, the result obtained is independent from it, because the equations with the complete fields are solved iteratively, for each timestep. Boundary conditions are imposed on each solution of the field equations at radial infinity thanks to a multidomain decomposition and a u = 1/r compactification in the last domain. The authors used ∼ 10^{5} SPH particles for each run, with an estimated accuracy level of 1–2%. Most of the CPU time was spent in calculating the values of quantities known from their spectral representation, at SPH particle positions. Another difficulty has been the determination of the domain boundary containing each neutron star, avoiding any Gibbs phenomena. Because the conformallyflat approximation discards gravitational waves, the dissipative effects of gravitational radiation back reaction were added by hand. The authors used the slowmotion approximation [227] to induce a shrinking of the binary systems, and the gravitational waves were calculated with the lowestorder quadrupole formulae. The code has passed many tests and, in particular, it has evolved several quasiequilibrium binary configurations without adding the radiation reaction force with resulting orbits that were nearly circular (change in separation lower than 4%). The code was thus able to follow irrotational neutron star binaries, including radiation reaction effects, up to the merger and the formation of a differentially rotating remnant, which is stable against immediate gravitational collapse for reasonably stiff equations of state. All the results agreed pretty well with previous relativistic calculations.
6.3.2 Blackholeneutronstar binaries
A similar combination of numerical techniques has been used by Faber et al. [74] to compute the dynamic evolution of merging blackholeneutronstar binaries. In addition to the conformallyflat approximation and similar to Taniguchi et al. [209], Faber et al. [74] considered only the case of an extremely large mass ratio between the black hole and the neutron star, thus holding the black hole position fixed and restricting the spectral computational grid to the neighborhood of the neutron star. The metric describing the space surrounding the black hole was thus, supposed to keep the form of a Schwarzschild black hole in isotropic coordinates. The neutron star was restricted to low compactness (only a few percent) in order to have systems that disrupt well outside the last stable orbit. The system was considered to be in corotation and, just as for neutron star binaries, the gravitational radiation reaction was added by hand. As stated above, the numerical methods used SPH discretization to treat dynamic evolution of matter, and the spectral library Lorene [99] to solve the Einstein field Poissonlike equations in the conformallyflat approximation. But here, the spectral domains associated with the neutron star did not extend to radial infinity (no compactified domain) and approximate boundary conditions were imposed, using a multipole expansion of the fields. The main reason being that the black hole central singularity could not be well described on the neutron star grid.
Faber et al. [74] have studied the evolution of neutronstarblackhole binaries with different polytropic indices for the neutron star matter equation of state, the initial data being obtained as solutions of the conformal thinsandwich decomposition of Einstein’s equations. They found that, at least for some systems, the mass transfer from the neutron star to the black hole plays an important role in the dynamics of the system. For most of these systems, the onset of tidal disruption occurred outside the last stable orbit, contrary to what had been previously proposed in analytical models. Moreover, they have not found any evidence for significant shocks within the body of the neutron star. This star possibly expanded during the mass loss, eventually losing mass outward and inward provided that it was not too far within the last stable orbit. Although the major part of released matter remained bound to the black hole, a significant fraction could be ejected with sufficient velocity to become unbound from the binary system.
6.3.3 Black hole binaries
Encouraging results concerning blackholebinary simulations with spectral methods have been first obtained by Scheel et al. [188]. They have used two coordinate frames to describe the motion of black holes in the spectral grid. Indeed, when using excision techniques (punctures are not regular enough to be well represented by spectral methods), excision boundaries are fixed on the numerical grid. This can cause severe problems when, due to the movement of the black hole, the excision surface becomes timelike and the whole evolution problem becomes illposed in the absence of boundary conditions. One solution seems to be the use of comoving coordinates, but the authors report that the GH formulation they use appear to be unstable with this setting. They, therefore, consider a first system of inertial coordinates (with respect to spatial infinity) to define the tensor components in the triad associated with these coordinates, and a second system of comoving (in some sense) coordinates. In the case of their blackholebinary tests [188], they define the comoving coordinates dynamically, with a feedback control system that adjusts the moving coordinate frame to control the location of each apparent horizon center.
The spectral code uses 44 domains of different types (spherical and cylindrical shells, rectangular blocks) to describe the binary system. Most of the numerical strategy to integrate Einstein’s equations is taken from the tests on the GH formulation of Lindblom et al. [137] and has already been detailed in Section 6.2.1. The important technical ingredient detailed by Scheel et al. [188] is the particular filtering of tensor fields in terms of spherical harmonics. The dualcoordinateframe representation can mix the tensor’s spherical harmonic components. So, in their filtering of the highestorder tensor sphericalharmonic coefficients, Scheel et al. [188] had to take into account this mixing by transforming spatial tensors into a rotatingframe tensor sphericalharmonic basis before filtering and then transforming back to an inertial frame basis. This method allowed them to evolve blackholebinary spacetimes for more than four orbits, until t ≳ 600 M_{ADM}.
However, a central problem has been the capability of the code to follow the merger phase, and even though the code was able to compute the inspiral quite accurately, it used to fail just before the black holes merged. The problem was that, when the black holes came close to each other, their horizons became extremely distorted and strong gradients would develop in the dynamic fields. This has been explained as a gauge effect, coming from the incapacity of the gauge condition to react and change the geometry when the two black holes begin to interact strongly, and can be seen as a coordinate singularity developing close to the merger. Nevertheless, a cure has been found, as explained in Scheel et al. [185]. The original gauge is kept until some given time and then smoothly changed to a new one, based on the gauge treatment by Pretorius [176, 175] (for the lapse): the gauge source function is evolved through a damped, driven wave equation, which drives the lapse toward unity and the shift vector toward zero near the horizons. Thus, the horizons expand in coordinate space and the dynamic fields are smoothed out near the horizons, preventing gauge singularities from developing. With this transition of the gauge condition, the evolution of the black holes can be tracked until the formation of a common horizon encompassing both black holes. Then, the evolution of this singledistorted dynamic black hole is achieved by first interpolating all variables onto a new computational domain containing only one excised region, then by choosing a new comoving coordinate system, and finally by again modifying the gauge condition to drive the shift vector to a timeindependent state.
These new gauge conditions have allowed Scheel et al. [185] to follow the inspiral during 16 orbits, and the merger and ringdown phase of an equalmass nonspinning blackholebinary system. They were able to compute the mass and the spin of the final black hole with very high accuracy (10^{−5} and 10^{−4} relative accuracy for the mass and spin, respectively), and to extract the physical waveform accurately to 0.01 radians in phase. This is the first spectral numerical simulation of the full evolution of a blackholebinary system.
7 Conclusions
We would like to conclude our overview of spectral methods in numerical relativity by pointing out a few items that we feel are particularly interesting.
7.1 Strengths and weaknesses
The main advantage of spectral methods, especially with respect to finite difference ones, is the very rapid convergence of the numerical approximation to the real function. This implies that very high accuracy can usually be reached with only a moderate number of points. This obviously makes the codes both faster and less demanding on memory. Various examples of convergence can be found in Section 2. However, this rapid convergence is only achieved for \({{\mathcal C}^\infty}\) functions. Indeed, when the functions are less continuous, spurious oscillations appear and the convergence only follows a power law. In the case of discontinuous functions, this is known as the Gibbs phenomenon (see the extreme case of Figure 11). Gibbslike phenomena are very likely to prevent codes from converging or to make time evolutions unstable. So spectral methods must rely heavily on the domain decomposition of space and the domains must be chosen so that the various discontinuities lie at the boundaries. Because of this, spectral methods are usually more difficult to implement than standard finite differences (see, for instance, the intricate mappings of [7]). The situation is even more complicated when the surfaces of discontinuities are not known in advance or have complicated shapes.
Spectral methods are also very efficient at dealing with problems that are related to coordinate singularities. Indeed, if the basis functions fulfill the regularity requirements, then all the functions will automatically satisfy them. In particular, it makes the use of spherical coordinates much easier than with other methods, as explained in Section 3.2.
Another nice feature is the fact that a function can be represented either by its coefficients or its values at the collocation points. Depending on the operation one has to perform, it is easier to work with the one representation or the other. When working in the coefficient space, one takes full advantage of the nonlocality of the spectral representation. A lot of operations that would be difficult otherwise can then be easily performed, like computing the ratio of two quantities vanishing at the same point (see, for instance, [81]).
7.2 Combination with other methods Spectral methods have also demonstrated that they can be a valuable tool when combined with other methods. For instance, when shocks are present, spectral methods alone have trouble dealing with discontinuities at the shock interface. However, this can be efficiently dealt with using Godunov methods. Such a combination has already been successfully applied to the simulation of the oscillations of compact stars in [71] and of core collapse [164].
Spectral methods have also been used in conjunction with a description of the fluid based on SPH (smoothed particle hydrodynamics) in the case of neutron star binaries [75] and for the merger of a neutron star with a black hole [74]. In both cases, the fluid is described by an ensemble of particles to which forces are applied. Such technique can account for complicated fluid configurations, like the spiral arms that are formed during the merger. Such arms would be tricky to follow by means of spectral methods alone. On the other hand, the equations describing gravity are nicely solved by spectral solvers.
7.3 Future developments
Finally, we would like to point out a few directions could lead to interesting results. Of course, we are not aware of what the other groups have planned for the future.
Appropriate choice of coordinates is evidently important. However, for binary systems, rather few results have been obtained using the natural choice of bispherical coordinates. So far, variations of such coordinates have only been used by Ansorg and collaborators and only in the context of initial data [8, 6, 7]. We believe that applying these coordinates, or similar coordinates, to evolutionary codes could lead to interesting results, in terms of both speed and accuracy.
The application of spectral methods to theories more complicated than general relativity is also imaginable. One of the possible fields of application is the study of branes, where there is an additional dimension to spacetime. The fact that spectral methods are accurate with relatively few degrees of freedom makes them a good candidate for simulating systems with extra dimensions. The addition of gauge fields is also something that could be studied with spectral methods, to investigate the possibility of “hairy” black holes, for instance. Of course, these are just a few ideas of what the future applications of spectral methods to the field of relativity might be. —
Footnotes
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