### 2.1 Spacelike (3+1) approaches

In the 3+1 (ADM) formulation [25158], spacetime is foliated into a set of nonintersecting spacelike hypersurfaces. There are two kinematic variables describing the evolution between these surfaces: the lapse function , which describes the rate of advance of time along a timelike unit vector normal to a surface, and the spacelike shift vector that describes the motion of coordinates within a surface.

The line element is written as

where is the 3-metric induced on each spacelike slice.

#### 2.1.1 1+1 Lagrangian formulation (May and White)

For historical reasons it is worth beginning the overview of the formulations with the pioneering numerical work in general relativistic hydrodynamics, namely the one-dimensional gravitational collapse code of May and White [247248]. Building on theoretical work by Misner and Sharp [263], May and White developed a time-dependent numerical code to solve the evolution equations describing adiabatic spherical collapse in general relativity. This code was based on a Lagrangian finite-difference scheme (see Section 4.1), in which the coordinates are co-moving with the fluid. Artificial viscosity terms were included in the equations to damp the spurious numerical oscillations caused by the presence of shock waves in the flow solution. May and White’s formulation became the starting point of a large number of numerical investigations in subsequent years and, hence, it is useful to describe its main features in some detail.

For a spherically-symmetric spacetime, the line element can be written as

being a radial (Lagrangian) coordinate, indicating the total rest mass enclosed inside the sphere .

The co-moving character of the coordinates leads, for a perfect fluid, to a stress-energy tensor whose nonvanishing components are . In these coordinates the local conservation equation for the baryonic mass, Equation (2), can be easily integrated to yield the metric potential :

The gravitational field equations, Equation (10), and the equations of motion, Equation (1), reduce to the following quasi-linear system of partial differential equations (see also [263]):

with the definitions and , satisfying . Additionally,
represents the total mass interior to radius at time . The final system, Equations (14) – (17), is closed with an EOS of the form given by Equation (9).

Hydrodynamics codes based on the original formulation of May and White and on later versions (e.g., [406]) have been used in many nonlinear simulations of supernova and neutron-star collapse (see, e.g., [262391] and references therein), as well as in perturbative computations of spherically-symmetric gravitational collapse within the framework of the linearized Einstein equations [346347]. However, the Lagrangian character of May and White’s formulation, together with other theoretical considerations concerning the particular coordinate gauge, has prevented its extension to multiple-dimensional calculations. In spite of this, for one-dimensional problems, the Lagrangian approach adopted by May and White has considerable advantages with respect to an Eulerian approach with spatially-fixed coordinates, most notably the reduction of numerical diffusion.

#### 2.1.2 3+1 Eulerian formulation (Wilson)

The use of Eulerian coordinates in multiple-dimensional numerical-relativistic hydrodynamics started with the pioneering work of Wilson [411417]. Introducing the basic dynamic variables , , and , representing the relativistic density, momenta, and generalized internal energy, respectively, and defined as

the equations of motion in Wilson’s formulation [411413] are:
with the “transport velocity” given by . We note that, in the original formulation [413], the momentum density equation, Equation (21), is solved only for the three spatial components, , and is obtained through the 4-velocity normalization condition .

A direct inspection of the system shows that the equations are written as a coupled set of advection equations. In doing so, the terms containing derivatives (in space or time) of the pressure are treated as source terms. This approach, hence, sidesteps an important guideline for the formulation of nonlinear hyperbolic systems of equations, namely the preservation of their conservation form. This is a necessary condition to guarantee correct evolution in regions of sharp entropy generation (i.e., shocks). Furthermore, some amount of numerical dissipation must be used to stabilize the solution across discontinuities. In this spirit, the first attempt to solve the equations of general-relativistic hydrodynamics in the original Wilson’s scheme [411] used a combination of finite-difference upwind techniques with artificial viscosity terms. Such terms adapted the classic treatment of shock waves introduced by von Neumann and Richtmyer [407] to the relativistic regime (see Section 4.1.1).

Wilson’s formulation has been (and still is) widely used in hydrodynamic codes developed by a variety of research groups. (The latest extensions made to incorporate magnetic fields are discussed in Section 3.1.1). Many different astrophysical scenarios were first investigated with these codes, including axisymmetric stellar core collapse [27927728338386319118], accretion onto compact objects [170317], numerical cosmology [727317] and, more recently, the coalescence of neutron-star binaries [416418242] (see [417] for an up-to-date summary of such studies). This formalism has also been employed, in the special-relativistic limit, in numerical studies of heavy-ion collisions [415249]. We note that in most of these investigations, the original formulation of the hydrodynamic equations was slightly modified by redefining the dynamic variables, Equation (19), with the addition of a multiplicative factor (the lapse function) and the introduction of the Lorentz factor, :

As mentioned before, the description of the evolution of self-gravitating matter fields in general relativity requires a joint integration of the hydrodynamic equations and the gravitational field equations (the Einstein equations). Using Wilson’s formulation for the fluid dynamics, such coupled simulations were first considered in [413], building on a vacuum numerical-relativity code specifically developed to investigate the headon collision of two black holes [382]. The resulting code was axially symmetric and aimed to integrate the coupled set of equations in the context of stellar core collapse [120].

More recently, Wilson’s formulation has been applied to the numerical study of the coalescence of neutron-star binaries in general relativity [416418242] (see Section 5.3.2). These studies adopt an approximation scheme for the gravitational field, by imposing the simplifying condition that the three-geometry (the 3-metric ) is conformally flat. The curvature of the three metric is then described by a position-dependent conformal factor times a flat-space Kronecker delta, and the line element, Equation (11), reads

Therefore, in this approximation scheme all radiation degrees of freedom are removed. Moreover, under the maximal-slicing condition (), the field equations reduce to a set of five Poisson-like elliptic equations in flat spacetime for the lapse, the shift vector, and the conformal factor. While in spherical symmetry this approach is no longer an approximation, being identical to Einstein’s theory, beyond spherical symmetry its quality degrades. In [193] it was shown by means of numerical simulations of extremely relativistic disks of dust that it has the same accuracy as the first post-Newtonian approximation. In less extreme situations, however, as in rotational stellar-core collapse, the approximation yields results comparable to those obtained using full general relativity (see [7492365]).

Wilson’s formulation showed some limitations in handling situations involving ultrarelativistic flows (), as first pointed out by Centrella and Wilson [73]. Norman and Winkler [291] performed a comprehensive numerical assessment of such formulation by means of special-relativistic hydrodynamics simulations. Figure 1 reproduces a plot from [291] in which the relative error of the density compression ratio in the relativistic shock reflection problem – the heating of a cold gas, which impacts at relativistic speeds with a solid wall and bounces back – is displayed as a function of the Lorentz factor of the incoming gas. The source of the data is [73]. This figure shows that for Lorentz factors of about 2 (), the threshold of the ultrarelativistic limit, the relative errors are between 5% and 7% (depending on the adiabatic exponent of the gas), showing a linear growth with .

Norman and Winkler [291] conclude that those large errors were mainly due to the way in which the artificial viscosity terms are included in the numerical scheme in Wilson’s formulation. These terms, commonly called in the literature (see Section 4.1.1), are only added to the pressure terms in some cases, namely at the pressure gradient in the source of the momentum equation, Equation (21), and at the divergence of the velocity in the source of the energy equation, Equation (22). However, [291] proposes that one add the terms in a consistent way, in order to consider the artificial viscosity as a real viscosity. Hence, the hydrodynamic equations should be rewritten for a modified stress-energy tensor of the following form:

In this way, for instance, the momentum equation takes the following form (in flat spacetime):
In Wilson’s original formulation, is omitted in the two terms containing the quantity . In general, is a nonlinear function of the velocity and, hence, the quantity in the momentum density of Equation (26) is a highly nonlinear function of the velocity and its derivatives. This fact, together with the explicit presence of the Lorentz factor in the convective terms of the hydrodynamic equations, as well as the pressure in the specific enthalpy, make the relativistic equations more strongly coupled than their Newtonian counterparts. As a result, Norman and Winkler proposed the use of implicit schemes as a way to describe more accurately such coupling. Their code, which in addition incorporates an adaptive grid, reproduces very accurate results even for ultrarelativistic flows with Lorentz factors of about ten in one-dimensional, flat spacetime simulations.

Recently, Anninos and Fragile [19] have compared state-of-the-art artificial viscosity schemes and high-order nonoscillatory central schemes (see Section 4.1.3) using Wilson’s formulation for the former class of schemes and a conservative formulation for the latter (similar to the one considered in [311309]; see Section 2.2.2). Using the three-dimensional Cartesian code cosmos, these authors found that earlier results for artificial viscosity schemes in shock tube tests or shock reflection tests are not improved, i.e. the numerical solution becomes increasingly unstable for shock velocities greater than . On the other hand, results for the shock-reflection problem with a second-order finite-difference central scheme show the suitability of such a scheme to handle ultrarelativistic flows (see Figure 8 of [19]), the underlying reason being the use of a conservative formulation of the hydrodynamic equations rather than the particular scheme employed (see Section 4.1.3).

#### 2.1.3 3+1 conservative Eulerian formulation (Ibáñez and colleagues)

In 1991, Martí, Ibáñez, and Miralles [237] presented a new formulation of the (Eulerian) general-relativistic hydrodynamics equations. This formulation was designed to take fundamental advantage of the hyperbolic and conservative character of the equations, contrary to the formulation discussed in the previous Section 2.1.2. From the numerical point of view, the hyperbolic and conservative nature of the relativistic Euler equations allows for the use of schemes based on the characteristic fields of the system, bringing to relativistic hydrodynamics existing tools of computational classical fluid dynamics. This procedure departs from earlier approaches, most notably in avoiding the need for artificial dissipation terms to handle discontinuous solutions [411413], as well as implicit schemes as proposed in [291].

If a numerical scheme written in conservation form converges, it automatically guarantees the correct Rankine–Hugoniot (jump) conditions across discontinuities - the shock-capturing property (see, e.g., [218]). Writing the relativistic hydrodynamic equations as a system of conservation laws, identifying the suitable vector of unknowns, and building up an approximate Riemann solver permitted the extension of state-of-the-art high-resolution shock-capturing schemes (HRSC in the following) from classical fluid dynamics into the realm of relativity [237].

Theoretical advances in the mathematical character of the relativistic hydrodynamic equations were first achieved studying the special relativistic limit. In Minkowski spacetime, the hyperbolic character of relativistic hydrodynamics and MHD is exhaustively studied by Anile and collaborators (see [15] and references therein) by applying Friedrichs’ definition of hyperbolicity [143] to a quasi-linear form of the system of hydrodynamic and MHD equations,

where are the Jacobian matrices of the system and is a suitable set of primitive (physical) variables (see below). The system (27) is hyperbolic in the time direction defined by the vector field with , if the following two conditions hold: (i) and (ii) for any such that , , the eigenvalue problem has only real eigenvalues and a complete set of right-eigenvectors . Besides verifying the hyperbolic character of the relativistic hydrodynamic equations, Anile and collaborators [15] obtained the explicit expressions for the eigenvalues and eigenvectors in the local rest frame, characterized by . In Font et al. [134] those calculations were extended to an arbitrary reference frame in which the motion of the fluid is described by the 4-velocity .

The approach followed in [134] for the equations of special-relativistic hydrodynamics was extended to general relativity in [36]. The choice of state vector (conserved quantities) in the 3+1 Eulerian formulation developed by Banyuls et al. [36] differs slightly from that of Wilson’s formulation [411]. It comprises the rest-mass density (), the momentum density in the -direction (), and the total energy density (), measured by a family of observers, which is the natural extension (for a generic spacetime) of the Eulerian observers in classical fluid dynamics. Interested readers are directed to [36] for more complete definitions and geometrical foundations.

In terms of the primitive variables , the conserved quantities are written as:

where the contravariant components of the three-velocity are defined as
and is the relativistic Lorentz factor with .

With this choice of variables the equations can be written in conservation form. Strict conservation is only possible in flat spacetime. For curved spacetimes there exist source terms, arising from the spacetime geometry. However, these terms do not contain derivatives of stress-energy–tensor components. More precisely, the resulting first-order flux-conservative hyperbolic system, well suited for numerical applications, reads:

with satisfying with . The state vector is given by
with . The vector of fluxes is
and the corresponding sources are

The local characteristic structure of the previous system of equations was presented in [36]. The eigenvalues (characteristic speeds) of the corresponding Jacobian matrices are all real (but not distinct, one showing a threefold degeneracy as a result of the assumed directional splitting approach) and a complete set of right eigenvectors exists. More precisely, for a fluid moving along the -direction, the eigenvalues read:

where is the speed of sound. We note that the Minkowskian limit of these expressions is recovered properly (see [101]) as well as the Newtonian one (, ). Hence, system (32) satisfies the definition of hyperbolicity. As will become apparent in Section 4.1.2, the knowledge of the spectral information of the flux-vector Jacobian matrices is essential in order to construct (upwind) HRSC schemes based on Riemann solvers. This information can be found in [36] (see also [137]).

Three-dimensional, Eulerian codes, which evolve the coupled system of Einstein and hydrodynamics equations following the conservative Eulerian approach discussed in this section have been developed by Font et al. [137] and by Baiotti et al. [27] (see Section 4.4 for further details). In particular, in [137] the spectral decomposition (eigenvalues and right-eigenvectors) of the general-relativistic hydrodynamics equations, valid for general spatial metrics, was derived, extending earlier results of [36] for nondiagonal metrics. A complete set of left-eigenvectors was presented by Ibáñez et al. [176]. Due to the paramount importance of the characteristic structure of the equations in the design of upwind HRSC schemes based upon Riemann solvers, we summarize all necessary information in Section 6.2 of the article.

The range of applications considered so far in relativistic astrophysics employing the above formulation of the hydrodynamics equations, Equations (32) – (35), includes the study of gravitational collapse (both stellar-core collapse to neutron stars and black-hole formation), accretion flows on to black holes, as well as neutron-star pulsations and neutron-star–binary mergers (see Section 5). The first applications in general relativity were performed, in one spatial dimension, in [237], using a slightly different form of the equations. Preliminary investigations of gravitational stellar collapse were attempted in [23553] by coupling the above formulation of the hydrodynamic equations to a hyperbolic formulation of the Einstein equations developed by [54]. Evolutions of fully-dynamic spacetimes in the context of stellar-core collapse, both in spherical symmetry and in axisymmetry, have also been done [178339969774]. These investigations are discussed in Section 5.1.1. In the special-relativistic limit this formulation has successfully been applied to simulate the evolution of (ultra-) relativistic extragalactic jets, using numerical models of increasing complexity (see, e.g., [2417]).