The hydrodynamic equations (either in Newtonian physics or in general relativity) constitute a nonlinear hyperbolic system and, hence, smooth initial data can transform into discontinuous data (the crossing of characteristics in the case of shocks) in a finite time during the evolution. As a consequence, classical finite difference schemes (see, e.g., [152, 287]) present important deficiencies when dealing with such systems. Typically, first-order accurate schemes are much too dissipative across discontinuities (excessive smearing) and second order (or higher) schemes produce spurious oscillations near discontinuities, which do not disappear as the grid is refined. To avoid these effects, standard finite difference schemes have been conveniently modified in various ways to ensure high-order, oscillation-free accurate representations of discontinuous solutions, as we discuss next.
with , v being the fluid velocity, the density, the spatial interval, and k a constant parameter whose value is conveniently adjusted in every numerical experiment. This parameter controls the number of zones in which shock waves are spread.
This type of generic recipe, with minor modifications, has been used in conjuction with standard finite difference schemes in all numerical simulations employing May and White's formulation, mostly in the context of gravitational collapse, as well as Wilson's formulation. So, for example, in May and White's original code  the artificial viscosity term, obtained in analogy with the one proposed by von Neumann and Richtmyer , is introduced in the equations accompanying the pressure, in the form
Further examples of similar expressions for the artificial viscosity terms, in the context of Wilson formulation, can be found in, e.g., [300, 123]. A state-of-the-art formulation of the artificial viscosity approach is reported in .
The main advantage of the artificial viscosity approach is its simplicity, which results in high computational efficiency. Experience has shown, however, that this procedure is both problem dependent and inaccurate for ultrarelativistic flows [208, 13]. Furthermore, the artificial viscosity approach has the inherent ambiguity of finding the appropriate form for Q that introduces the necessary amount of dissipation to reduce the spurious oscillations and, at the same time, avoids introducing excessive smearing in the discontinuities. In many instances both properties are difficult to achieve simultaneously. A comprehensive numerical study of artificial-viscosity-induced errors in strong shock calculations in Newtonian hydrodynamics (including also proposed improvements) was presented by Noh .
For the sake of simplicity let us consider in the following an initial value problem for a one-dimensional scalar hyperbolic conservation law,
and let us introduce a discrete numerical grid of space-time points . An explicit algorithm written in conservation form updates the solution from time to the next time level as
where is a consistent numerical flux function (i.e., ) of p + q +1 arguments and and are the time step and cell width respectively. Furthermore, is an approximation of the average of u (x, t) within the numerical cell :
The class of all weak solutions is too wide in the sense that there is no uniqueness for the initial value problem. The numerical method should, hence, guarantee convergence to the physically admissible solution . This is the vanishing-viscosity limit solution, that is, the solution when , of the ``viscous version'' of the scalar conservation law, Equation (39):
Mathematically, the solution one needs to search for is characterized by the so-called entropy condition (in the language of fluids, the condition that the entropy of any fluid element should increase when running into a discontinuity). The characterization of these entropy-satisfying solutions for scalar equations was given by Oleinik . For hyperbolic systems of conservation laws it was developed by Lax .
The Lax-Wendroff theorem  cited above does not establish whether the method converges. To guarantee convergence, some form of stability is required, as Lax first proposed for linear problems (Lax equivalence theorem ; see, e.g., ). Along this direction, the notion of total-variation stability has proven very successful, although powerful results have only been obtained for scalar conservation laws. The total variation of a solution at time , TV , is defined as
A numerical scheme is said to be TV-stable if TV is bounded for all at any time for each initial data. In the case of nonlinear, scalar conservation laws it can be proved that TV-stability is a sufficient condition for convergence , as long as the numerical schemes are written in conservation form and have consistent numerical flux functions. Current research has focused on the development of high-resolution numerical schemes in conservation form satisfying the condition of TV-stability, such as the so-called total variation diminishing (TVD) schemes  (see below).
Let us now consider the specific system of hydrodynamic equations as formulated in Equation (30), and let us consider a single computational cell of our discrete spacetime. Let be a region (simply connected) of a given four-dimensional manifold , bounded by a closed three-dimensional surface . We further take the 3-surface as the standard-oriented hyper-parallelepiped made up of two spacelike surfaces plus timelike surfaces that join the two temporal slices together. By integrating system (30) over a domain of a given spacetime, the variation in time of the state vector within is given - keeping apart the source terms - by the fluxes through the boundary . The integral form of system (30) is
which can be written in the following conservation form, well-adapted to numerical applications:
A numerical scheme written in conservation form ensures that, in the absence of sources, the (physically) conserved quantities, according to the partial differential equations, are numerically conserved by the finite difference equations.
The computation of the time integrals of the interface fluxes appearing in Equation (45) is one of the distinctive features of upwind HRSC schemes. One needs first to define the so-called numerical fluxes, which are recognized as approximations to the time-averaged fluxes across the cell interfaces, which depend on the solution at those interfaces, , during a time step,
At the cell interfaces the flow can be discontinuous and, following the seminal idea of Godunov , the numerical fluxes can be obtained by solving a collection of local Riemann problems, as is depicted in Figure 2 . This is the approach followed by the so-called Godunov-type methods [117, 75]. Figure 2 shows how the continuous solution is locally averaged on the numerical grid, a process that leads to the appearance of discontinuities at the cell interfaces. Physically, every discontinuity decays into three elementary waves: a shock wave, a rarefaction wave, and a contact discontinuity. The complete structure of the Riemann problem can be solved analytically (see  for the solution in Newtonian hydrodynamics and [165, 231] in special relativistic hydrodynamics) and, accordingly, used to update the solution forward in time.
For reasons of numerical efficiency and, particularly in multi-dimensions, the exact solution of the Riemann problem is frequently avoided and linearized (approximate) Riemann solvers are preferred. These solvers are based on the exact solution of Riemann problems corresponding to a linearized version of the original system of equations. After extensive experimentation, the results achieved with approximate Riemann solvers are comparable to those obtained with the exact solver (see  for a comprehensive overview of Godunov-type methods, and  for an excellent summary of approximate Riemann solvers in special relativistic hydrodynamics).
In the frame of the local characteristic approach, the numerical fluxes appearing in Equation (45) are computed according to some generic flux-formula that makes use of the characteristic information of the system. For example, in Roe's approximate Riemann solver  it adopts the following functional form:
where and are the values of the primitive variables at the left and right sides, respectively, of a given cell interface. They are obtained from the cell centered quantities after a suitable monotone reconstruction procedure.
The way these variables are computed determines the spatial order of accuracy of the numerical algorithm and controls the amplitude of the local jumps at every cell interface. If these jumps are monotonically reduced, the scheme provides more accurate initial guesses for the solution of the local Riemann problems (the average values give only first-order accuracy). A wide variety of cell reconstruction procedures is available in the literature. Among the slope limiter procedures (see, e.g., [287, 153]) most commonly used for TVD schemes  are the second order, piecewise-linear reconstruction, introduced by van Leer  in the design of the MUSCL scheme (Monotonic Upstream Scheme for Conservation Laws), and the third order, piecewise parabolic reconstruction developed by Colella and Woodward  in their Piecewise Parabolic Method (PPM). Since TVD schemes are only first-order accurate at local extrema, alternative reconstruction procedures for which some growth of the total variation is allowed have also been developed. Among those, we mention the total variation bounded (TVB) schemes  and the essentially non-oscillatory (ENO) schemes .
Alternatively, high-order methods for nonlinear hyperbolic systems have also been designed using flux limiters rather than slope limiters, as in the FCT scheme of Boris and Book . In this approach, the numerical flux consists of two pieces, a high-order flux (e.g., the Lax-Wendroff flux) for smooth regions of the flow, and a low-order flux (e.g., the flux from some monotone method) near discontinuities, with the limiter (see [287, 153] for further details).
The last term in the flux-formula, Equation (49), represents the numerical viscosity of the scheme, and it makes explicit use of the characteristic information of the Jacobian matrices of the system. This information is used to provide the appropriate amount of numerical dissipation to obtain accurate representations of discontinuous solutions without excessive smearing, avoiding, at the same time, the growth of spurious numerical oscillations associated with the Gibbs phenomenon. In Equation (49), are the eigenvalues and right-eigenvectors of the Jacobian matrix of the flux vector, respectively. Correspondingly, the quantities are the jumps of the so-called characteristic variables across each characteristic field. They are obtained by projecting the jumps of the state vector variables with the left-eigenvectors matrix:
The ``tilde'' in Equations (49) and (50) indicates that the corresponding fields are averaged at the cell interfaces from the left and right (reconstructed) values.
During the last few years most of the standard Riemann solvers developed in Newtonian fluid dynamics have been extended to relativistic hydrodynamics: Eulderink , as discussed in Section 2.2.1, explicitly derived a relativistic Roe Riemann solver ; Schneider et al.  carried out the extension of Harten, Lax, van Leer, and Einfeldt's (HLLE) method [117, 75]; Martí and Müller  extended the PPM method of Woodward and Colella ; Wen et al.  extended Glimm's exact Riemann solver; Dolezal and Wong  put into practice Shu-Osher ENO techniques; Balsara  extended Colella's two-shock approximation, and Donat et al.  extended Marquina's method . Recently, much effort has been spent concerning the exact special relativistic Riemann solver and its extension to multi-dimensions [165, 231, 238, 239]. The interested reader is addressed to  and references therein for a comprehensive description of such solvers in special relativistic hydrodynamics.
In the context of special and general relativistic MHD, Koide et al. [141, 142] applied a second-order central scheme with nonlinear dissipation developed by Davis  to the study of black hole accretion and formation of relativistic jets. One-dimensional test simulations in special relativistic hydrodynamics performed by the author and coworkers  using the SLIC (slope limiter centred) scheme (see  for details) showed its capabilities to yield satisfactory results, comparable to those obtained by HRSC schemes based on Riemann solvers, even well inside the ultrarelativistic regime. The slopes of the original central scheme were limited using high-order reconstruction procedures such as PPM , which was essential to keep the inherent diffusion of central schemes at discontinuities at reasonable levels. Very recently, Del Zanna and Bucciantini  assessed a third-order convex essentially non-oscillatory central scheme in multi-dimensional special relativistic hydrodynamics. Again, these authors obtained results as accurate as those of upwind HRSC schemes in standard tests (shock tubes, shock reflection test). Yet another central scheme has been assessed by  in one-dimensional special and general relativistic hydrodynamics, where similar results to those of  are reported. These authors also validate their central scheme in simulations of spherical accretion onto a Schwarzschild black hole, and further provide a comparison with an artificial viscosity based scheme.
It is worth emphasizing that early pioneer approaches in the field of relativistic hydrodynamics [208, 54] used standard finite difference schemes with artificial viscosity terms to stabilize the solution at discontinuities. However, as discussed in Section 2.1.2, those approaches only succeeded in obtaining accurate results for moderate values of the Lorentz factor (). A key feature lacking in those investigations was the use of a conservative approach for both the system of equations and the numerical schemes. A posteriori, and in the light of the results reported by [63, 13, 92], it appears that this was the ultimate reason preventing the extension of the early computations to the ultrarelativistic regime.
The alternative of using high-order central schemes instead of upwind HRSC schemes becomes apparent when the spectral decomposition of the hyperbolic system under study is not known. The straightforwardness of a central scheme makes its use very appealing, especially in multi-dimensions where computational efficiency is an issue. Perhaps the most important example in relativistic astrophysics is the system of (general) relativistic MHD equations. Despite some progress in recent years (see, e.g., [20, 143]), much more work is needed concerning their solution with HRSC schemes based upon Riemann solvers. Meanwhile, an obvious choice is the use of central schemes [141, 142].
There are, essentially, two basic ways of handling source terms. The first approach is based on unsplit methods by which a single finite difference formula advances the entire equation over one time step (as in Equation (45)):
The temporal order of this basic algorithm can be improved by introducing successive sub-steps to perform the time update (e.g., predictor-corrector, Shu and Osher's conservative high order Runge-Kutta schemes, etc.)
Correspondingly, the second approach is based on fractional step or splitting methods . The basic idea is to split the equation into different pieces (transport + sources) and apply appropriate methods for each piece independently. In the first-order Godunov splitting, , the operator solves the homogeneous PDE in the first step to yield the intermediate value . Then, in the second step, the operator solves the ordinary differential equation to yield the final value . In order to achieve second-order accuracy (assuming each independent method is second order), a common fractional step method is the Strang splitting, where . Therefore, this method advances by a half time step the solution for the ODE containing the source terms, and by a full time step the conservation law (the PDE) in between each source step.
We note that in some cases the source terms may become stiff, as in phenomena that either occur on much faster timescales than the hydrodynamic time step , or act over much smaller spatial scales than the grid resolution . Stiff source terms may easily lead to numerical difficulties. The interested reader is directed to  (and references therein) for further information on various approaches to overcome the problems of stiff source terms.
|Numerical Hydrodynamics in General Relativity
José A. Font
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