The hydrodynamic and MHD equations (either in Newtonian physics or in general relativity) constitute nonlinear hyperbolic systems and, hence, smooth initial data can transform into discontinuous data (cross of characteristics in the case of shocks) in a finite time during the evolution. As a consequence, conventional finite-difference schemes (see, e.g., [218, 219, 398]) 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 under grid refinement. 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.
The idea of modifying the hydrodynamic equations by introducing artificial viscosity terms to damp the amplitude of spurious oscillations near discontinuities was originally proposed by von Neumann and Richtmyer  in the context of the (classical) Euler equations. The basic idea is to introduce a purely artificial dissipative mechanism whose form and strength are such that the shock transition becomes smooth, extending over a small number of intervals of the space variable. In their original work von Neumann and Richtmyer proposed the following expression for the viscosity term:
with , being the fluid velocity, the density, the spatial interval, and a constant parameter whose value needs to be 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 conjunction 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., [411, 171, 417]. In particular, a state-of-the-art formulation of the artificial viscosity approach is reported in . Correspondingly, the interested reader is also directed to the recent work by [86, 20, 108] for details on diverse modern implementations of artificial viscosity terms in nonconservative formulations of the GRMHD equations.
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 [291, 19]. Furthermore, the artificial viscosity approach has the inherent ambiguity of finding the appropriate form for that introduces the necessary amount of dissipation to reduce the spurious oscillations and, at the same time, avoids introducing excessive smearing at 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 .
In finite-difference schemes, convergence properties under grid refinement must be enforced to ensure that the numerical results are correct (i.e., if a scheme with an order of accuracy is used, the global error of the numerical solution has to tend to zero as as the cell width tends to zero). For hyperbolic systems of conservation laws, schemes written in conservation form are preferred since, according to the Lax–Wendroff theorem , they guarantee that the convergence, if it exists, is to one of the weak solutions of the original system of equations. Such weak solutions are generalized solutions that satisfy the integral form of the conservation system. They are classical solutions (continuous and differentiable) away from discontinuities and have a finite number of discontinuities.
For the sake of simplicity, let us consider in the following an initial value problem for a one-dimensional scalar hyperbolic conservation law,
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 (70):
Mathematically, the solution one needs to search for is characterized by the entropy condition (in the language of fluids, the condition that the entropy of any fluid element should increase when crossing 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 proven 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 total variation diminishing (TVD) schemes  (see below).
Let us now consider the specific system of hydrodynamic equations as formulated in Equation (32) 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 three-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 (32) 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 (32) is
Besides its convergence properties, 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 (76) is one of the distinctive features of upwind HRSC schemes. One needs first to define the 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 timestep,
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 Godunov-type methods [164, 112]. 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 of any of the following type: shock waves, rarefaction waves, and contact discontinuities (see, e.g., ). The complete structure of the Riemann problem can be solved analytically (see  for the solution in Newtonian hydrodynamics, [238, 322] in special-relativistic hydrodynamics, and [340, 150] in special-relativistic MHD) and, accordingly, used to update the solution forward in time.
For reasons of numerical efficiency and, particularly in multiple 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. The spectral decomposition of the flux-vector Jacobian matrices is on the basis of all solvers (extending ideas used for linear hyperbolic systems). 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 (76) 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:
For a purely linear system, Equation (80) provides the exact expression for the flux in terms of the conserved variables. Therefore, the above expression for the Roe flux of conserved variables is the natural extension (after linearization) of the upwind flux for characteristic variables (see, e.g., ), once the quasilinear system is written in diagonal form, namely
The last term in the flux formula, Equation (80), 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 (80), are, respectively, the eigenvalues and right eigenvectors of the Jacobian matrix of the flux vector. Correspondingly, the quantities are the jumps of the characteristic variables across each characteristic field. They are obtained by projecting the jumps of the state-vector variables with the left-eigenvector matrix:
The way the cell-reconstructed 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., [398, 219]) 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 nonoscillatory (ENO) schemes  and extensions thereof.
Alternatively, high-order methods for nonlinear hyperbolic systems have also been designed using flux limiters rather than slope limiters, as in the Flux-Corrected Transport (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 [398, 219] for further details).
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’s Riemann solver ; Schneider et al.  carried out the extension of Harten, Lax, van Leer, and Einfeldt’s (HLLE) method [164, 112]; 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; Donat et al.  extended Marquina’s method . Recently, much effort has been spent concerning the exact special relativistic Riemann solver and its extension to multiple dimensions [238, 322, 333, 334]. The interested reader is directed to  and references therein for a comprehensive description of such solvers in special-relativistic hydrodynamics.
Upwind HRSC schemes are general enough to be applicable to any hyperbolic system as long as the wave structure of the equations is known. As discussed before, the relativistic (and classical) MHD system of equations show degeneracies in the eigenvectors of the flux-vector Jacobian matrices. This fact makes it hazardous to solve them with linearized Riemann solvers based on the full spectral decomposition of the flux-vector Jacobians. For the case of special-relativistic MHD, approaches based on such full-wave decomposition have been put forward by [199, 34, 198]. Advances in this front have been significant, as even the exact solution of the Riemann problem in special-relativistic MHD has been obtained recently [340, 150]. In addition, there has been a successful attempt by  to develop a GRMHD code in which a full-wave decomposition (Roe-type) Riemann solver based on a single, renormalized set of right and left eigenvectors has been implemented. This set of eigenvectors is regular for any physical state, including degeneracies (see  for details). Such a Riemann solver is invoked in the code of  after a (local) linear coordinate transformation based on the procedure developed by  that allows one to use special-relativistic Riemann solvers in general relativity, and which has been properly extended to include magnetic fields (see  for details on such extension and Section 6.1 for details of the procedure in the purely hydrodynamic case). A similar approach is followed in the GRMHD code of Komissarov .
Due to the numerical degeneracies of the GRMHD system most approaches to solve those equations use, however, incomplete Riemann solvers, i.e., simpler alternative approaches to compute the numerical fluxes such as the Harten–Lax–van Leer (HLL) single-state Riemann solver of  (see also  for an improved HLL scheme at contact discontinuities (HLLC) for ideal special-relativistic MHD), or high-order central (symmetric) schemes, which entirely sidestep the use of the Jacobian matrix eigenvector structure. Such symmetric schemes, which have proven recently to yield results with an accuracy comparable to those provided by full-wave decomposition Riemann solvers in simulations involving both hydrodynamic and magnetohydrodynamic flows, are briefly discussed next.
The use of high-order nonoscillatory symmetric (central) TVD schemes for solving hyperbolic systems of conservation laws emerged in the mid 1980s [83, 338, 427, 287] (see also  and  and references therein) as an alternative approach to upwind HRSC schemes. Symmetric schemes are based on either high-order schemes (e.g., Lax–Wendroff) with additional dissipative terms [83, 338, 427], or on nonoscillatory high-order extensions of first-order central schemes (e.g., Lax–Friedrichs) [287, 226]. One of the nicest properties of central schemes is that they exploit the conservation form of the Lax–Wendroff or Lax–Friedrichs schemes. Therefore, they yield the correct propagation speeds of all nonlinear waves appearing in the solution. Furthermore, central schemes sidestep the use of Riemann solvers, which results in enhanced computational efficiency, especially in multidimensional problems. Its use is, thus, not limited to those systems of equations in which the characteristic information is explicitly known or in which the Riemann problem is prohibitively expensive to compute.
For illustrative purposes let us write the numerical flux function resulting in the fully-discrete central scheme of Kurganov and Tadmor , to use in the conservation form algorithm given by Equation (76):
Conservative central schemes have been gradually developed during the last few years to reach a mature status where a number of characteristic-information-free central schemes of high order can be applied to any nonlinear hyperbolic system of conservation laws. The typical results obtained for the Euler equations show a quality comparable to that of upwind HRSC schemes, at the expense of a small loss of sharpness of the solution at discontinuities . An up-to-date summary of the status and applications of this approach is discussed in [398, 211, 392].
In recent years there have been various successful attempts to apply high-order central schemes to solve the relativistic hydrodynamics equations as well, including those by  (Lax–Wendroff scheme with conservative TVD dissipation terms),  (Lax–Friedrichs or HLL schemes with third-order ENO reconstruction algorithms),  (nonoscillatory central differencing), and [230, 361] (semidiscrete central scheme of Kurganov–Tadmor ). On the other hand, symmetric schemes (such as HLL, Kurganov–Tadmor, or Liu–Osher schemes) are being currently employed by a growing number of groups in GRMHD [196, 197, 149, 108, 366, 20, 24, 286].
In the context of special and general-relativistic MHD, Koide et al. [196, 197] applied a second-order central scheme with nonlinear dissipation developed by  to the study of black-hole accretion and formation of relativistic jets, investigating issues such as the magnetic extraction of rotational energy of the black hole . More recently  assessed a state-of-the-art third-order, convex, essentially nonoscillatory, central scheme  in multidimensional special-relativistic hydrodynamics, later extended to relativistic MHD by . 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 considered by [19, 20] in one-dimensional special and general-relativistic hydrodynamics and MHD, where results similar to those reported by [89, 90] are discussed. The scheme of Kurganov–Tadmor (see Equation (83)) was assessed by  in the context of special-relativistic hydrodynamics, using standard numerical experiments such as shock tubes, the shock reflection test, and the relativistic version of the flat-faced step test. As for the other central schemes analyzed by [89, 19] the results were comparable to those obtained by HRSC schemes based on Riemann solvers, even well inside the ultrarelativistic regime. Lucas–Serrano et al.  used high-order reconstruction procedures such as those provided by the PPM scheme  and the piecewise hyperbolic method (PHM) , which proved essential for keeping the inherent diffusion of central schemes at discontinuities at reasonably low levels. The scheme also produced accurate results in the case of full general-relativistic hydrodynamic simulations involving dynamic spacetimes, as shown by  for oscillations of rapidly rotating neutron stars and the merger of neutron-star binaries. The scheme of Kurganov–Tadmor is also the adopted choice in the GRMHD simulations in dynamic spacetimes of [108, 366]. Finally, a MUSCL-type scheme with HLL numerical fluxes is used in the code of , which allows a systematic investigation of GRMHD processes in accretion tori around black holes. Such HLL fluxes are also the choice used in the recent approaches of [396, 91], which also implement a weighted, essentially nonoscillatory (WENO) method to build up fifth-order convergent numerical codes. Such high-order schemes may be suitable to solve the total (kinetic, thermal, and magnetic) energy equation in GRMHD codes, which deal with the challenging regime posed by high-Mach number flows.
It is worth emphasizing that early pioneer approaches in the field of relativistic hydrodynamics [291, 73] 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, accordingly, for the numerical schemes. In light of the results reported in recent investigations employing central schemes [89, 19, 230], it appears that this was the ultimate reason preventing the extension of early computations to the ultrarelativistic regime.
The alternative of using high-order component-wise central schemes instead of upwind HRSC schemes becomes apparent when the spectral decomposition of the hyperbolic system under study is unknown. The straightforwardness of a central scheme makes its use very appealing, especially in multiple dimensions where computational efficiency is an issue. Perhaps the most important example in relativistic astrophysics is the system of general-relativistic MHD equations for which, despite some progress, has been achieved in recent years (see, e.g., [34, 199, 24, 23]), much more work is needed concerning their solution with full-wave-decomposition HRSC schemes based upon Riemann solvers. Meanwhile, an obvious choice is the use of central schemes.
Most “conservation laws” involve source terms on the right-hand side (RHS) of the equations. In hydrodynamics, for instance, those terms arise when considering external forces such as gravity, which make the RHS of the momentum and energy equations no longer zero (see Section 2). Other effects leading to the appearance of source terms are radiative heat transfer (accounted for in the energy equation) and ionization (resulting in a collection of nonhomogeneous continuity equations for the mass of each species, which is not conserved separately). The incorporation of the source terms in the solution procedure is a common issue to all numerical schemes considered so far. Since a detailed discussion on the numerical treatment of source terms is beyond the scope of this article, we simply provide some basic information in this section, directing the interested reader to [398, 219] and references therein for further details.
There are, essentially, two 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 (76)):
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 . This method advances by a half timestep the solution for the ODE containing the source terms, and by a full timestep 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 time scales than the hydrodynamic timestep , 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.
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