When simulating relativistic heavy ion collisions, the use of a realistic equation of state is essential for an adequate description of the phenomenon. However, as these simulations have been performed with FCT based difference schemes (see, e.g., [166]), this poses no specific numerical problem. The simulation of flows obeying elaborated microphysics with HRSC methods needs in some cases the extension of the present relativistic Riemann solvers to handle general equations of state. This is the case of the Roe-Eulderink method (extensible by the procedure developed in the classical case by Glaister [64]), and rPPM and rGlimm both relying on an exact solution of the Riemann problem for ideal gases with constant adiabatic exponent (which can also be extended following the procedure of Colella & Glaz [32] for classical hydrodynamics). We expect the second generation of SRHD codes to be capable of treating general equations of state and various source/sink terms routinely.

Concerning the usage of complex equations of state (EOS) a limitation must be pointed out which is associated with the Riemann solvers used in HRSC methods, even in the Newtonian limit. These problems are especially compounded in situations where there are phase transitions present. In this case the EOS may have a discontinuous adiabatic exponent and may even be non-convex. The Riemann solver of Colella & Glaz [32] often fails in these situations, because it is derived under the assumption of convexity in the EOS. When convexity is not present the character of the solution to the Riemann problem changes. Situations where phase transitions cause discontinuities in the adiabatic index or non-convexity of the EOS are encountered, e.g., in simulations of neutron star formation, of the early Universe, and of relativistic heavy ion collisions.

Another interesting area that deserves further research is the application of relativistic HRSC methods in simulations of reactive multi-species flows, especially as such flows still cause problems for the Newtonian CFD community (see, e.g., [149]). The structure of the solution to the Riemann problem becomes significantly more complex with the introduction of reactions between multiple species. Riemann solvers that incorporate source terms [97], and in particular source terms due to reactions, have been proposed for classical flows [11, 79], but most HRSC codes still rely on operator splitting.

A SRHD simulation of a relativistic jet based on a combined HLL-AMR scheme was performed by Duncan & Hughes [46]. Plewa et al. [148] have modeled the deflection of highly supersonic jets propagating through non-homogeneous environments using the HRSC scheme of Martí et al. [111] combined with the AMR implementation AMRA of Plewa [147]. Komissarov & Falle [85] have combined their numerical scheme with the adaptive grid code Cobra, which has been developed by Mantis Numerics Ltd. for industrial applications [54], and which uses a hierarchy of grids with a constant refinement factor of two between subsequent grid levels.

A very interesting and powerful procedure was proposed by Balsara [8] and has been implemented by Pons et al. [150]. This procedure allows one to exploit all the developments in the field of special relativistic Riemann solvers in general relativistic hydrodynamics. The procedure relies on a local change of coordinates at each zone interface such that the spacetime metric is locally flat. In that locally flat spacetime any special relativistic Riemann solver can be used to calculate the numerical fluxes, which are then transformed back. The transformation to an orthonormal basis is valid only at a single point in spacetime. Since the use of Riemann solvers requires the knowledge of the behavior of the characteristics over a finite volume, the use of the local Lorentz basis is only an approximation. The effects of this approximation will only become known through the study of the performance of these methods in situations where the structure of the spacetime varies rapidly in space and perhaps time as well. In such a situation finer grids and improved time advancing methods will definitely be required. The implementation is simple and computationally inexpensive.

Characteristic formulations of the Einstein field equations are able to handle the long term numerical description of single black hole spacetimes in vacuum [15]. In order to include matter in such an scenario, Papadopoulos & Font [138] have generalized the HRSC approach to cope with the hydrodynamic equations in such a null foliation of spacetime. Actually, they have presented a complete (covariant) re-formulation of the equations in GR, which is also valid for spacelike foliations in SR. They have extensively tested their method calculating, among other tests, shock tube problem 1 (see Section 6.2.1), but posed on a light cone and using the appropriate transformations of the exact solution [108] to account for advanced and retarded times.

Other developments in GRHD in the past included finite element methods for simulating spherically symmetric collapse in general relativity [103], general relativistic pseudo-spectral codes based on the (3+1) ADM formalism [7] for computing radial perturbations [70] and 3D gravitational collapse of neutron stars [19], and general relativistic SPH [102]. The potential of these methods for the future is unclear, as none of them is specifically appropriate for ultra-relativistic speeds and strong shock waves which are characteristic of most astrophysical applications.

Peitz & Appl [139] have addressed the difficult issue of non-ideal GRHD, which is of particular importance, e.g., for the simulation of accretion discs around compact objects, rotating relativistic fluid configurations, and the evolution of density fluctuations in the early universe. They have accounted for dissipative effects by applying the theory of extended causal thermodynamics, which eliminates the causality violating infinite signal speeds arising from the conventional Navier-Stokes equation. Peitz & Appl have not implemented their model numerically yet.

Evans & Hawley [52] extended the second-order accurate, Newtonian, artificial-viscosity transport method of Hawley et al. [75] to the evolution of the MHD induction equation. Special relativistic 2D MHD test problems with Lorentz factors up to have been investigated by Dubal [45] with a code based on FCT techniques (see Section 4).

In a series of papers Koide and coworkers [82,
81,
128,
129,
83] have investigated relativistic magnetized jets using a
symmetric TVD scheme (see Section
3). Koide, Nishikawa & Mutel [82] simulated a 2D RMHD slab jet, whereas Koide [81] investigated the effect of an oblique magnetic field on the
propagation of a relativistic slab jet. Nishikawa et al. [128,
129] extended these simulations to 3D and considered the propagation
of a relativistic jet with a Lorentz factor
*W*
= 4.56 along an aligned and an oblique external magnetic field.
The 2D and 3D simulations published up to now only cover the very
early propagation of the jet (up to 20 jet radii) and are
performed with moderate spatial resolution on an equidistant
Cartesian grid (up to 101 zones per dimension, i.e., 5 zones per
beam radius).

Van Putten [180, 181] has proposed a method for accurate and stable numerical simulations of RMHD in the presence of dynamically significant magnetic fields in two dimensions and up to moderate Lorentz factors. The method is based on MHD in divergence form using a 2D shock-capturing method in terms of a pseudo-spectral smoothing operator (see Section 4). He applied this method to 2D blast waves [183] and astrophysical jets [182, 184].

Steps towards the extension of linearized Riemann solvers to ideal RMHD have already been taken. Romero [158] has derived an analytical expression for the spectral decomposition of the Jacobian in the case of a planar relativistic flow field permeated by a transversal magnetic field (nonzero field component only orthogonal to flow direction). Van Putten [178] has studied the characteristic structure of the RMHD equations in (constraint free) divergence form. Finally, Komissarov [84] has presented a robust Godunov-type scheme for RMHD, which is based on a linear Riemann solver, has second-order accuracy in smooth regions, enforces magnetic flux conservation, and which can cope with ultra-relativistic flows.

We end with the simulations performed by Koide, Shibata & Kudoh [83] on magnetically driven axisymmetric jets from black hole accretion disks. Their GRMHD code is an extension of the special relativistic MHD code developed by Koide et al. [82, 81, 128]. The necessary modifications of the code were quite simple, because in the (nonrotating) black hole's Schwarzschild spacetime the GRMHD equations are identical to the SRMHD equations in general coordinates, except for the gravitational force terms and the geometric factors of the lapse function. With the pioneering work of Koide, Shibata & Kudoh the epoch of exciting GRMHD simulations has just begun.

Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr-1999-3
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