- the accuracy and robustness in describing high Lorentz factor flows with strong shocks,
- the effort required to extend to multi-dimensions, and
- the effort required to extend to RMHD and GRHD.

These aspects are summarized in Table 12 for most of the numerical methods discussed in this review.

Since their introduction in numerical RHD in the early 1990s, Riemann-solver-based HRSC methods have demonstrated their ability to describe accurately (i.e., in a stable way and without excessive smearing) relativistic flows of arbitrarily large Lorentz factors and strong discontinuities reaching the same quality as in classical hydrodynamics. In addition (as it is the case for classical flows, too), HRSC methods show the best performance compared to any other method (e.g., artificial viscosity, FCT or SPH). This last assertion applies also to the symmetric HRSC relativistic algorithms developed recently.

Nevertheless, a lot of effort has been put into improving non-HRSC methods. Using a consistent formulation of artificial viscosity has significantly enhanced the capability of SPH (e.g., [262]) and of finite difference schemes. A good example of the latter case is the algorithm recently proposed in [10], but the 40% overshoot in the post-shock density in Problem 2 confirms the need for an implicit treatment of the equations as originally proposed by [214]. Concerning relativistic SPH, recent investigations using a conservative formulation of the hydrodynamic equations [53, 262, 204] have reached an unprecedented accuracy compared to previous SPH simulations, although some issues still remain. Besides the strong smearing of shocks, the description of contact discontinuities and of thin structures moving at ultra-relativistic speeds needs to be improved (see Section 6.2).

Concerning FCT, codes based on a conservative formulation of the RHD equations have been able to handle special relativistic flows with discontinuities at all flow speeds, although the quality of the results is lower than that of HRSC methods in all cases [257, 245, 247].

The extension to multi-dimensions is straightforward for most relativistic codes. Finite difference techniques are easily extended using directional splitting. HRSC methods based on exact solutions of the Riemann problem [181, 295] benefit from the development of a multi-dimensional relativistic Riemann solver [235]. The adaptive grid, artificial viscosity, implicit code of Norman and Winkler [214], and the relativistic Glimm method of Wen et al. [295] are restricted to one-dimensional flows. The latter method produces the best results in all the tests analyzed in Section 6.

The symmetric TVD scheme proposed by Davis [68] and extended to GRMHD (see below) by Koide et al. [138] combines several characteristics making it very attractive. It is written in conservation form and is TVD, i.e., it is converging to the physical solution. In addition, it does not require spectral information, and hence allows for a simple extension to RMHD. Quite similar statements can be made about the approach proposed by van Putten [287]. In contrast to FCT schemes (which are also easily extended to general systems of equations), both Koide et al.’s and van Putten’s methods are very stable when simulating mildly relativistic flows (maximum Lorentz factors 4) with discontinuities. Their only drawback is an excessive smearing of the latter. Expectations concerning the correct description of ultrarelativistic MHD flows by means of symmetric TVD schemes may be met in the near future by global third-order symmetric schemes [72].

Concerning the extension of Riemann-solver-based HRSC schemes to RMHD, we mention the efforts by Balsara [14] and Komissarov [143] in 1D and 2D RMHD (see Section 8.2.4).

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