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., [261]) 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 [213
]. Concerning
relativistic SPH, recent investigations using a conservative
formulation of the hydrodynamic equations [53
, 261
, 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 [256, 244, 246].
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 [234
]. The adaptive grid,
artificial viscosity, implicit code of Norman and
Winkler [213], 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
) 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).
|