In the mid-eighties, Norman and
Winkler [213] proposed a
reformulation of the difference equations of SRHD with an
artificial viscosity consistent with the relativistic dynamics of
non-perfect fluids. The strong coupling introduced in the equations
by the presence of the viscous terms in the definition of
relativistic momentum and total energy densities required an
implicit treatment of the difference equations. Accurate results
across strong relativistic shocks with large Lorentz factors were
obtained in combination with adaptive mesh techniques. However, no
multi-dimensional version of this code was developed.
Attempts to integrate the RHD equations avoiding
the use of artificial viscosity were performed in the early
nineties. Dubal [77] developed a 2D code
for relativistic magneto-hydrodynamics based on an explicit
second-order Lax-Wendroff scheme incorporating a flux-corrected
transport (FCT) algorithm [33
]. Following a
completely different approach Mann [172
] proposed a
multi-dimensional code for GRHD based on smoothed particle
hydrodynamics (SPH) techniques [199
], which he applied
to relativistic spherical collapse [174
]. When tested
against 1D relativistic shock tubes all these codes performed
similar to the code of Wilson. More recently, Dean et
al. [69
] have applied flux
correcting algorithms for the SRHD equations in the context of
heavy ion collisions. Recent developments in relativistic SPH
methods [53
, 261
] are discussed in
Section 4.2.
A major breakthrough in the simulation of
ultra-relativistic flows was accomplished when high-resolution
shock-capturing (HRSC) methods, specially designed to solve
hyperbolic systems of conservations laws, were applied to solve the
SRHD equations [179, 176
, 83
, 84
].