  In this paper, we investigate the recently developed lattice Boltzmann model
for relativistic hydrodynamics. To this purpose, we perform simulations of
shock waves in quark-gluon plasma in the low and high viscosities regime, using
three different computational models, the relativistic lattice Boltzmann (RLB),
the Boltzmann Approach Multi-Parton Scattering (BAMPS), and the viscous sharp
and smooth transport algorithm (vSHASTA). From the results, we conclude that
the RLB model departs from BAMPS in the case of high speeds and high
temperature(viscosities), the departure being due to the fact that the RLB is
based on a quadratic approximation of the Maxwell-J\"uttner distribution, which
is only valid for sufficiently low temperature and velocity. Furthermore, we
have investigated the influence of the lattice speed on the results, and shown
that inclusion of quadratic terms in the equilibrium distribution improves the
stability of the method within its domain of applicability. Finally, we assess
the viability of the RLB model in the various parameter regimes relevant to
ultra-relativistic fluid dynamics.
