  We develop a relativistic lattice Boltzmann model capable of describing
relativistic fluid dynamics at ultra-high velocities, with Lorentz factors up
to $\gamma \sim 10$. To this purpose, we first build a new lattice kinetic
scheme by expanding the Maxwell-J\"uttner distribution function in an
orthogonal basis of polynomials and applying an appropriate quadrature,
providing the discrete versions of the relativistic Boltzmann equation and the
equilibrium distribution. To achieve ultra-high velocities, we include a flux
limiter scheme, and introduce the bulk viscosity by a suitable extension of the
discrete relativistic Boltzmann equation. The model is validated by performing
simulations of shock waves in viscous quark-gluon plasmas and comparing with
existing models, finding very good agreement. To the best of our knowledge, we
for the first time successfully simulate viscous shock waves in the highly
relativistic regime. Moreover, we show that our model can also be used for
near-inviscid flows even at very high velocities. Finally, as an astrophysical
application, we simulate a relativistic shock wave, generated by, say, a
supernova explosion, colliding with a massive interstellar cloud, e.g.
molecular gas.
