  We develop a relativistic lattice Boltzmann (LB) model, providing a more
accurate description of dissipative phenomena in relativistic hydrodynamics
than previously available with existing LB schemes. The procedure applies to
the ultra-relativistic regime, in which the kinetic energy (temperature) far
exceeds the rest mass energy, although the extension to massive particles
and/or low temperatures is conceptually straightforward. In order to improve
the description of dissipative effects, the Maxwell-Juettner distribution is
expanded in a basis of orthonormal polynomials, so as to correctly recover the
third order moment of the distribution function. In addition, a time dilatation
is also applied, in order to preserve the compatibility of the scheme with a
cartesian cubic lattice. To the purpose of comparing the present LB model with
previous ones, the time transformation is also applied to a lattice model which
recovers terms up to second order, namely up to energy-momentum tensor. The
approach is validated through quantitative comparison between the second and
third order schemes with BAMPS (the solution of the full relativistic Boltzmann
equation), for moderately high viscosity and velocities, and also with previous
LB models in the literature. Excellent agreement with BAMPS and more accurate
results than previous relativistic lattice Boltzmann models are reported.
