  We present a new method for the numerical solution of the radiative-transfer
equation (RTE) in multidimensional scenarios commonly encountered in
computational astrophysics. The method is based on the direct solution of the
Boltzmann equation via an extension of the Lattice Boltzmann (LB) equation and
allows to model the evolution of the radiation field as it interacts with a
background fluid, via absorption, emission, and scattering. As a first
application of this method, we restrict our attention to a frequency
independent ("grey") formulation within a special-relativistic framework, which
can be employed also for classical computational astrophysics. For a number of
standard tests that consider the performance of the method in optically thin,
optically thick and intermediate regimes with a static fluid, we show the
ability of the LB method to produce accurate and convergent results matching
the analytic solutions. We also contrast the LB method with commonly employed
moment-based schemes for the solution of the RTE, such as the M1 scheme. In
this way, we are able to highlight that the LB method provides the correct
solution for both non-trivial free-streaming scenarios and the intermediate
optical-depth regime, for which the M1 method either fails or provides
inaccurate solutions. When coupling to a dynamical fluid, on the other hand, we
present the first self-consistent solution of the RTE with LB methods within a
relativistic-hydrodynamic scenario. Finally, we show that besides providing
more accurate results in all regimes, the LB method features smaller or
comparable computational costs compared to the M1 scheme.
