  A comprehensive characterization of lattice Boltzmann (LB) schemes to perform
warm fluid numerical simulations of particle wakefield acceleration (PWFA)
processes is discussed in this paper. The LB schemes we develop hinge on the
moment matching procedure, allowing the fluid description of a warm
relativistic plasma wake generated by a driver pulse propagating in a neutral
plasma. We focus on fluid models equations resulting from two popular closure
assumptions of the relativistic kinetic equations, i.e., the local equilibrium
and the warm plasma closure assumptions. The developed LB schemes can thus be
used to disclose insights on the quantitative differences between the two
closure approaches in the dynamics of PWFA processes. Comparisons between the
proposed schemes and available analytical results are extensively addressed.
