High-order quadrature-based lattice Boltzmann models for the flow of   ultrarelativistic rarefied gases
  We present a systematic procedure for the construction of relativistic lattice Boltzmann models (R-SLB) appropriate for the simulation of flows of massless particles. Quadrature rules are used for the discretization of the momentum space in spherical coordinates. The models are optimized for one-dimensional flows. The applications considered in this paper are the Sod shock tube and the one-dimensional boost invariant expansion (Bjorken flow). Our models are tested against exact solutions in the inviscid and ballistic limits. At intermediate relaxation times (finite viscosity), we compare with the results obtained using the Boltzmann approach of multiparton scattering model for the Sod shock tube problem, as well as with a semi-analytic solution for the non-ideal Bjorken flow. In all cases our models give remarkably good results. We define a convergence test in order to find the quadrature order necessary to obtain convergence at a predefined accuracy. We show that, while in the hydrodynamic regime the number of velocities is comparable to that required for the more popular collision-streaming type models, as we go towards the ballistic regime, the size of the velocity set must be substantially increased in order to accurately reproduce the analytic profiles. 
Victor E. Ambrus and Robert Blaga
physics.flu-dyn