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In this paper, we aim to address several important issues about the recentlydeveloped lattice Boltzmann (LB) model for relativistic hydrodynamics [M.Mendoza et al., Phys. Rev. Lett. 105, 014502 (2010); Phys. Rev. D 82, 105008(2010)]. First, we study the conservation law of particle number in therelativistic LB model. Through the Chapman-Enskog analysis, it is shown that inthe relativistic LB model the conservation equation of particle number is aconvection-diffusion equation rather than a continuity equation, which makesthe evolution of particle number dependent on the relaxation time. Furthermore,we investigate the origin of the discontinuities appeared in the relativisticproblems with high viscosities, which were reported in a recent study [D. Huppet al., Phys. Rev. D 84, 125015 (2011)]. A multiple-relaxation-time (MRT)relativistic LB model is presented to examine the influences of differentrelaxation times on the discontinuities. Numerical experiments show thediscontinuities can be eliminated by setting the relaxation time $\tau_e$(related to the bulk viscosity) to be sufficiently smaller than the relaxationtime $\tau_v$ (related to the shear viscosity). Meanwhile, it is found that therelaxation time $\tau_\varepsilon$, which has no effect on the conservationequations at the Navier-Stokes level, will affect the numerical accuracy of therelativistic LB model. Moreover, the accuracy of the relativistic LB model forsimulating moderately relativistic problems is also investigated.

Lattice Boltzmann method for relativistic hydrodynamics: Issues on conservation law of particle number and discontinuities