Lattice Boltzmann method for relativistic hydrodynamics: Issues on   conservation law of particle number and discontinuities
  In this paper, we aim to address several important issues about the recently developed 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 the relativistic LB model. Through the Chapman-Enskog analysis, it is shown that in the relativistic LB model the conservation equation of particle number is a convection-diffusion equation rather than a continuity equation, which makes the evolution of particle number dependent on the relaxation time. Furthermore, we investigate the origin of the discontinuities appeared in the relativistic problems with high viscosities, which were reported in a recent study [D. Hupp et al., Phys. Rev. D 84, 125015 (2011)]. A multiple-relaxation-time (MRT) relativistic LB model is presented to examine the influences of different relaxation times on the discontinuities. Numerical experiments show the discontinuities can be eliminated by setting the relaxation time $\tau_e$ (related to the bulk viscosity) to be sufficiently smaller than the relaxation time $\tau_v$ (related to the shear viscosity). Meanwhile, it is found that the relaxation time $\tau_\varepsilon$, which has no effect on the conservation equations at the Navier-Stokes level, will affect the numerical accuracy of the relativistic LB model. Moreover, the accuracy of the relativistic LB model for simulating moderately relativistic problems is also investigated. 
Q. Li, K. H. Luo, and X. J. Li
physics.comp-ph gr-qc