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.

💡 Research Summary

The paper addresses two fundamental shortcomings of the recently proposed relativistic lattice Boltzmann (LB) model: (1) the incorrect form of the particle‑number conservation law and (2) the appearance of spurious discontinuities in high‑viscosity relativistic flows. By performing a Chapman‑Enskog expansion the authors demonstrate that, contrary to the expected continuity equation (∂ₜn + ∇·(n u)=0), the LB scheme yields a convection‑diffusion equation (∂ₜn + ∇·(n u)=D ∇²n) where the diffusion coefficient D is proportional to the single relaxation time τ. Consequently, the particle number is not strictly conserved; its evolution depends on τ, and artificial diffusion becomes severe when τ is large (i.e., in the high‑viscosity regime).

To investigate the origin of the discontinuities reported in earlier work (Hupp et al., Phys. Rev. D 84, 125015, 2011), the authors develop a multiple‑relaxation‑time (MRT) relativistic LB model. In the MRT framework each kinetic moment—associated with shear viscosity, bulk viscosity, and energy transport—has its own relaxation time: τ_v (shear), τ_e (bulk), and τ_ε (energy). Numerical experiments on relativistic shock‑tube and shear‑flow problems reveal that setting the bulk‑viscosity relaxation τ_e significantly smaller than the shear‑viscosity relaxation τ_v (e.g., τ_e ≈ 0.1 τ_v) eliminates the unphysical jumps in pressure and density that plague the single‑relaxation‑time (SRT) formulation. The authors attribute this to the suppression of an artificial bulk‑viscous pressure wave that otherwise amplifies with increasing τ.

Although τ_ε does not appear in the Navier‑Stokes‑level conservation equations, the study finds that it influences numerical accuracy. Large τ_ε values distort the temperature and energy distribution, leading to higher global errors. Therefore, τ_ε must be tuned alongside τ_e and τ_v to achieve optimal performance.

The paper further evaluates the model’s accuracy for moderately relativistic flows (Lorentz factor γ≈1.2–2). With the optimized relaxation‑time hierarchy (τ_e ≪ τ_v, moderate τ_ε), the MRT LB results agree with high‑resolution finite‑difference solutions within 5 % relative error for density, velocity, and pressure fields. In more extreme relativistic regimes (γ≈5) the method still outperforms the original SRT LB, though residual discretization errors remain.

In summary, the authors (i) expose a fundamental flaw in the particle‑number conservation of the original relativistic LB model, (ii) propose an MRT extension that decouples bulk and shear viscous effects, (iii) demonstrate that a small bulk‑viscosity relaxation time removes spurious discontinuities, and (iv) show that careful selection of τ_e, τ_v, and τ_ε yields accurate simulations for moderately relativistic problems. These findings provide a practical guideline for researchers employing LB techniques in relativistic hydrodynamics, especially in astrophysical and high‑energy‑density contexts where high viscosity and strong gradients are common.