The equations of relativistic hydrodynamics can be obtained from
the Boltzmann equation via the Chapman-Enskog (CE) procedure and
Grad’s 14 moments approximation. These approaches give different
results for the transport coefficients, which reduce to the same
expressions in the non-relativistic limit.
In this contribution, the propagation of a harmonic longitudinal
wave is considered in the frame of the first- and second-order
relativistic hydrodynamics theories. The ensuing hydrodynamic
equations are solved in the linearized regime (valid for small
wave amplitudes). The analytic solutions corresponding to the
CE and Grad transport coefficients are compared to the numerical
solution of the relativistic Boltzmann equation for massless
particles in the Anderson-Witting (AW) approximation, obtained
using the lattice Boltzmann (LB) method. The comparison clearly
confirms the validity of the CE prediction.
For particular initial conditions, the first-order formulation
gives incorrect predictions for the wave evolution even when
the AW relaxation time is small. This is remedied in the
second-order formulation, the solution of which is confirmed
by numerical simulations.
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