  Prediction of non-equilibrium flows is critical to space flight. In the
present work we demonstrate that the recently developed spectral
multiple-relaxation-time (SMRT) lattice Boltzmann (LB) model is theoretically
equivalent to Grad's eigen-system [Grad, H., In Thermodynamics of Gases (1958)]
where the eigen-functions obtained by tensor decomposition of the Hermite
polynomials are also those of the linearized Boltzmann equation. Numerical
results of shock structure simulation using Maxwell molecular model agree very
well with those of a high-resolution fast spectral method code up to Mach 7
provided that the relaxation times of the irreducible tensor components match
their theoretical values. If a reduced set of relaxation times are used such as
in the Shakhov model and lumped-sum relaxation of Hermite modes, non-negligible
discrepancies starts to occur as Mach number is raised, indicating the
necessity of the fine-grained relaxation model. Together with the proven
advantages of LB, the LB-SMRT scheme offers a competitive alternative for
non-equilibrium flow simulation.
