  A thermal lattice Boltzmann model is constructed on the basis of the
ellipsoidal statistical Bhatnagar-Gross-Krook (ES-BGK) collision operator via
the Hermite moment representation. The resulting lattice ES-BGK model uses a
single distribution function and features an adjustable Prandtl number.
Numerical simulations show that using a moderate discrete velocity set, this
model can accurately recover steady and transient solutions of the ES-BGK
equation in the slip-flow and early transition regimes in the small Mach number
limit that is typical of microscale problems of practical interest. In the
transition regime in particular, comparisons with numerical solutions of the
ES-BGK model, direct Monte Carlo and low-variance deviational Monte Carlo
simulations show good accuracy for values of the Knudsen number up to
approximately 0.5. On the other hand, highly non-equilibrium phenomena
characterized by high Mach numbers, such as viscous heating and force-driven
Poiseuille flow for large values of the driving force, are more difficult to
capture quantitatively in the transition regime using discretizations chosen
with computational efficiency in mind such as the one used here, although
improved accuracy is observed as the number of discrete velocities is
increased.
