  An original spectral study of the compressible hybrid lattice Boltzmann
method (HLBM) on standard lattice is proposed. In this framework, the mass and
momentum equations are addressed using the lattice Boltzmann method (LBM),
while finite difference (FD) schemes solve an energy equation. Both systems are
coupled with each other thanks to an ideal gas equation of state. This work
aims at answering some questions regarding the numerical stability of such
models, which strongly depends on the choice of numerical parameters. To this
extent, several one- and two-dimensional HLBM classes based on different energy
variables, formulation (primitive or conservative), collision terms and
numerical schemes are scrutinized. Once appropriate corrective terms
introduced, it is shown that all continuous HLBM classes recover the
Navier-Stokes Fourier behavior in the linear approximation. However, striking
differences arise between HLBM classes when their discrete counterparts are
analysed. Multiple instability mechanisms arising at relatively high Mach
number are pointed out and two exhaustive stabilization strategies are
introduced: (1) decreasing the time step by changing the reference temperature
$T_{ref}$ and (2) introducing a controllable numerical dissipation $\sigma$ via
the collision operator. A complete parametric study reveals that only HLBM
classes based on the primitive and conservative entropy equations are found
usable for compressible applications. Finally, an innovative study of the
macroscopic modal composition of the entropy classes is conducted. Through this
study, two original phenomena, referred to as shear-to-entropy and
entropy-to-shear transfers, are highlighted and confirmed on standard
two-dimensional test cases.
