  The lattice Boltzmann method (LBM) is known to suffer from stability issues
when the collision model relies on the BGK approximation, especially in the
zero viscosity limit and for non-vanishing Mach numbers. To tackle this
problem, two kinds of solutions were proposed in the literature. They consist
in changing either the numerical discretization (finite-volume,
finite-difference, spectral-element, etc) of the discrete velocity Boltzmann
equation (DVBE), or the collision model. In this work, the latter solution is
investigated in details. More precisely, we propose a comprehensive comparison
of (static relaxation time based) collision models, in terms of stability, and
with preliminary results on their accuracy, for the simulation of isothermal
high-Reynolds number flows in the (weakly) compressible regime. It starts by
investigating the possible impact of collision models on the macroscopic
behavior of stream-and-collide based D2Q9-LBMs, which clarifies the exact
physical properties of collision models on LBMs. It is followed by extensive
linear and numerical stability analyses, supplemented with an accuracy study
based on the transport of vortical structures over long distances. In order to
draw conclusions as general as possible, the most common moment spaces (raw,
central, Hermite, central Hermite and cumulant), as well as regularized
approaches, are considered for the comparative studies. LBMs based on dynamic
collision mechanisms (entropic collision, subgrid scale models, explicit
filtering, etc) are also briefly discussed.
