  Among the existing models for compressible fluids, the one by Kataoka and
Tsutahara (KT model, Phys. Rev. E 69, 056702, 2004) has a simple and rigorous
theoretical background. The drawback of this KT model is that it can cause
numerical instability if the local Mach number exceeds 1. The precise mechanism
of this instability has not yet been clarified. In this paper, we derive
entropy functions whose local equilibria are suitable to recover the Euler-like
equations in the framework of the lattice Boltzmann method for the KT model.
Numerical examples are also given, which are consistent with the above
theoretical arguments, and show that the entropy condition is not fully
guaranteed in KT model. The negative entropy may be the inherent cause for the
non-physical oscillations in the vicinity of the shock. In contrast to these
Karlin's microscopic entropy approach, the corresponding subsidiary entropy
condition in the LBM calculation could also be deduced explicitly from the
macroscopic version, which provides some insights on the numerical instability
of the lattice Boltzmann model for shock calculation.
