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Among the existing models for compressible fluids, the one by Kataoka andTsutahara (KT model, Phys. Rev. E 69, 056702, 2004) has a simple and rigoroustheoretical background. The drawback of this KT model is that it can causenumerical instability if the local Mach number exceeds 1. The precise mechanismof this instability has not yet been clarified. In this paper, we deriveentropy functions whose local equilibria are suitable to recover the Euler-likeequations in the framework of the lattice Boltzmann method for the KT model.Numerical examples are also given, which are consistent with the abovetheoretical arguments, and show that the entropy condition is not fullyguaranteed in KT model. The negative entropy may be the inherent cause for thenon-physical oscillations in the vicinity of the shock. In contrast to theseKarlin's microscopic entropy approach, the corresponding subsidiary entropycondition in the LBM calculation could also be deduced explicitly from themacroscopic version, which provides some insights on the numerical instabilityof the lattice Boltzmann model for shock calculation.

Entropy and weak solutions in the thermal model for the compressible Euler equations