Abstract The current finite difference lattice Boltzmann method (FDLBM) gives a fixed specific heat ratio because internal energy is limited to the translational freedom of the space. Yan et al. and Kataoka et al. clarified the conditions for deriving models with arbitrary specific heat ratio and proposed Euler models. However, these model applications to numerical simulations showed the weakness in the numerical stability. In this paper, a two-dimensional FDLBM Navier Stokes model and a three-dimensional FDLBM Euler model, which allow arbitrary values to be set for the specific heat ratio, were proposed. These models stably performed numerical simulations from subsonic to supersonic ranges.