Second-order approximation of extended thermodynamics of a monatomic gas and hyperbolicity region

The rational extended thermodynamics theory describes non-equilibrium phenomena for rarefied gases, and it is usually approximated in the neighborhood of an equilibrium state. Consequently, the hyperbolicity of its differential system holds only in some domain of the state variables (called hyperbolicity region). In this paper, we present a second-order approximation with respect to non-equilibrium variables, in the case of a monatomic gas theory with 13 fields. We verify that, in the case of one-dimensional space, the radius of the hyperbolicity region is larger than the corresponding radius of the first-order approximation. Moreover, when the model involves three-dimensional field variables, we prove that the equilibrium state for differential systems with quadratic approximation is inside the hyperbolicity region. This fact is in contrast with the first-order models that, in some cases of three-dimensional field variables, present the equilibrium point at the boundary of the hyperbolicity region.