Thermal lattice Boltzmann method based on a theoretically simple derivation of the Taylor expansion

We propose an approach to derive the thermal lattice Boltzmann method that is based on the Taylor expansion in variables of temperature as well as velocity and a direct calculation upon the Gaussian quadrature based hypothesis. This enables us to overcome the isothermal limitation and the low-order accuracy simultaneously. A systematic framework is explained for constructing numerically stable lattice Boltzmann models. The stability of this one-dimensional lattice Boltzmann model is demonstrated with a shock tube simulation.