  A univariate polynomial equation is presented. It provides models of the
thermal lattice Boltzmann equation. The models can be accurate up to any
required level and can be applied to regular lattices, which allow efficient
and accurate approximate solutions of the Boltzmann equation. We derive models
satisfying the complete Galilean invariant and providing accuracy of the
4th-order moment and beyond. We simulate thermal shock tube problems to
illustrate the accuracy of our models and to show the remarkably enhanced
stability obtained by our models and our discretized equilibrium distributions.
