A thermodynamic lattice Boltzmann model whose equilibrium distribution function include high order terms of velocity and internal energy terms is presented, and a new kind of implement of gravity and temperature boundary is also proposed by which thermodynamics phenomenon permits to be simulated. Based on the HPP model, a three-overlapped HPP lattice Boltzmann model is introduced. The Benard pattern of thermal convection phenomenon has been simulated successfully by using this model on SUN workstation computer.

Lattice method is a new method of numerical simulation for fluid dynamics developed recently.Its essential idea is the model-rebuilding.Based on this method two models are composed for linear and quasi-linear parabolic equations.Several typical numerical examples are studied using the models and satisfactory results are achieved.