The Navier-Stokes Equation Associated with the Discrete Boltzmann Equation

This chapter is a summary of the work concerning the Navier-Stokes equation derived from the discrete Boltzmann equation. It considers a model of gas whose molecular velocities are restricted to a set of m constant vectors v₁,v_m in IRⁿ. The purpose of this chapter is to study the hydrodynamical equations derived from the discrete Boltzmann equation by applying the Chapman-Enskog method. This chapter shows that the Navier-Stokes equation is transformed into a symmetric system by change of the dependent variable. It is known that the Navier-Stokes equation can be transformed into a coupled system of a symmetric hyperbolic system and a symmetric strongly parabolic system, by changing the dependent variable from w t o u.