Generalized Orthogonal Polynomial Solutions of the Boltzmann Equation

Solutions of the Boltzmann equation of the form f(r, v, t) = f(0)(r, v, t) Σ ap(r, t)Mp(v; r, t) are considered, where f(0) is some zero-order approximation to f. It is shown that, to obtain reasonable equations for the ap and to relate them to the moments of f, the Mp should be a polynomial set in velocity-space, orthogonalized with f(0) as the weighting factor. It is shown how the Burnett method and the Grad method are derivable when the problem is a near-equilibrium one; and how the half-range polynomial method results when particular symmetry conditions are imposed. An illustration of the method is given for the case of the equilibration of a two-temperature gas mixture.