  In this paper a spectral-Lagrangian method for the Boltzmann equation for a
multi-energy level gas is proposed. Internal energy levels are treated as
separate species and inelastic collisions (leading to internal energy
excitation and relaxation) are accounted for. The formulation developed can
also be used for the case of a mixture of monatomic gases without internal
energy (where only elastic collisions occur). The advantage of the
spectral-Lagrangian method lies in the generality of the algorithm in use for
the evaluation of the elastic and inelastic collision operators. The
computational procedure is based on the Fourier transform of the partial
elastic and inelastic collision operators and exploits the fact that these can
be written as weighted convolutions in Fourier space with no restriction on the
cross- section model. The conservation of mass, momentum and energy during
collisions is enforced through the solution of constrained optimization
problems. Numerical solutions are obtained for both space homogeneous and space
in- homogeneous problems. Computational results are compared with those
obtained by means of the DSMC method in order to assess the accuracy of the
proposed spectral-Lagrangian method.
