Spectral Lagrangian  methods for  non-linear Boltzmann type eqautions
Irene Gamba,UT Austin


We present a deterministic spectral solver for the non-linear Boltzmann Transport Equation (energy conservative and non-conservative) for rather general collision kernels. The computation of the non-linear Boltzmann Collision integral and the lack of appropriate conservation properties due to spectral methods has been taken care by framing the conservation properties in the form of a constrained minimization problem which is solved easily using a Lagrange multiplier method. We benchmark our code with several examples of models for Maxwell type of interactions, (elastic or inelastic) for which explicit solution formulas are known. The numerical moments are compared with exact moments formulas and the numerical non-equilibrium probability distributions functions are compared to the general asymptotic results. In the case of space inhomogeneous boundary value problems, the numerical method captures the discoutinuous behahior of
the probability distribution functions solution of the BTE with diffusive boundary conditions and sudden changes in boundary temperature, as predicted by Y. Sone for solutions of  the BTE and computed by Aoki, et al .91, using alternative models.