Spectral-Lagrangian solvers for non-linear Boltzmann type
equations: numerics and analysis
Irene Gamba
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.
This work is in collaboration with my ICES student Harsha
Tharkabhushanam.