  We present new results building on the conservative deterministic spectral
method for the space homogeneous Boltzmann equation developed by Gamba and
Tharkabhushaman. This approach is a two-step process that acts on the weak form
of the Boltzmann equation, and uses the machinery of the Fourier transform to
reformulate the collisional integral into a weighted convolution in Fourier
space. A constrained optimization problem is solved to preserve the mass,
momentum, and energy of the resulting distribution.
  Within this framework we have extended the formulation to the case of more
general case of collision operators with anisotropic scattering mechanisms,
which requires a new formulation of the convolution weights. We also derive the
grazing collisions limit for the method, and show that it is consistent with
the Fokker-Planck-Landau equations as the grazing collisions parameter goes to
zero.
