  The computation of the radiative transfer equation is expensive mainly due to
two stiff terms: the transport term and the collision operator. The stiffness
in the former comes from the fact that particles (such as photons) travels at
the speed of light, while that in the latter is due to the strong scattering in
the diffusive regime. We study the fully implicit scheme for this equation to
account for the stiffness. The main challenge in the implicit treatment is the
coupling between the spacial and velocity coordinates that requires the large
size of the to-be-inverted matrix, which is also ill-conditioned and not
necessarily symmetric. Our main idea is to utilize the spectral structure of
the ill-conditioned matrix to construct a pre-conditioner, which, along with an
exquisite split of the spatial and angular dependence, significantly improve
the condition number and allows matrix-free treatment. We also design a fast
solver to compute this pre-conditioner explicitly in advance. Meanwhile, we
reformulate the system via an even-odd parity, which results in a symmetric and
positive definite matrix that can be inverted using conjugate gradient method.
This idea can also be implemented to the original non-symmetric system whose
inversion is solved by GMRES. A qualitative comparison with the conventional
methods, including Krylov iterative method pre-conditioned with diffusive
synthetic acceleration and asymptotic preserving scheme via even-odd
decomposition, is also discussed.
