Implicit Asymptotic Preserving Method for Linear Transport Equations
  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. 
Qin Li and Li Wang
math.NA