On robustly convergent and efficient iterative methods for anisotropic radiative transfer

This paper considers the iterative solution of linear systems arising from discretization of the anisotropic radiative transfer equation with discontinuous elements on the sphere. In order to achieve robust convergence behavior in the discretization parameters and the physical parameters we develop preconditioned Richardson iterations in Hilbert spaces. The preconditioner is constructed in two steps where the first step ensures mesh independence while the second step aims at reducing the influence of the optical parameters. We show convergence of the resulting scheme when applied to the second-order formulation of the radiative transfer equation, viz. the even-parity equations. We discuss in detail the efficient implementation and application of the discrete operators. In particular, for the considered discontinuous spherical elements, the scattering operator becomes dense and we show that $\mathcal{H}$- or $\mathcal{H}^2$-matrix compression can be applied in a black-box fashion to obtain almost linear or linear complexity when applying the corresponding approximations. The effectiveness of the proposed method is shown in numerical examples.