In [Turek, S.: A Generalized Mean Intensity Approach for the Numerical Solution of the Radiative Transfer Equation, Preprint, 1994], the concept of the generalized mean intensity has
been proposed as a special numerical approach to the (linear) radiative transfer
equations which can result in a significant reduction of the dimension of the discretized
system, without eliminating any information for the specific intensities. Moreover, in
combination with Krylov-space methods (CG, Bi-CGSTAB, etc.), robust and very efficient
solvers as extensions of the classical approximate Lambda-iteration have been
developed. In this paper, the key tool is the combination of special renumbering
techniques together with finite difference-like discretization strategies for the arising
transport operators which are based on short-characteristic upwinding techniques of
variable order and which can be applied to highly unstructured meshes with locally
varying mesh widths. We demonstrate how such special upwinding schemes can be constructed
of first order, and particularly of second order accuracy, always leading to lower
triangular system matrices. As a consequence, the global matrix assembling can be avoided
(`on-the-fly`), so that the storage cost are almost optimal, and the solution of the
corresponding convection-reaction subproblems for each direction can be obtained very
efficiently. As a further consequence, this approach results in a direct solver in the
case of no scattering, while in the case of non-vanishing absorption and scattering
coefficients the resulting convergence rates for the full systems depend only on their
ratio and the absolute size of these physical quantities, but not on the grid size or
mesh topology. We demonstrate these results via prototypical configurations and we
examine the resulting accuracy and efficiency for different computational domains, meshes
and problem parameters.