The radiative transfer equation (RTE) arises in many different areas of science and engineering. In this paper, we propose and investigate a discrete-ordinate discontinuous-streamline diffusion (DODSD) method for solving the RTE, which is a combination of the discrete-ordinate technique and the discontinuous-streamline diffusion method. Different from the discrete-ordinate discontinuous Galerkin (DODG) method for the RTE, an artificial diffusion parameter is added to the test functions in the spatial discretization. Stability and error estimates in certain norms are proved. Numerical results show that the proposed method can lead to a more accurate approximation in comparison with the DODG method.

For the classic diffusion description of radiative transfer, the specific intensity can be represented by a small angular deviation of the local Planckian equilibrium. In a transparent media, the angular anisotropy becomes strong and one has to solve the general transfer equation. We propose a hierarchy of models that can describe the regime that lies between those two limits. Every member of this family is hyperbolic, flux-limited, and possesses a locally dissipated entropy. This hierarchy also formally recovers the diffusion limit. This study demonstrates that the two-polynomial model is already capable of capturing strong anisotropies.

This paper provides mathematical analysis of optical tomography in a situation when the examined object, for example the human brain, is strongly scattering with non-scattering inclusions. Light propagation in biological tissue is often modelled by the diffusion approximation of the radiative transfer equation. To be justified, the diffusion approximation demands that the medium is strongly scattering. Naturally, this is not true for non-scattering inclusions, for which some other model is needed. This is found through geometrical optics. Combination of the two models leads to an elliptic partial differential equation with boundary conditions on the outer boundary as well as on the boundaries of the non-scattering regions. The well-posedness of this forward problem is the main concern of this work.

AMS 2000 Mathematics subject classification: Primary 35Q60; 35R30