Radiation Transport in Scattering Media

The production, processing, and also the use of glass and ceramic materials as fireproof materials are considerably influenced by processes of radiative transfer. The cooling of hot glass melts essentially happens due to radiation.

In life sciences, photon and electron beams have already been applied for some time now for diagnostic and therapeutic purposes. Computer tomography in cancer diagnostics or dosimetry in cancer therapy, e.g., are acknowledged therapy methods. Optical tomography is explored worldwide in order to detect structures and properties of the human tissue.

Equation Describes Radiative Transfer

Such and similar processes from the fields of industry and medicine are described by the radiative transfer equation - an integro-differential equation. In the case of isotropic respectively linearly anisotropic scattering, this integro-differential equation can be transformed into an integral equation. Apart from the smaller number of unknowns, such a formulation as integral equation has enormous advantages with respect to the numerical solution (secured stability, symmetry, well-conditioned systems, application of fast iteration methods).

However, the discretization of the integral equation results in large, non-sparse systems of equations. It has been shown that, in the case of a very strong scattering, the resulting system of equations is ill-conditioned (small errors, e.g., truncation errors, can falsify the solution extremely), requiring adequate preconditioners.

Integral Formulation, Matrix Compression, Discrete Ordinate Method

Due to the non-sparse matrix, it does not make sense in practical applications to store the matrix in its original form. Matrix compression methods, as they are used in the case of boundary element methods, appear as efficient solution methods and have been applied to radiative transfer.

Due to the exponential damping, the efficiency of this solution method in the case of radiative transfer is strongly depending on optical thickness. The method, which has been developed on the basis of the integral formulation, has been compared with the Discrete Ordinate (DO) Method, which is currently very popular in publications. In the case of an optically thick and strongly scattering medium, the formulation as integral equation is not recommendable. However, in the case of an optically thin medium, where the DO method requires a large number of discrete directions, thus becoming inefficient, the integral formulation in combination with the matrix compression leads to very good results.

In the case of local heat sources, the integral formulation is also superior to the DO method and should be considered as an alternative for the solution of practical problems.