Numerical solution of the unsteady radiative transfer equation

Abstract

In problems involving the transport of radiation through matter, it is necessary to solve the radiative transfer equation numerically. The scope for analytic solutions is restricted because of the complexity of the radiation-material interaction problem; this is particularly severe if the material is compressible. A study is made of various difference schemes within the framework of the discrete S N method (1) with a view to obtaining a well-behaved solution. In addition to the usual requirements of consistency, stability and accuracy, there are further desirable features. Firstly, it is preferable to keep in the difference scheme, the photon conservation property as expressed by the basic transport equation. Secondly, the solution should be non-negative, not only because such values are unphysical, but also because of their effect on the radiation-material coupling. Such a coupling is usually non-linear, and negative solutions can lead to instabilities in the numerical solution. Finally, it is preferable that the method should lead to simple calculations. Results are given for an example involving the radiative heating of a semi-infinite slab. It is shown that the diffusion limit imposes a particularly severe test, and the method used by Lathrop in the computer code DTF-IV (6) is favoured. A second example demonstrates a disturbing oscillatory behaviour arising from Diamond differencing which is basic to the DTF-IV method. A modification to this method designed to overcome this problem is examined.