An evaluation of the difference formulation for photon transport in a two level system

In this paper, we extend the difference formulation for radiation transport to the case of a single atomic line. We examine the accuracy, performance and stability of the difference formulation within the framework of the Symbolic Implicit Monte Carlo method. The difference formulation, introduced for thermal radiation by some of the authors, has the unique property that the transport equation is written in terms that become small for thick systems. We find that the difference formulation has a significant advantage over the standard formulation for a thick system. The correct treatment of the line profile, however, requires that the difference formulation in the core of the line be mixed with the standard formulation in the wings, and this may limit the advantage of the method. We bypass this problem by using the gray approximation. We develop three Monte Carlo solution methods based on different degrees of implicitness for the treatment of the source terms, and we find only conditional stability unless the source terms are treated fully implicitly.