Advances have been made in the area of Monte Carlo methods for this problem. Encouraged by the success and robustness of implicit Monte Carlo techniques in local thermodynamic equilibrium, an implicit method was developed for line transport. While Implicit Monte Carlo (IMC)[3] works very well for most cases, it contains an effective scattering term that is inefficient when dealing with optically thick problems and that becomes negative for an atomic line in the gain regime. The Symbolic Implicit Monte Carlo (SIMC) method [4] was born in the notion that you can track and score spontaneously emitted particles with an unknown symbolic weight that is determined at the end of a time step. This removes the source of the ineffiency, the effective scattering term, from the IMC method.
The extension of the SIMC method to thermal radiation in local thermodynamic equilibrium was first published in Ref. [5]. A key feature of that work was noting that, once effective scattering is removed, one may track particles containing weight vectors in frequency space instead of using statistical sampling. Little demonstration of the advantage of weight vectors is offered in Ref. [5]. We demonstrate in this paper that a significant advantage results if one desires spectral information from the problem output.
The stability and accuracy of the IMC and SIMC methods have been thoroughly analyzed by N'Kaoua and Sentis [6], for the case of linear transport. Their analysis applies directly to the line transport problems considered in this paper.
The goal of this paper is to compare the three methods (IMC, SIMC and SIMC with weight vectors in frequency space) for slab geometry. The codes that implement these methods model a two level atom in slab geometry with collisional coupling between levels and with incident radiation. We study the methods for a variety of problems in an attempt to compare and contrast the techniques under a wide range of conditions. We look at computational efficiency, accuracy and convergence of results as a function of time step, zone size and variable width zoning strategies. Sensitivity to spatial biasing for problems involving high opacity, which can have a significant impact on computational efficiency, is also explored.
Three test problems of study are presented in this paper. The first problem examines the performance of the methods for high opacity. The second test problem studies an opaque cold slab heated by monochromatic incident radiation that is off line center. The third test problem looks at a case where the only source of photons is a central zone in a slab that is collisionally pumped. This test problem has a partial analytical solution that is used to verify the accuracy of the numerical methods.
In this investigation, we demonstrate that the SIMC method does well in reducing noise for high opacity problems, and that the weight vector extension to SIMC provides a further substantial reduction in noise for problems where spectral information is desired. We also show that a geometric progression of zone sizes near an interface, with spatial biasing for spontaneous emission, is effective in improving the performance of these methods. Finally, we show that IMC is less susceptible to teleportation error than SIMC, but that this advantage evaporates as the time step size is reduced in order to obtain better temporal accuracy.