Comparison of implicit and symbolic implicit Monte Carlo line transport with frequency weight vector extension

Abstract

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

“…We discuss the numerical accuracy and efficiency, and report on the numerical stability of each of the three Monte Carlo solution methods we developed in the previous section, with emphasis on exploring the stability characteristics of the fully explicit version, itself free of a matrix solve at the end of each integration cycle, relative to the SIMC treatment of the standard formulation for a range of optical thicknesses. We do not address the issue of teleportation error [9] in this work. For the sake of brevity, we refer to each of the Monte Carlo solution methods we developed above for the difference formulation for atomic line transport as one of a trio of ''difference methods'' and to SIMC for the standard formulation as the ''standard method''.…”

Symbolic implicit Monte Carlo radiation transport in the difference formulation: a piecewise constant discretization

“…A low-order discretization, that keeps the temperature and the opacity of the medium constant in each cell, does not yield the correct diffusion limit under such conditions. This ''teleportation'' effect was discussed in a previous paper [14]. Its deleterious effect can be reduced by using a heuristic known as tilting [16].…”

“…Deviations from a straight line are indicative of boundary layers when they occur within a mean free path or so of a surface, but otherwise they indicate serious problems in the numerical solution. Teleportation errors result in a wrong slope and cause curvature in the interior solution [14]; this is the reason why we limit zone thickness to one mean free path in our piecewise constant treatment of the material temperature.…”

“…The first is the selection of a Monte Carlo technique that is numerically stable and provides correct treatment of the stiff coupling between the radiation and the material in the thick limit. Several authors have shown that the radiation-matter coupling is properly treated by the symbolic implicit Monte Carlo method (SIMC) [10,11], producing a correct implicit solution of the radiation field and the material temperature at the end of a time step [12], while effective scattering techniques [5,13,14] possess a significant deficiency in this regard.…”

Asymptotic diffusion limit of the symbolic Monte-Carlo method for the transport equation

“…We prove in this paper that the original method gives a wrong solution in those regions and we improve the SIMC method to obtain a consistent discretization of the diffusion equation. The difficulty of obtaining the correct diffusion behavior with a Monte-Carlo method has been known for a long time (see [8] for a recent contribution): we present a rigorous analysis of this weakness.…”