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

Abstract

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.

An extension of implicit Monte Carlo diffusion: Multigroup and the difference formulation

“…These instabilities do not occur in SIMC with an implicit treatment of the reference temperature. [9,13]. However, comparable instabilities have been noted in SIMC when the reference temperature is treated explicitly.…”

“…However, comparable instabilities have been noted in SIMC when the reference temperature is treated explicitly. [13] Several improvements to IMD could be explored, including extension of the method to two or three dimensional orthogonal meshes. However, unstructured polyhedral meshes, which are sometimes used in high energy density physics, pose difficult problems because of the possible creation of positive off diagonal elements in the coefficient matrix [1].…”

“…T d can be any temperature, but the stability of the method is heavily dependent on this choice [13]. This is a potentially useful approach because if the radiation is near equilibrium with the matter at temperature T d , then e d will be small.…”

“…If the reference Planckian is set equal to the Planckian of the current material state, computational cost is shifted to regions where the material and photon energy densities are out of equilibrium. The difference formulation has been explored primarily in SIMC and has been shown to signif cantly increase the f gure of merit compared to the standard solution [13,9]. It has also been shown to yield promising results when used with IMC [14], but has not been explored in conjunction with Implicit Monte Carlo Diffusion (IMD) [1] or Discrete Diffusion Monte Carlo (DDMC) [11].…”

Accurate and Efficient Radiation Transport in Optically Thick Media – by Means of the Symbolic Implicit Monte Carlo Method in the Difference Formulation

“…Relaxing the need for implicit treatment of the source terms was one of the hopes of the authors, given that implicit treatment requires the solution of a non-linear system of equations for each time step. Our experiences in this regard, documented in [Daf05], were made even more difficult by the T 4 term for thermal emission. Explicit treatment of the ∂B/∂x source terms was abandoned as a result, but may be worth revisiting in mixed physics applications where the time step size is limited for reasons outside of transport physics.…”