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

Abstract

Implicit Monte Carlo (IMC) and Implicit Monte Carlo Diffusion (IMD) are approaches to the numerical solution of the equations of radiative transfer. IMD was previously derived and numerically tested on grey, or frequency-integrated problems [1]. In this research, we extend Implicit Monte Carlo Diffusion (IMD) to account for frequency dependence, and we implement the difference formulation [2] as a source manipulation variance reduction technique. We derive the relevant probability distributions and present the frequency dependent IMD algorithm, with and without the difference formulation. The IMD code with and without the difference formulation was tested using both grey and frequency dependent benchmark problems. The Su and Olson semi-analytic Marshak wave benchmark was used to demonstrate the validity of the code for grey problems [3]. The Su and Olson semi-analytic picket fence benchmark was used for the frequency dependent problems [4]. The frequency dependent IMD algorithm reproduces the results of both Su and Olson benchmark problems. Frequency group ref nement studies indicate that the computational cost of ref ning the group structure is likely less than that of group ref nement in deterministic solutions of the radiation diffusion methods. Our results show that applying the difference formulation to the IMD algorithm can result in an overall increase in the f gure of merit for frequency dependent problems. However, the creation of negatively weighted particles from the difference formulation can cause signif cant numerical instabilities in regions of the problem with sharp spatial gradients in the solution. An adaptive implementation of the difference formulation may be necessary to focus its use in regions that are at or near thermal equilibrium.

A hybrid transport-diffusion Monte Carlo method for frequency-dependent radiative-transfer simulations

“…(6) that when a particle undergoes a collision, with probability 1 À f n it scatters isotropically with a new frequency sampled according to Eq. (9). It is this effective scattering that represents radiation absorption and re-emission within a time step in IMC.…”

“…Cleveland, Gentile, and Palmer [9] have also recently developed a version of DDMC (which they refer to as Implicit Monte Carlo Diffusion) that accounts for frequency dependence. However, our new DDMC method differs from theirs in two important ways.…”

Mitigating Teleportation Error in Frequency-Dependent Hybrid Implicit Monte Carlo Diffusion Methods

“…HIMCD dynamically applies Implicit Monte Carlo Diffusion (Gentile, 2001;Cleveland et al, 2010) to regions of a problem that are opaque and diffusive while applying standard Implicit Monte Carlo (IMC) (Fleck and Cum mings, 1971) to regions where the diffusion approximation is invalid. Teleportation er ror arises in Monte Carlo simulations when a source is represented with the wrong spatial distribution causing radiation energy to propagate unphysically fast through a material (McKinley et al, 2003).…”

“…Much of this work will build from the large literature base of the IMD and DDMC methods. This includes the previous work on frequency dependent IMD (Cleveland et al, 2010), frequency-dependent Hybrid DDMC (Densmore et al, 2012), and more recent work with regards to HIMCD and Hy brid DDMC with regards to the radiation hydrodynamics equations (Cleveland and Gentile, 2013;Wollaeger et al, 2013).…”

“…DDMC (Densmore et al, 2012) differs primarily in the way that the photons are tracked in time. IMD tracks photons discretely in time based on the temporal discretization, and DDMC tracks photon continuously in time (Cleveland et al, 2010). This work focuses on IMD, but the reader should note that much of what is discussed is similarly applicable to DDMC.…”