Four Decades of Implicit Monte Carlo

Abstract

In 1971, Fleck and Cummings derived a system of equations to enable robust Monte Carlo simulations of time-dependent, thermal radiative transfer problems. Denoted the "Implicit Monte Carlo" (IMC) equations, their solution remains the de facto standard of high-fidelity radiative transfer simulations. Over the course of 44 years, their numerical properties have become better understood, and accuracy enhancements, novel acceleration methods, and variance reduction techniques have been suggested. In this review, we rederive the IMC equations-explicitly highlighting assumptions as they are made-and outfit the equations with a Monte Carlo interpretation. We put the IMC equations in context with other approximate forms of the radiative transfer equations and present a new demonstration of their equivalence to another well-used linearization solved with deterministic transport methods for frequency-independent problems. We discuss physical and numerical limitations of the IMC equations for asymptotically small time steps, stability characteristics and the potential of maximum principle violations for large time steps, and solution behaviors in an asymptotically thick diffusive limit. We provide a new stability analysis for opacities with general monomial dependence on temperature. We consider spatial accuracy limitations of the IMC equations and discussion acceleration and variance reduction techniques.

A new Implicit Monte-Carlo scheme for photonics (without teleportation error and without tilts)

“…To be more precise, in this paper, we are even interested in being able to accurately capture a particular regime: in diffusive media, system (1) 4π dω = aT 4 m and Φ r (T r ) = aT 4 r , the second equation is equivalent to T m = T r : the radiative and matter temperatures are at equilibrium. In the above equation, δ ∼ 0 is a small parameter characterising what is commonly called the equilibrium 1 diffusion 2 limit [2,3,4]. The limit can be defined by introducing a characteristic length X , a characteristic time T and a characteristic collision rate λ and assuming we have…”

“…System (1) and its limit (4) are relevant to model photons incoming into cold media [1,5,6]. At this stage of the discussion, the problem of interest in this paper may seem to lack generality in the sense we here focus on what is commonly called the grey approximation 3 with isotropic scattering 4 and the opacities are independent of the temperature 5 . Still, it is enough to focus on the main numerical difficulties we aim at considering in this paper: a high-dimensional (6 dimensions in 3D) and highly nonlinear (via B ∝ T 4 m ) system (1) degenerating toward (4) in diffusive media.…”

Light Curves and Spectra from a Unimodal Core-collapse Supernova

“…The SuperNu code has an implementation of Implicit Monte Carlo (IMC) (Fleck & Cummings 1971;Wollaber 2016) for thermal radiative transfer, and Discrete Diffusion Monte Carlo (DDMC) (Densmore et al 2007(Densmore et al , 2012Abdikamalov et al 2012) to accelerate IMC in optically thick regions of phase space. SuperNu has features specialized for homologous outflows and structured opacity (Wollaeger et al 2013;Wollaeger & Van Rossum 2014).…”

Monte Carlo radiative transfer

“…However, these conditions are very restrictive and limit the applicability of IMC. More information about the maximum principle violation, and about efforts to alleviate it within the IMC framework as well as other drawbacks, such as accurately reproducing the diffusion limit, the introduction of damped oscillations or teleportation errors, are summarized by Wollaber (2016).…”