Frequency-dependent Radiative Transfer Research Articles

Discontinuous Galerkin finite element method for the continuum radiative transfer problem inside axis-symmetric circumstellar envelopes

Context. The study of the continuum radiative transfer problem inside circumstellar envelopes is both a theoretical and numerical challenge, especially in the frequency-dependent and multi-dimensional case. While approximate methods are easier to handle numerically, they often fail to accurately describe the radiation field inside complex geometries. For these cases, it is necessary to directly solve the radiative transfer equation numerically. Aims. We investigate the accuracy of the discontinuous Galerkin finite element method (DGFEM hereafter) applied to the frequency-dependent two-dimensional radiative transfer problem, and coupled with the radiative equilibrium equation. We next used this method in the context of axis-symmetric circumstellar envelopes. Methods. The DGFEM is a variant of finite element methods. It employs discontinuous elements and flux integrals along their boundaries, ensuring local flux conservation. However, as opposed to the classical finite element methods, the solution is discontinuous across element edges. We implemented this approach in a code and tested its accuracy by comparing our results with the benchmarks from the literature. Results. For all the tested cases, the temperatures profiles agree within one percent. Additionally, the emerging spectral energy distributions (SEDs) and images, obtained by ray-tracing techniques from the DGFEM emissivity, agree on average within 5% and 10%, respectively. Conclusions. We show that the DGFEM can accurately describe the continuum radiative transfer problem inside axis-symmetric circumstellar envelopes. Consecutively the emerging SEDs and images are also well reproduced. The DGFEM provides an alternative method (other than Monte-Carlo methods for instance) for solving the radiative transfer equation, and it could be used in cases that are more difficult to handle with the other methods.

A Unified Gas-Kinetic Particle Method for Frequency-Dependent Radiative Transfer Equations with Isotropic Scattering Process on Unstructured Mesh

A Unified Gas-Kinetic Particle Method for Frequency-Dependent Radiative Transfer Equations with Isotropic Scattering Process on Unstructured Mesh

Unified gas-kinetic particle method for frequency-dependent radiation transport

This paper proposes a unified gas-kinetic particle (UGKP) method for the frequency-dependent photon transport process. The photon transport is a typical multiscale process governed by the nonlinear radiative transfer equations (RTE). The flow regime of photon transport varies from the ballistic regime to the diffusive regime with respect to optical depth and photon frequency. The UGKP method is an asymptotic preserving (AP) scheme and a regime adaptive scheme. The teleportation error is significantly reduced, and the computational efficiency is remarkably improved in the diffusive regime. Distinguished from the standard multigroup treatment, the proposed UGKP method solves the frequency space in a non-discretized way. Therefore the Rosseland diffusion system can be precisely preserved in the optically thick regime. Taking advantage of the local integral solution of RTE, the distribution of the emitted photon can be constructed from its macroscopic moments. The Monte Carlo particles in the UGKP method need only be tracked before their first collision events, and a re-sampling process is performed to close the photon distribution for each time step. The high computational cost of calculating excessive scattering events can be saved, especially in the optically thick regime. The proposed UGKP method is implicit and removes the light speed constraint on the time step. The particle tracking approach combining the implicit formulation makes the proposed UGKP method an efficient solution algorithm for frequency-dependent radiative transfer problems. We demonstrate with numerical examples the capability of the proposed multi-frequency UGKP method.

State-of-the-art simulations of line-driven accretion disc winds: realistic radiation hydrodynamics leads to weaker outflows

ABSTRACT Disc winds are a common feature in accreting astrophysical systems on all scales. In active galactic nuclei (AGNs) and accreting white dwarfs (AWDs), specifically, radiation pressure mediated by spectral lines is a promising mechanism for driving these outflows. Previous hydrodynamical simulations have largely supported this idea, but relied on highly approximate treatments of ionization and radiative transfer. Given the sensitivity of line driving to the ionization state and radiation field in the outflow, here, we present a new method for carrying out 2.5D radiation hydrodynamic simulations that takes full account of the frequency-dependent radiative transfer through the wind, the corresponding ionization state, and the resulting radiative accelerations. Applying our method to AWDs, we find that it is much harder to drive a powerful line-driven outflow when the interaction between matter and radiation is treated self-consistently. This conclusion is robust to changes in the adopted system parameters. The fundamental difficulty is that discs luminous enough to drive such a wind are also hot enough to overionize it. As a result, the mass-loss rates in our simulations are much lower than those found in earlier, more approximate calculations. We also show that the ultraviolet spectra produced by our simulations do not match those observed in AWDs. We conclude that, unless the overionization problem can be mitigated (e.g. by subgrid clumping or a softer-than-expected radiation field), line driving may not be a promising mechanism for powering the outflows from AWDs. These conclusions are likely to have significant implications for disc winds in AGN also.

A maximum principle preserving implicit Monte Carlo method for frequency-dependent radiative transfer equations

In this paper, we develop a maximum principle preserving Monte Carlo method for the frequency-dependent (multi-group) radiative transfer equations. In order to deal with the nonlinear coupling between the radiation intensity and background material, Newton's iteration is involved into the discretization of the nonlinear system. We employ the maximum principle of the material temperature as the stopping criterion of the nonlinear iterations. The classical implicit Monte Carlo method can be regarded as one iteration of the new iterative method. We demonstrate that if the nonlinear iteration of the new method converges, it will become a fully implicit discretization of the multi-group radiative transfer equations. Moreover, the radiation temperature will also satisfy the maximum principle under certain conditions. Several frequency-dependent radiative transfer examples are given to show the effectiveness of the new method.

Three-component modelling of O-rich AGB star winds

Stellar winds of cool and pulsating asymptotic giant branch (AGB) stars enrich the interstellar medium with large amounts of processed elements and various types of dust. We present the first study on the influence of gas-to-dust drift on ab initio simulations of stellar winds of M-type stars driven by radiation pressure on forsterite particles. Our study is based on our radiation hydrodynamic model code T-800 that includes frequency-dependent radiative transfer, dust extinction based on Mie scattering, grain growth and ablation, gas-to-dust drift using one mean grain size, a piston that simulates stellar pulsations, and an accurate high spatial resolution numerical scheme. To enable this study, we calculated new gas opacities based on the EXOMOL database, and we extended the model code to handle the formation of minerals that may form in M-type stars. We determine the effects of drift by comparing drift models to our new and extant non-drift models. Three out of four new drift models show high drift velocities, 87–310 km s−1. Our new drift model mass-loss rates are 1.7–13 per cent of the corresponding values of our non-drift models, but compared to the results of two extant non-drift models that use the same stellar parameters, these same values are 0.33–1.5 per cent. Meanwhile, a comparison of other properties such as the expansion velocity and grain size show similar values. Our results, which are based on single-component forsterite particles, show that the inclusion of gas-to-drift is of fundamental importance in stellar wind models driven by such transparent grains. Assuming that the drift velocity is insignificant, properties such as the mass-loss rate may be off from more realistic values by a factor of 50 or more.

A fully asymptotic preserving decomposed multi-group method for the frequency-dependent radiative transfer equations

The opacity of FRTE depends on not only the material temperature but also the frequency, whose values may vary several orders of magnitude for different frequencies. The gray radiation diffusion and frequency-dependent diffusion equations are two simplified models that can approximate the solution to FRTE in the thick opacity regime. The frequency discretization for the two limit models highly affects the numerical accuracy. However, classical frequency discretization for FRTE considers only the absorbing coefficient. In this paper, we propose a new decomposed multi-group method for frequency discretization that is not only AP in both gray radiation diffusion and frequency-dependent diffusion limits, but also the frequency discretization of the limiting models can be tuned. Based on the decomposed multi-group method, a full AP scheme in frequency, time, and space is proposed. Several numerical examples are used to verify the performance of the proposed scheme.

A Unified Gas-Kinetic Particle Method for Frequency-Dependent Radiative Transfer Equations with Isotropic Scattering Process on Unstructured Mesh

A Unified Gas-Kinetic Particle Method for Frequency-Dependent Radiative Transfer Equations with Isotropic Scattering Process on Unstructured Mesh

Self-ionizing Galactic Winds

We present hydrodynamical simulations of galactic winds from star-forming galaxies including nonequilibrium ionization and frequency-dependent radiative transfer, processes that have remained largely unaccounted for in galactic wind studies. We consider radiation from massive stars, the metagalactic UV/X-ray background, and the self-radiation of the supernovae heated gas. We compare our results to classical galactic wind solutions and show the importance of our newly included physical processes toward observations of ions such as O iii, O vi, O vii, and O viii plus the observable soft X-ray spectra. Nonequilibrium ionization is reflected in overionized gas compared to equilibrium solutions, leading to much enhanced column densities of highly ionized species. The wind produces excess soft X-ray (E ≳ 100 eV) radiation that is several orders of magnitude higher compared to the metagalactic background. This radiation ionizes the higher ions (such as O vii) somewhat, but affects the lower ions (such as O iii) significantly. We predict that the observable X-ray spectra should contain the signatures of such nonequilibrium effects, especially in X-ray lines such as O vii and O viii. Simple estimates suggest that both the temperature and density of the winds may be overestimated by factors of a few to almost 2 orders of magnitude using simple equilibrium models. We conclude that both the nonequilibrium ionization and the radiation from the wind itself need to be considered for proper modeling of the optical/UV/X-ray emitting plasma in galactic winds.

Thethesanproject: Lyman-α emission and transmission during the Epoch of Reionization

ABSTRACTThe visibility of high-redshift Lyman-alpha emitting galaxies (LAEs) provides important constraints on galaxy formation processes and the Epoch of Reionization (EoR). However, predicting realistic and representative statistics for comparison with observations represents a significant challenge in the context of large-volume cosmological simulations. The thesan project offers a unique framework for addressing such limitations by combining state-of-the-art galaxy formation (IllustrisTNG) and dust models with the arepo-rt radiation-magnetohydrodynamics solver. In this initial study, we present Lyman-alpha centric analysis for the flagship simulation that resolves atomic cooling haloes throughout a $(95.5\, \text{cMpc})^3$ region of the Universe. To avoid numerical artefacts, we devise a novel method for accurate frequency-dependent line radiative transfer in the presence of continuous Hubble flow, transferable to broader astrophysical applications as well. Our scalable approach highlights the utility of LAEs and red damping-wing transmission as probes of reionization, which reveal nontrivial trends across different galaxies, sightlines, and frequency bands that can be modelled in the framework of covering fractions. In fact, after accounting for environmental factors influencing large-scale ionized bubble formation such as redshift and UV magnitude, the variation across galaxies and sightlines mainly depends on random processes including peculiar velocities and self-shielded systems that strongly impact unfortunate rays more than others. Throughout the EoR local and cosmological optical depths are often greater than or less than unity such that the exp (− τ) behaviour leads to anisotropic and bimodal transmissivity. Future surveys will benefit by targeting both rare bright objects and Goldilocks zone LAEs to infer the presence of these (un)predictable (dis)advantages.

Non-equilibrium ionization and radiative transport in an evolving supernova remnant

ABSTRACT We present numerical simulations of the evolution of a supernova (SN) remnant expanding into a uniform background medium with density $n_\mathrm{ H} = 1.0\, \mbox{ cm}^{-3}$ and temperature of 104 K. We include a dynamically evolving non-equilibrium ionization (NEI) network (consisting of all the ions of H, He, C, N, O, Ne, Mg, Si, S, Fe), frequency dependent radiation transfer (RT), thermal conduction, and a simple dust evolution model, all intra-coupled to each other and to the hydrodynamics. We assume spherical symmetry. Photoionization, radiation losses, photo-heating, charge-exchange heating/cooling, and radiation pressure are calculated on the fly depending on the local radiation field and ion fractions. We find that the dynamics and energetics (but not the emission spectra) of the SN remnants can be well modelled by collisional equilibrium cooling curves even in the absence of non-equilibrium cooling and radiative transport. We find that the effect of precursor ionizing radiation at different stages of SN remnant are dominated by rapid cooling of the shock and differ from steady-state shocks. The predicted column densities of different ions such as N+, C3+, and N4+, can be higher by up to several orders of magnitude compared to steady-state shocks. We also present some high-resolution emission spectra that can be compared with the observed remnants to obtain important information about the physical and chemical states of the remnant, as well as constrain the background interstellar medium.

Three-component modelling of C-rich AGB star winds – V. Effects of frequency-dependent radiative transfer including drift★

ABSTRACT Stellar winds of cool carbon stars enrich the interstellar medium with significant amounts of carbon and dust. We present a study of the influence of two-fluid flow on winds where we add descriptions of frequency-dependent radiative transfer (RT). Our radiation hydrodynamic models in addition include stellar pulsations, grain growth and ablation, gas-to-dust drift using one mean grain size, dust extinction based on both the small particle limit (SPL) and Mie scattering, and an accurate numerical scheme. We calculate models at high spatial resolution using 1024 gridpoints and solar metallicities at 319 frequencies, and we discern effects of drift by comparing drift models to non-drift models. Our results show differences of up to 1000 per cent in comparison to extant results. Mass-loss rates and wind velocities of drift models are typically, but not always, lower than in non-drift models. Differences are larger when Mie scattering is used instead of the SPL. Amongst other properties, the mass-loss rates of the gas and dust, dust-to-gas density ratio, and wind velocity show an exponential dependence on the dust-to-gas speed ratio. Yields of dust in the least massive winds increase by a factor 4 when drift is used. We find drift velocities in the range $10\!-\!67\, \mbox{km}\, \mbox{s}^{-1}$, which is drastically higher than in our earlier works that use grey RT. It is necessary to include an estimate of drift velocities to reproduce high yields of dust and low wind velocities.

Radiation-hydrodynamic Simulations of Spherical Protostellar Collapse for Very Low-mass Objects

Abstract We perform radiation-hydrodynamical simulations of protostellar collapse in spherical symmetry, with a special focus on very low-mass objects, i.e., brown dwarfs and sub-brown dwarfs. The inclusion of a realistic equation of state, which includes the effect of hydrogen dissociation, allows for a modeling of the complete process from the beginning of the collapse until the formation of the protostar. We solve the frequency-dependent radiative transfer equation without any diffusion approximation, using realistic dust and gas opacities. Our results show that the properties of the protostar are essentially independent of the initial conditions, which had previously only been confirmed for higher mass ranges. For very low-mass initial conditions, however, we find that the first core phase of the collapse shows some significant differences in the time evolution, with the first core lifetime increasing dramatically because of the reduced accretion rate from the surrounding envelope. We consider the observational implications of this. We also investigate the opposite case of a collapse without any first core phase, which may occur for very unstable initial conditions. In the Appendix, we describe a severe numerical problem that causes an unphysical expansion after the formation of the protostar, which may affect other attempts at similar calculations of self-gravitational collapse. We explain the origin of the unphysical behavior and present a solution that can be used in similar investigations.

Sub-parsec-scale dynamics of a dusty gas disc exposed to anisotropic AGN radiation with frequency-dependent radiative transfer

We explore the gas dynamics near the dust sublimation radius of active galactic nucleus (AGN). For the purpose, we perform axisymmetric radiation hydrodynamic simulations of a dusty gas disk of radius $\approx 1\,\mathrm{pc}$ around a supermassive black hole of mass $10^{7}\,\mathrm{M_{\odot}}$ taking into account (1) anisotropic radiation of accretion disk, (2) X-ray heating by corona, (3) radiative transfer of infrared (IR) photons reemitted by dust, (4) frequency dependency of direct and IR radiations, and (5) separate temperatures for gas and dust. As a result, we find that for Eddington ratio $\approx 0.77$, a nearly neutral, dense ($\approx 10^{6\operatorname{-}8}\;\mathrm{cm^{-3}}$), geometrically-thin ($h/r<0.06$) disk forms with a high velocity ($\approx 200 \sim 3000\;\mathrm{km/s}$) dusty outflow launched from the disk surface. The disk temperature is determined by the balance between X-ray heating and various cooling, and the disk is almost supported by thermal pressure. Contrary to \citet{krolik07:_agn}, the radiation pressure by IR photons is not effective to thicken the disk, but rather compresses it. Thus, it seems difficult for a radiation-supported, geometrically-thick, obscuring torus to form near the dust sublimation radius as far as the Eddington ratio is high ($\sim 1$). The mass outflow rate is $0.05$-$0.1\;\mathrm{M_{\odot}}/\mathrm{yr}$ and the column density of the outflow is $N_{\mathrm{H}}\lesssim 10^{21}\;\mathrm{cm^{-2}}$. To explain observed type-II AGN fraction, it is required that outflow gas is extended to larger radii ($r\gtrsim 10\;\mathrm{pc}$) or that a denser dusty wind is launched from smaller radii ($r\sim 10^{4}\;R_{g}$).

Formula omitted]-adaptive discontinuous Galerkin methods for simplified [formula omitted] approximations of frequency-dependent radiative transfer

We investigate the performance of a class of hp-adaptive discontinuous Galerkin methods for the numerical solution of simplified PN approximations of radiative transfer in non-gray semitransparent media. By introducing an optical scale and using asymptotic expansions in the radiative transfer equation we formulate the simplified PN approximations. The optical spectrum is decomposed in frequency bands and the simplified PN equations are solved for each frequency band. As a numerical solver for the simplified PN equations we consider a high-order discontinuous Galerkin method. The solver belongs to a class of finite element methods whose approximate solutions are discontinuous across inter-element boundaries; this property renders the method ideally suited for the hp-adaptivity. An error estimator is shown to provide reliable and practically useful upper bounds for the numerical errors independent of the optical scales used in the simulations. The proposed method is simple, fast and highly accurate. The performance of the method is analyzed on several applications in frequency-dependent radiative transfer. The aim of such a method compared to the conventional finite element methods is to solve the simplified PN equations efficiently and with a high level of accuracy on unstructured meshes with different elements. The obtained results demonstrate the ability of the proposed method to capture the main radiative features.

An asymptotic preserving unified gas kinetic scheme for frequency-dependent radiative transfer equations

This paper presents an extension of previous work (Sun et al., 2015 [22]) of the unified gas kinetic scheme (UGKS) for the gray radiative transfer equations to the frequency-dependent (multi-group) radiative transfer system. Different from the gray radiative transfer equations, where the optical opacity is only a function of local material temperature, the simulation of frequency-dependent radiative transfer is associated with additional difficulties from the frequency-dependent opacity. For the multiple frequency radiation, the opacity depends on both the spatial location and the frequency. For example, the opacity is typically a decreasing function of frequency. At the same spatial region the transport physics can be optically thick for the low frequency photons, and optically thin for high frequency ones. Therefore, the optical thickness is not a simple function of space location. In this paper, the UGKS for frequency-dependent radiative system is developed. The UGKS is a finite volume method and the transport physics is modeled according to the ratio of the cell size to the photon's frequency-dependent mean free path. When the cell size is much larger than the photon's mean free path, a diffusion solution for such a frequency radiation will be obtained. On the other hand, when the cell size is much smaller than the photon's mean free path, a free transport mechanism will be recovered. In the regime between the above two limits, with the variation of the ratio between the local cell size and photon's mean free path, the UGKS provides a smooth transition in the physical and frequency space to capture the corresponding transport physics accurately. The seemingly straightforward extension of the UGKS from the gray to multiple frequency radiation system is due to its intrinsic consistent multiple scale transport modeling, but it still involves lots of work to properly discretize the multiple groups in order to design an asymptotic preserving (AP) scheme in all regimes. The current scheme is tested in a few frequency-dependent radiation problems, and the results are compared with the solutions from the well-defined implicit Monte Carlo (IMC) method. The UGKS is much more efficient than IMC, and the computational times of both schemes for all test cases are listed. The UGKS seems to be the first discrete ordinate method (DOM) for the accurate capturing of multiple frequency radiative transport physics from ballistic particle motion to the diffusive wave propagation.

The influence of frequency-dependent radiative transfer on the structures of radiative shocks

Radiative shocks are shocks in a gas where the radiative energy and flux coming from the very hot post-shock material are non-negligible in the shock's total energy budget, and are often large enough to heat the material ahead of the shock. Many simulations of radiative shocks, both in the contexts of astrophysics and laboratory experiments, use a grey treatment of radiative transfer coupled to the hydrodynamics. However, the opacities of the gas show large variations as a function of frequency and this needs to be taken into account if one wishes to reproduce the relevant physics. We have performed radiation hydrodynamics simulations of radiative shocks in Ar using multigroup (frequency dependent) radiative transfer with the heracles code. The opacities were taken from the odalisc database. We show the influence of the number of frequency groups used on the dynamics and morphologies of subcritical and supercritical radiative shocks in Ar gas, and in particular on the extent of the radiative precursor. We find that simulations with even a low number of groups show significant differences compared to single-group (grey) simulations, and that in order to correctly model such shocks, a minimum number of groups is required. Results appear to eventually converge as the number of groups increases above 50. We were also able to resolve in our simulations of supercritical shocks the adaptation zones which connect the cooling layer to the final post-shock state and the precursor. Inside these adaptation zones, we find that the radiative flux just ahead of the shock in one or several high-opacity groups can heat the gas to a temperature higher than the post-shock temperature. Through the use of Hugoniot curves, we have checked the consistency of our radiation hydrodynamics scheme by showing that conservation of mass, momentum and energy (including radiative flux) holds throughout the computational domain for all our simulations. We conclude that a minimum number of frequency groups are required to correctly simulate radiating flows in gases whose opacities present large variations as a function of frequency.

PROTOPLANETARY DISK STRUCTURE WITH GRAIN EVOLUTION: THE ANDES MODEL

We present a self-consistent model of a protoplanetary disk: 'ANDES' ('AccretioN disk with Dust Evolution and Sedimentation'). ANDES is based on a flexible and extendable modular structure that includes 1) a 1+1D frequency-dependent continuum radiative transfer module, 2) a module to calculate the chemical evolution using an extended gas-grain network with UV/X-ray-driven processes surface reactions, 3) a module to calculate the gas thermal energy balance, and 4) a 1+1D module that simulates dust grain evolution. For the first time, grain evolution and time-dependent molecular chemistry are included in a protoplanetary disk model. We find that grain growth and sedimentation of large grains to the disk midplane lead to a dust-depleted atmosphere. Consequently, dust and gas temperatures become higher in the inner disk (R < 50 AU) and lower in the outer disk (R > 50 AU), in comparison with the disk model with pristine dust. The response of disk chemical structure to the dust growth and sedimentation is twofold. First, due to higher transparency a partly UV-shielded molecular layer is shifted closer to the dense midplane. Second, the presence of big grains in the disk midplane delays the freeze-out of volatile gas-phase species such as CO there, while in adjacent upper layers the depletion is still effective. Molecular concentrations and thus column densities of many species are enhanced in the disk model with dust evolution, e.g., CO2, NH2CN, HNO, H2O, HCOOH, HCN, CO. We also show that time-dependent chemistry is important for a proper description of gas thermal balance.

Full SED fitting with the KOSMA-τPDR code

We revised the treatment of interstellar dust in the KOSMA-\tau\ PDR model code to achieve a consistent description of the dust-related physics in the code. The detailed knowledge of the dust properties is then used to compute the dust continuum emission together with the line emission of chemical species. We coupled the KOSMA-\tau\ PDR code with the MCDRT (multi component dust radiative transfer) code to solve the frequency-dependent radiative transfer equations and the thermal balance equation in a dusty clump under the assumption of spherical symmetry, assuming thermal equilibrium in calculating the dust temperatures, neglecting non-equilibrium effects. We updated the calculation of the photoelectric heating and extended the parametrization range for the photoelectric heating toward high densities and UV fields. We revised the computation of the H2 formation on grain surfaces to include the Eley-Rideal effect, thus allowing for high-temperature H2 formation. We demonstrate how the different optical properties, temperatures, and heating and cooling capabilities of the grains influence the physical and chemical structure of a model cloud. The most influential modification is the treatment of H2 formation on grain surfaces that allows for chemisorption. This increases the total H2 formation significantly and the connected H2 formation heating provides a profound heating contribution in the outer layers of the model clumps. The contribution of PAH surfaces to the photoelectric heating and H2 formation provides a boost to the temperature of outer cloud layers, which is clearly traced by high-J CO lines. Increasing the fraction of small grains in the dust size distribution results in hotter gas in the outer cloud layers caused by more efficient heating and cooler cloud centers, which is in turn caused by the more efficient FUV extinction.

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

Discrete Diffusion Monte Carlo (DDMC) is a technique for increasing the efficiency of Implicit Monte Carlo radiative-transfer simulations in optically thick media. In DDMC, particles take discrete steps between spatial cells according to a discretized diffusion equation. Each discrete step replaces many smaller Monte Carlo steps, thus improving the efficiency of the simulation. In this paper, we present an extension of DDMC for frequency-dependent radiative transfer. We base our new DDMC method on a frequency-integrated diffusion equation for frequencies below a specified threshold, as optical thickness is typically a decreasing function of frequency. Above this threshold we employ standard Monte Carlo, which results in a hybrid transport-diffusion scheme. With a set of frequency-dependent test problems, we confirm the accuracy and increased efficiency of our new DDMC method.