Time-dependent Radiative Transfer Equation Research Articles

A Micro-Macro Decomposed Reduced Basis Method for the Time-Dependent Radiative Transfer Equation

A Micro-Macro Decomposed Reduced Basis Method for the Time-Dependent Radiative Transfer Equation

Exact analytical solution of the one-dimensional time-dependent radiative transfer equation with linear scattering

The radiative transfer equation (RTE) is a cornerstone for describing the propagation of electromagnetic radiation in a medium, with applications spanning atmospheric science, astrophysics, remote sensing, and biomedical optics. Despite its importance, an exact analytical solution to the RTE has remained elusive, necessitating the use of numerical approximations such as Monte Carlo, discrete ordinate, and spherical harmonics methods. In this paper, we present an exact solution to the one-dimensional time-dependent RTE. We delve into the moments of the photon distribution, providing a clear view of the transition to the diffusion regime. This analysis offers a deeper understanding of light propagation in the medium. Furthermore, we demonstrate that the one-dimensional RTE is equivalent to the Klein-Gordon equation with an imaginary mass term determined by the inverse reduced scattering length. Contrary to naive expectations of superluminal solutions, we find that our solution is strictly causal under appropriate boundary conditions, determined by the light transport problem. We validate the found solution using Monte Carlo simulations and benchmark the performance of the latter. Our analysis reveals that even for highly forward scattering, dozens of random light scatterings are required for an accurate estimate, underscoring the complexity of the problem. Moreover, we propose a method for faster convergence by adjusting the parameters of Monte Carlo sampling. We show that a Monte Carlo method sampling photon scatterings with input parameters (μs,g), where μs is the inverse scattering length and g is the scattering anisotropy parameter, is equivalent to that with (μs(1−g)/2,−1). This equivalence leads to a significantly faster convergence to the exact solution, offering a substantial improvement of the Monte Carlo method for the one-dimensional RTE. Our findings not only contribute to the theoretical understanding of the RTE but also have potential implications for improving the numerical methods used to approximate it.

Stochastic Galerkin Methods for Time-Dependent Radiative Transfer Equations with Uncertain Coefficients

Stochastic Galerkin Methods for Time-Dependent Radiative Transfer Equations with Uncertain Coefficients

A comparison of numerical methods for computing the reionization of intergalactic hydrogen and helium by a central radiating source

ABSTRACT We compare numerical methods for solving the radiative transfer equation in the context of the photoionization of intergalactic gaseous hydrogen and helium by a central radiating source. Direct integration of the radiative transfer equation and solutions using photon packets are examined, both for solutions to the time-dependent radiative transfer equation and in the infinite-speed-of-light approximation. The photon packet schemes are found to be more generally computationally efficient than a direct integration scheme. While all codes accurately describe the growth rate of hydrogen and helium ionization zones, it is shown that a fully time-dependent method is required to capture the gas temperature and ionization structure in the near zone of a source when an ionization front expands at a speed close to the speed of light. Applied to quasi-stellar objects in the Epoch of Reionization (EoR), temperature differences as high as 5 × 104 K result in the near zone for solutions of the time-dependent radiative transfer equation compared with solutions in the infinite-speed-of-light approximation. Smaller temperature differences are found following the nearly full photoionization of helium in gas in which the hydrogen was already ionized and the helium was singly ionized. Variations found in the temperature and ionization structure far from the source, where the gas is predominantly neutral, may affect some predictions for 21-cm EoR experiments.

High order asymptotic preserving discontinuous Galerkin methods for gray radiative transfer equations

In this paper, we will develop a class of high order asymptotic preserving (AP) discontinuous Galerkin (DG) methods for nonlinear time-dependent gray radiative transfer equations (GRTEs). Inspired by the works in [6,55], we propose to penalize the nonlinear GRTEs under the micro-macro decomposition framework by adding a weighted linear diffusive term. A hyperbolic, namely Δt=O(h) in the transport regime where Δt and h are the time step and mesh size respectively, and unconditional stability instead of parabolic time step restriction Δt=O(h2) in the diffusive regime are obtained, which are also free from the photon mean free path. We further employ a Picard iteration with a predictor-corrector procedure, to decouple the resulting global nonlinear system to a linear system with local nonlinear algebraic equations within each outer iterative loop. For the resulting scheme, only an implicit system for the macroscopic variable needs to be solved, while the microscopic variable can be updated explicitly. Besides, the nonlinear implicit system is decoupled to a linear positive definite system followed by nonlinear algebraic equations due to the Picard iteration. Namely, high dimension, implicit treatment and nonlinearity are all decoupled. Our scheme is shown to be AP and asymptotically accurate (AA). Numerical tests for one and two spatial dimensional problems are performed to demonstrate that our scheme is high order accurate, effective and efficient.

General Relativistic Radiation Magnetohydrodynamics Simulations of Black Hole Accretion Disks: Comparison of Methods Based on Variable Eddington Tensor and Based on M1 Closure

We perform 2.5-dimensional general relativistic radiation magnetohydrodynamics simulations of black hole accretion disks and disk winds in the range of mass accretion rate from 0.1 to 104.5 times the Eddington limit. In this paper, we compare the results of the INAZUMA code, in which the frequency-integrated time-dependent radiation transfer equation is solved in order to evaluate the Eddington tensor, with those of the first momentum (M1) approximation method. In both methods, accretion disks and disk winds appear, and there is no remarkable difference in accretion rate, outflow rate, or luminosity. However, the significant difference in the radiation field appears around the rotation axis. In the M1 method, the radial component of the radiation flux tends to be amplified owing to unphysical radiation collisions. Such an enhancement of the outward radiation flux does not appear in INAZUMA. Also, the problem of radiation not reaching the rotation axis occurs with M1, but not with INAZUMA. Our results indicate that the radiation transfer equation should be solved to obtain the accurate radiation field in the optically thin region around the rotation axis.

Nonlinear iterative projection methods with multigrid in photon frequency for thermal radiative transfer

This paper presents nonlinear iterative methods for the fundamental thermal radiative transfer (TRT) model defined by the time-dependent multifrequency radiative transfer (RT) equation and the material energy balance (MEB) equation. The iterative methods are based on the nonlinear projection approach and use multiple grids in photon frequency. They are formulated by the high-order RT equation on a given grid in photon frequency and low-order moment equations on a hierarchy of frequency grids. The material temperature is evaluated in the subspace of lowest dimensionality from the MEB equation coupled to the effective grey low-order equations. The algorithms apply various multigrid cycles to visit frequency grids. Numerical results are presented to demonstrate convergence of the multigrid iterative algorithms in TRT problems with large number of photon frequency groups.

A Numerical Scheme for General Relativistic Radiation Magnetohydrodynamics Based on Solving a Grid-based Boltzmann Equation

Abstract We develop a general relativistic radiation magnetohydrodynamics (GR-RMHD) code inazuma in which the time-dependent radiation transfer equation (frequency-integrated Boltzmann equation) is solved in curved spacetime. The Eddington tensor is derived from the specific intensity, and we solve the zeroth and first moment equations in order to update the radiation fields. Therefore, our code can solve the radiation field around relativistic compact objects more appropriately than an approximation method like the M1 closure scheme. The numerical scheme of magnetohydrodynamics is the same as that of our previous code. In some test calculations for propagating radiation and radiation hydrodynamics in flat spacetime, our code shows similar results to our previous work. Radiation propagation in curved spacetime is also properly solved for. We also show the radiation transport from the super-Eddington accretion disk around the black hole. The disk structure, such as the density, velocity, and temperature, is fixed by the model obtained using the GR-RMHD simulation with the M1 method. We found that the difference between our scheme and the M1 method appears in the optically thin outflow region around the rotation axis while the radiation field is almost the same in the optically thick disk region.

Light transport with the equation of radiative transfer: The Fourier Continuation – Discrete Ordinates (FC–DOM) Method

We present a method for the treatment of the time dependent radiative transfer equation under the discrete ordinate approximation. The novelty of the proposed approach stems, in part, from the incorporation of a spectral method for the calculation of the spatial differential operators based on the Fourier Continuation procedure introduced recently by Bruno and co–authors. This is a spatially dispersionless and high order method, which can handle arbitrary geometries, including those encountered in the forward model of light transport in optical tomography. We validate our theoretical results by comparison with analytic and experimental outcomes of the fluence measurements on tissue-like phantoms. The method makes it possible to calculate the time of flight of photons in random media efficiently and with high accuracy.

An Improved Accurate Solver for the Time-Dependent RTE in Underwater Optical Wireless Communications

Underwater optical wireless communication (UOWC) has been widely advocated as a viable way to satisfy these high-speed links constraints in the marine medium through the use of the visible spectrum. Nevertheless, UOWC faces several limitations, such as the path-loss due to the absorption and scattering phenomena, caused by underwater particles. Thus, quantifying this path-loss is of paramount importance in the design of futuristic UOWC systems. To this end, several approaches have been used in this regard, namely the Beer–Lambert’s law, Monte Carlo simulation, as well as radiative transfer equation (RTE). This last mentioned evaluates the optical path-loss of the light wave in an underwater channel in terms of the absorption and scattering coefficients as well as the scattering phase function (SPF). In this paper, an improved numerical solver to evaluate the time-dependent RTE for UOWC is proposed. The proposed numerical algorithm was improved based on the previously proposed ones, by making use of an improved finite difference scheme, a modified scattering angular discretization, as well as an enhancement of the quadrature method by involving a more accurate seven-point quadrature scheme in order to calculate the weight coefficients corresponding to the RTE integral term. Importantly, we applied the RTE solver to three different volume scattering functions, namely the single-term Henyey–Greenstein (HG) phase function, the two-term HG phase function, and the Fournier–Forand phase function, over both Harbor-I and Harbor-II water types. Based on the normalized received power evaluated through the proposed algorithm, the bit error rate performance of the UOWC system is investigated in terms of system and channel parameters. The enhanced algorithm gives a tightly close performance to its Monte Carlo counterpart by adjusting the numerical cumulative distribution function computation method as well as optimizing the number of scattering angles.

High-order spherical harmonics-nodal collocation scheme for the numerical solution of the time-dependent radiative transfer equation.

The validity of the spherical harmonics-nodal collocation approximation to the stationary Boltzmann equation has been established in multidimensional neutron transport problems. This is a high-order approximation method that allows a coarse spatial discretization without losing accuracy. We extend the method to solve the time-dependent radiative transfer equation for absorbing media with anisotropic scattering, also incorporating into the model reflecting boundary conditions, due to refractive index mismatching. The formalism is then applied to numerical test cases in one and two spatial dimensions that, using typical values in optical tomography for the physical parameters, check the accuracy and convergence of the method.

Enhancing early-time diffusion through beam collimation in pulse propagation in sparse discrete random media.

Solutions of the time-dependent radiative transfer equation (RTE) are used to describe propagation of a pulsed collimated beam through a random medium consisting of discrete scatterers of sizes large compared to the wavelengths-a situation particularly relevant to free-space optical communication through atmospheric obscurants. The RTE is solved in the spherical-harmonics basis with no approximations other than a truncation N in the angular momenta; the results confirm convergence of the solution for a fixed beam width and growing N. The obtained time-resolved radiance includes both the usual "late-time diffusion" (LTD), responsible for the well known reduction of the "coherence bandwidth" and, thus, a serious limitation in the transfer rate, and the more recently identified "early-time diffusion" (ETD) component, attenuated at a rate significantly lower than for the coherent (ballistic) signal and characterized by a very short rise time, allowing a high-rate data transfer. The ratio of the ETD to the LTD signal for the considered collimated beams is much (orders of magnitude) higher than in the previously examined problem of an omnidirectional source, increasing its potential usefulness in communication and related imaging applications.

Accurate and efficient computation of the 3D radiative transfer equation in highly forward-peaked scattering media using a renormalization approach

This paper considers an accurate and efficient numerical scheme for solving the radiative transfer equation (RTE) based on the discrete ordinates method (DOM) in highly forward-peaked scattering media. The DOM approximates the scattering integral of the RTE including the phase function into a quadrature sum with the total number of discrete angular directions (TND) by a quadrature set. Due to large numerical errors of the scattering integral based on the DOM in highly forward-peaked scattering, the phase function is renormalized to satisfy its normalization conditions. Although the renormalization approaches of the phase function improve the accuracy of the numerical results of the RTE, the computational efficiency of the RTE is still required. This paper develops the first order renormalization approach using the double exponential formula for three quadrature sets: level symmetric even, even and odd, and Lebedev sets in a wide range of the TND from 48 to 1454. Numerical errors of the three-dimensional time-dependent RTE are investigated by the analytical solutions of the RTE. The investigation shows that the level symmetric even set with the TND of 48 using the developed approach provides the most accurate results of the RTE among the quadrature sets in the range of the TND, while to obtain the same accuracy by the conventional zeroth order renormalization approach, the TND needs to be larger than 360. The results suggest the large reduction of computational loads by the developed approach to less than 10 percent from those in the conventional approach.

Theoretical and Numerical Analysis of an Initial-Boundary Value Problem for the Radiative Transfer Equation with Fresnel Matching Conditions

A Cauchy problem for the time-dependent radiative transfer equation in a three-dimensional multicomponent medium with generalized matching conditions describing Fresnel reflection and refraction at the interface of the media is considered. The unique solvability of the problem is proven, a Monte Carlo method for solving the initial-boundary value problem is developed, and computational experiments for different implementations of the algorithm are conducted.

Dusty Cloud Acceleration by Radiation Pressure in Rapidly Star-forming Galaxies

Abstract We perform two-dimensional and three-dimensional radiation hydrodynamic simulations to study cold clouds accelerated by radiation pressure on dust in the environment of rapidly star-forming galaxies dominated by infrared flux. We utilize the reduced speed of light approximation to solve the frequency-averaged, time-dependent radiative transfer equation. We find that radiation pressure is capable of accelerating the clouds to hundreds of kilometers per second while remaining dense and cold, consistent with observations. We compare these results to simulations where acceleration is provided by entrainment in a hot wind, where the momentum injection of the hot flow is comparable to the momentum in the radiation field. We find that the survival time of the cloud accelerated by the radiation field is significantly longer than that of a cloud entrained in a hot outflow. We show that the dynamics of the irradiated cloud depends on the initial optical depth, temperature of the cloud, and intensity of the flux. Additionally, gas pressure from the background may limit cloud acceleration if the density ratio between the cloud and background is . In general, a 10 pc-scale optically thin cloud forms a pancake structure elongated perpendicular to the direction of motion, while optically thick clouds form a filamentary structure elongated parallel to the direction of motion. The details of accelerated cloud morphology and geometry can also be affected by other factors, such as the cloud lengthscale, reduced speed of light approximation, spatial resolution, initial cloud structure, and dimensionality of the run, but these have relatively little affect on the cloud velocity or survival time.

Light propagation model of titanium dioxide suspensions in water using the radiative transfer equation

Constructions of numerical schemes for solving the radiative transfer equation (RTE) are crucial to evaluate light propagation inside photocatalytic systems. We develop accurate and efficient schemes of the three-dimensional and time-dependent RTE for numerical phantoms modeling aqueous titanium dioxide suspensions, in which the anisotropy of the forward-directed scattering varies and the strength of absorption is comparable to that of scattering. To improve the accuracy and efficiency of the numerical solutions, the forward-directed phase function is renormalized in the zeroth or first order with a small number of discrete angular directions. Then, we investigate the influences of the forward-directed scattering on the numerical solutions by comparing with the analytical solutions. The investigation shows that with the anisotropy factor less than approximately 0.7 corresponding to the moderate forward-directed scattering, the numerical solutions of the RTE using the both of the zeroth and first order renormalization approaches are accurate due to the reductions of the numerical errors of the phase function. With the anisotropy factor more than approximately 0.7 corresponding to the highly forward-directed scattering, the first order renormalization approach still provides the accurate results, while the zeroth order approach does not due to the large errors of the phase function. These results suggest that the developed scheme using the first order renormalization can provide accurate and efficient calculations of light propagation in photocatalytic systems.

Improving the thin-disk models of circumstellar disk evolution. The 2+1-dimensional model

Circumstellar disks of gas and dust are naturally formed from contracting pre-stellar molecular cores during the star formation process. To study various dynamical and chemical processes that take place in circumstellar disks prior to their dissipation and transition to debris disks, the appropriate numerical models capable of studying the long-term disk chemodynamical evolution are required. We present a new 2+1-dimensional numerical hydrodynamics model of circumstellar disk evolution, in which the thin-disk model is complemented with the procedure for calculating the vertical distributions of gas volume density and temperature in the disk. The reconstruction of the disk vertical structure is performed at every time step via the solution of the time-dependent radiative transfer equations coupled to the equation of the vertical hydrostatic equilibrium. We perform a detailed comparison between circumstellar disks produced with our previous 2D model and with the improved 2+1D approach. The structure and evolution of resulting disks, including the differences in temperatures, densities, disk masses and protostellar accretion rates, are discussed in detail. The new 2+1D model yields systematically colder disks, while the in-falling parental clouds are warmer. Both effects act to increase the strength of disk gravitational instability and, as a result, the number of gravitationally bound fragments that form in the disk via gravitational fragmentation as compared to the purely 2D thin-disk simulations with a simplified thermal balance calculation.

A new scheme of the time-domain fluorescence tomography for a semi-infinite turbid medium

A new scheme for reconstruction of a fluorophore target embedded in a semi-infinite medium was proposed and evaluated. In this scheme, we neglected the presence of the fluorophore target for the excitation light and used an analytical solution of the time-dependent radiative transfer equation (RTE) for the excitation light in a homogeneous semi-infinite media instead of solving the RTE numerically in the forward calculation. The inverse problem for imaging the fluorophore target was solved using the Landweber–Kaczmarz method with the concept of the adjoint fields. Numerical experiments show that the proposed scheme provides acceptable results of the reconstructed shape and location of the target. The computation times of the solution of the forward problem and the whole reconstruction process were reduced by about 40 and 15%, respectively.

Modeling of light propagation in the human neck for diagnoses of thyroid cancers by diffuse optical tomography.

Diffuse optical tomography using near-infrared light in a wavelength range from 700 to 1000nm has the potential to enable non-invasive diagnoses of thyroid cancers; some of which are difficult to detect by conventional methods such as ultrasound tomography. Diffuse optical tomography needs to be based on a physically accurate model of light propagation in the neck, because it reconstructs tomographic images of the optical properties in the human neck by inverse analysis. Our objective here was to investigate the effects of three factors on light propagation in the neck using the 2D time-dependent radiative transfer equation: (1) the presence of the trachea, (2) the refractive-index mismatch at the trachea-tissue interface, and (3) the effect of neck organs other than the trachea (spine, spinal cord, and blood vessels). There was a significant influence of reflection and refraction at the trachea-tissue interface on the light intensities in the region between the trachea and the front of the neck surface. Organs other than the trachea showed little effect on the light intensities measured at the front of the neck surface although these organs affected the light intensities locally. These results indicated the necessity of modeling the refractive-index mismatch at the trachea-tissue interface and the possibility of modeling other neck organs simply as a homogeneous medium when the source and detectors were far from large blood vessels.

A NUMERICAL SCHEME FOR SPECIAL RELATIVISTIC RADIATION MAGNETOHYDRODYNAMICS BASED ON SOLVING THE TIME-DEPENDENT RADIATIVE TRANSFER EQUATION

ABSTRACT We develop a numerical scheme for solving the equations of fully special relativistic, radiation magnetohydrodynamics (MHDs), in which the frequency-integrated, time-dependent radiation transfer equation is solved to calculate the specific intensity. The radiation energy density, the radiation flux, and the radiation stress tensor are obtained by the angular quadrature of the intensity. In the present method, conservation of total mass, momentum, and energy of the radiation magnetofluids is guaranteed. We treat not only the isotropic scattering but also the Thomson scattering. The numerical method of MHDs is the same as that of our previous work. The advection terms are explicitly solved, and the source terms, which describe the gas–radiation interaction, are implicitly integrated. Our code is suitable for massive parallel computing. We present that our code shows reasonable results in some numerical tests for propagating radiation and radiation hydrodynamics. Particularly, the correct solution is given even in the optically very thin or moderately thin regimes, and the special relativistic effects are nicely reproduced.