Time-dependent Radiation Transport Research Articles

A sweep-based low-rank method for the discrete ordinate transport equation

The dynamical low-rank (DLR) approximation is an efficient technique to approximate the solution to matrix differential equations. Recently, the DLR method was applied to radiation transport calculations to reduce memory requirements and computational costs. This work extends the low-rank scheme for the time-dependent radiation transport equation in 2-D and 3-D Cartesian geometries with discrete ordinates discretization in angle (SN method). The reduced system that evolves on a low-rank manifold is constructed via the “unconventional” basis update & Galerkin integrator to avoid a substep that is backward in time, which could be unstable for dissipative problems. The resulting system preserves the information on angular direction by applying separate low-rank decompositions in each octant where angular intensity has the same sign as the direction cosines. Then, transport sweeps and source iteration can efficiently solve this low-rank-SN system. The numerical results in 2-D and 3-D Cartesian geometries demonstrate that the low-rank solution requires less memory and computational time than solving the full rank equations using transport sweeps without losing accuracy.

Stochastic radiative transfer in random media. II. Coupling of radiation to material.

We study the mechanism of the impact of random media on the stochastic radiation transport based on a one-dimensional (1D) planar model. To this end, we use a random sampling of mixtures combined with a deterministic solution of the time-dependent radiation transport equationcoupled to a material temperature equation. Compared to purely absorbing cases [C.-Z. Gao etal., Phys. Rev. E 102, 022111 (2020)10.1103/PhysRevE.102.022111], we find that material temperatures can significantly suppress the impact of mixing distribution and size, which is understood from the analysis of energy transport channels. By developing a steady-state stochastic transport model, it is found that the mechanism of transmission of radiation is distance dependent, which is closely related to the mean free path of photons l_{p}. Furthermore, we suggest that it is the relationship between l_{p} and L (the width of random medium) that determines the impact of random media on the stochastic radiation transport, which is further corroborated by additional simulations. Most importantly, combining the proposed simple relationship and 1D simulations, we resolve the existing disputable issue of the impact of random media in previous multidimensional works, showing that multidimensional results are essentially consistent and the observed weak or remarkable impact of random media is mainly due to the distinctly different relationship between l_{p} and L. Our results may have practical implications in relevant experiments of stochastic radiative transfer.

A Sweep-Based Low-Rank Method for the Discrete Ordinate Transport Equation

The dynamical low-rank (DLR) approximation is an efficient technique to approximate the solution to matrix differential equations. Recently, the DLR method was applied to radiation transport calculations to reduce memory requirements and computational costs. This work extends the low-rank scheme for the time-dependent radiation transport equation in 2-D and 3-D Cartesian geometries with discrete ordinates discretization in angle (SN method). The reduced system that evolves on a low-rank manifold is constructed via an unconventional basis update and Galerkin integrator to avoid a substep that is backward in time, which could be unstable for dissipative problems. The resulting system preserves the information on angular direction by applying separate low-rank decompositions in each octant where angular intensity has the same sign as the direction cosines. Then, transport sweeps and source iteration can efficiently solve this low-rank-SN system. The numerical results in 2-D and 3-D Cartesian geometries demonstrate that the low-rank solution requires less memory and computational time than solving the full rank equations using transport sweeps without losing accuracy.

A Shock-in-Jet Synchrotron Mirror Model for Blazars

Reinhard Schlickeiser has made groundbreaking contributions to various aspects of blazar physics, including diffusive shock acceleration, the theory of synchrotron radiation, the production of gamma-rays through Compton scattering in various astrophysical sources, etc. This paper, describing the development of a self-consistent shock-in-jet model for blazars with a synchrotron mirror feature, is therefore an appropriate contribution to a Special Issue in honor of Reinhard Schlickeiser’s 70th birthday. The model is based on our previous development of a self-consistent shock-in-jet model with relativistic thermal and non-thermal particle distributions evaluated via Monte-Carlo simulations of diffusive shock acceleration, and time-dependent radiative transport. This model has been very successful in modeling spectral variability patterns of several blazars, but has difficulties describing orphan flares, i.e., high-energy flares without a significant counterpart in the low-frequency (synchrotron) radiation component. As a solution, this paper investigates the possibility of a synchrotron mirror component within the shock-in-jet model. It is demonstrated that orphan flares result naturally in this scenario. The model’s applicability to a recently observed orphan gamma-ray flare in the blazar 3C279 is discussed and it is found that only orphan flares with mild (≲ a factor of 2–3) enhancements of the Compton dominance can be reproduced in a synchrotron-mirror scenario, if no additional parameter changes are invoked.

A high-order/low-order (HOLO) algorithm for preserving conservation in time-dependent low-rank transport calculations

Dynamical low-rank (DLR) approximation methods have previously been developed for time-dependent radiation transport problems. One crucial drawback of DLR is that it does not conserve important quantities of the calculation, which limits the applicability of the method. Here we address this conservation issue by solving a low-order equation with closure terms computed via a high-order solution calculated with DLR. We observe that the high-order solution well approximates the closure term, and the low-order solution can be used to correct the conservation bias in the DLR evolution. We also apply the linear discontinuous Galerkin method for the spatial discretization. We then demonstrate with the numerical results that this so-called high-order/low-order (HOLO) algorithm is conservative without sacrificing computational efficiency and accuracy.

An Implicit Finite Volume Scheme to Solve the Time-dependent Radiation Transport Equation Based on Discrete Ordinates

Abstract We describe a new algorithm to implicitly solve the time-dependent, frequency-integrated radiation transport (RT) equation, which is coupled to an explicit solver for equations of magnetohydrodynamics (MHD) using Athena++. The radiation field is represented by specific intensities along discrete rays, which are evolved using a conservative finite volume approach for both Cartesian and curvilinear coordinate systems. All terms for spatial transport of photons and interactions between gas and radiation are calculated implicitly together. An efficient Jacobi-like iteration scheme is used to solve the implicit equations. This removes any time-step constraint due to the speed of light in RT. We evolve the specific intensities in the lab frame to simplify the transport step. The lab frame specific intensities are transformed to the comoving frame via Lorentz transformation when the source term is calculated. Therefore, the scheme does not need any expansion in terms of v/c. The radiation energy and momentum source terms for the gas are calculated via direct quadrature in the angular space. The time step for the whole scheme is determined by the normal Courant–Friedrichs–Lewy condition in the MHD module. We provide a variety of test problems for this algorithm, including both optically thick and thin regimes, and for both gas and radiation pressure-dominated flows to demonstrate its accuracy and efficiency.

A low-rank method for two-dimensional time-dependent radiation transport calculations

The low-rank approximation is a complexity reduction technique to approximate a tensor or a matrix with a reduced rank, which has been applied to the simulation of high dimensional problems to reduce the memory required and computational cost. In this work, a dynamical low-rank approximation method is developed for the time-dependent radiation transport equation in 1-D and 2-D Cartesian geometries. Using a finite volume discretization in space and a spherical harmonics basis in angle, we construct a system that evolves on a low-rank manifold via an operator splitting approach. Numerical results on five test problems demonstrate that the low-rank solution requires less memory and computational time than solving the full rank equations with the same accuracy. It is furthermore shown that the low-rank algorithm can obtain high-fidelity results by increasing the number of basis functions while keeping the rank fixed.

A Fast Algorithm for Time-Dependent Radiative Transport Equation Based on Integral Formulation

In this work, we introduce a fast numerical algorithm to solve the time-dependent radiative transport equation (RTE). Our method uses the integral formulation of RTE and applies the treecode algorithm to reduce the computational complexity from O(M^{2+1/d} ) to O(M^{1+1/d} log M ), where M is the number of points in the physical domain. The error analysis is presented and numerical experiments are performed to validate our algorithm.

Time-dependent modelling of nanofluid-based direct absorption parabolic trough solar collectors

In this paper we propose a time-dependent, three-dimensional model for the efficiency of a nanofluid-based direct-absorption parabolic trough solar collector under a turbulent flow regime. The model consists of a system of equations: a partial differential equation for conservation of energy, and a time-dependent radiative transport equation describing the propagation of solar radiation through the nanofluid. Writing the model in dimensionless form reveals four controlling dimensionless numbers: one describing the relative importance of conduction and advection and three describing the heat loss to the surroundings. Realistic parameter values are applied to reduce the model further and these indicate that two of the dimensionless groups have a much smaller impact on the performance of the solar collector. We use the resulting solution for the temperature to calculate an analytic expression for the collector’s efficiency. This expression permits optimisation of design parameters such as particle loading, incoming radiative intensity, receiver dimensions, the inlet temperature, and solar concentration ratio.

Lattice Boltzmann method for solving non-equilibrium radiative transport problems

Recently developed lattice Boltzmann equation (LBE) framework (Bindra et al., 2012; McCulloch and Bindra, 2016) for radiation transport is extended to solve time-dependent non-equilibrium radiative and neutron transport problems. Dynamics of radiation and material energy exchange is modeled by coupling radiation transport equation with the material energy equation in a one-dimensional isotropically scattering homogenous medium. The LBE solutions are compared against the existing semi-analytical P1 benchmark and show good agreement for different times. Furthermore, a two-dimensional D2Q8 lattice Boltzmann method is proposed for solving the neutron transport equation in a heterogenous media (e.g., checkerboard lattice with pure scattering and absorbing cells). The results obtained with LBE are in good agreement with the existing discrete ordinate method results for the benchmark problem.

Numerical Modeling of Photon Migration in Human Neck Based on the Radiative Transport Equation

Biomedical optical imaging has a possibility of a comprehensive diagnosis of thyroid cancer in conjunction with ultrasound imaging. For improvement of the optical imaging, this study develops a higher order scheme for solving the time-dependent radiative transport equation (RTE) by use of the finite-difference and discrete-ordinate methods. The accuracy and efficiency of the developed scheme are examined by comparison with the analytical solutions of the RTE in homogeneous media. Then, the developed scheme is applied to describing photon migration in the human neck model. The numerical simulations show complex behaviors of photon migration in the human neck model due to multiple diffusive reflection near the trachea.

Gray radiation transport models for two-dimensional binary stochastic media with material temperature coupling using spherical harmonics

Due to the high computational cost of time-dependent radiation transport calculations, most multi-dimensional simulations of stochastic media have used the lowest angle order approximation, the P1 approximation. Here spherical harmonics of order n=5 are used to solve the transport equation in two-dimensional binary stochastic media. The results are consistent with earlier P1 simulations. Transport solutions using constant and temperature-dependent opacities and heat capacities are shown and analyzed. The standard closure poorly approximates the mean radiation field in these test problems. For one physical case, a less common closure is better. To best fit the most general cases, a new procedure is presented. In all cases, the approximate transport solutions work best in dilute systems where one material comprises less than about 10% of the total. The conclusions reached here should be independent of the transport solution method whether one uses spherical harmonics or discrete ordinates.

An imperfection of time-dependent diffusion models for a determination of scattering medium optical properties

In the present work the absorption and reduced scattering coefficients of the homogeneous scattering medium are determined using three various types of the diffusion model (classical, refined and extrapolated) in the case of the scattering slab geometry. The expression for the refined diffusion model in this case was derived for the first time. These coefficients are determined from experimental temporal distributions of transmitted photons at different slab thickness (from 10mm to 110mm) with fitting procedure. The specific dependencies of the scattering and absorption coefficients on the slab thickness were obtained for each of the examined models, although it is physically impossible for the same substance. This effect is caused by the fact that the significant features of the scattering indicatrix are lost when the diffusion approximation has been applied to the time-dependent radiation transport equation. This explanation was confirmed by Monte Carlo simulation, where the similar dependencies were obtained. There are presented several remarks regarding the experimental measurements when diffusion models are used.

Green's function of the time-dependent radiative transport equation in terms of rotated spherical harmonics

The time-dependent radiative transport equation is solved for the three-dimensional spatially uniform infinite medium which is illuminated by a point unidirectional source using a spherical harmonics transform under rotation. Apart from the numerical evaluation of a spherical Hankel transform which connects the spatial distance with the radial distance in Fourier space, the dependence on all variables is found analytically. For the special case of a harmonically modulated source, even the spherical Hankel transform can be carried out analytically. Additionally, a special solution for the isotropically scattering infinite medium is given. The Monte Carlo method is used for a successful verification of the derived solution.

Adaptive source biasing sampling for time-dependent radiation transport problems

Monte Carlo method is used to simulate the radiation transport problems which have characteristics of high temperature, high pressure, multi-media and large deformation of the grid. Due to the nonuniform particle distribution and great difference in the sizes of the grid, it makes the statistical errors of the grid fluxes fluctuate wildly. The error accumulation after many time steps may even influence the truth of the calculated results. So the global source biasing technique is developed. The importance function used here for source biasing sampling is based on the grid fluxes and errors of the previous time step. This is much faster than the traditional methods for solving the adjoint equation. The numerical tests showed that the neutron flux errors in the important grids are obviously reduced by using global source biasing which adaptively assigns the number of particles in each grid.

Fully implicit 1D radiation hydrodynamics: Validation and verification

A fully implicit finite difference scheme has been developed to solve the hydrodynamic equations coupled with radiation transport. Solution of the time-dependent radiation transport equation is obtained using the discrete ordinates method and the energy flow into the Lagrangian meshes as a result of radiation interaction is fully accounted for. A tridiagonal matrix system is solved at each time step to determine the hydrodynamic variables implicitly. The results obtained from this fully implicit radiation hydrodynamics code in the planar geometry agrees well with the scaling law for radiation driven strong shock propagation in aluminium. For the point explosion problem the self similar solutions are compared with results for pure hydrodynamic case in spherical geometry. Results obtained when radiation interaction is also accounted agree with those of point explosion with heat conduction for lower input energies. Having, thus, benchmarked the code, self convergence of the method w.r.t. time step is studied in detail for both the planar and spherical problems. Spatial as well as temporal convergence rates are ≃1 as expected from the difference forms of mass, momentum and energy conservation equations. This shows that the asymptotic convergence rate of the code is realized properly.

Moment Analysis of Angular APproximation Methods for Time-Dependent Radiation Transport

We extend moment analysis, a technique developed for investigating the accuracy of discrete-ordinates spatial discretization schemes, to time-dependent radiation transport and apply it to several angular approximation methods. Specifically, we examine the diffusion approximation, the P 1/3 approximation, and three time-dependent generalizations of the simplified PNapproximation: the SP 2 , SP 3 , and SSP 3 approximations. We show that all of the these methods preserve the correct flux-weighted average of x but not the correct flux-weighted average of (x-xa)2, where x is the spatial variable and xais an arbitrary point. We also demonstrate that, for general cross sections and large elapsed time, the error in the flux-weighted average of (x-xa)2 is smallest in magnitude for the SP 2 and approximations. In addition, we present a simple improvement to the SP 2 approximation that allows this method to produce the correct flux-weighted average of (x-xa)2. We present numerical results that test this analysis. From these results, we find that the angular approximation methods with the most accurate solutions also have the most accurate values for the flux-weighted average of (x-xa)2. In particular, the SP 2 and SP 3 approximations are two of the most accurate methods at large elapsed times, while the improved SP 2 approximation is one of the most accurate methods at all times. We also observe, however, that an accurate value for the flux-weighted average of (x-xa)2 is not always accompanied by an accurate solution. Consequently, we conclude that an accurate flux-weighted average of (x-xa)2 is a necessary rather than sufficient condition for an overall accurate angular approximation method.

Infinite integrals of Whittaker and Bessel functions with respect to their indices

We obtain several new closed-form expressions for the evaluation of a family of infinite-domain integrals of the Whittaker functions Wκ,μ(x) and Mκ,μ(x) and the modified Bessel functions Iμ(x) and Kμ(x) with respect to the index μ. The new family of definite integrals is useful in a variety of contexts in mathematical physics. In particular, the integral involving Kμ(x) represents a new example of the Kontorovich–Lebedev transform. We discuss the relationship between the results derived here and the previously known integrals of Whittaker and Bessel functions. In some cases, we obtain entirely new expressions, and in other cases, we generalize previously known results. An application to time-dependent radiation transport theory is also discussed.

On solutions to the Pn equations for thermal radiative transfer

We present results for the spherical harmonics ( P n ) method for solving problems of time-dependent thermal radiative transport. We prove a theorem that demonstrates that in the streaming limit, the spatially and temporally continuous P n equations will allow negative energy densities for any finite order of n. We also develop an implicit numerical method for solving the P n equations to explore the impact of the theorem. The numerical method uses a high-resolution Riemann solver to produce an upwinded discretization. We employ a quasi-linear approach to integrate the nonlinearites added to make the scheme non-oscillatory. We use the backward Euler method for time integration and treat the material interaction terms fully nonlinearly. Reflecting boundary conditions for the P n equations are presented and we show how to implement this boundary condition using ghost cells. The implicit method was able to produce robust results to thermal transport problems in one and two dimensions. The numerical method is used to analyze the accuracy of various P n expansion orders on several problems. In two-dimensional problems the numerical P n solutions contained negative radiation energy densities as predicted by our theorem. The numerical results showed that the material temperature also became negative, a result outside the scope of the theorem. Our numerical method can handle these negative values, but they would cause problems in a radiation-hydrodynamics calculation.

A New Algorithm for Two‐Dimensional Transport for Astrophysical Simulations. I. General Formulation and Tests for the One‐Dimensional Spherical Case

We derive new equations using the mixed-frame approach for one- and two-dimensional (axisymmetric) time-dependent radiation transport and the associated couplings with matter. Our formulation is multi-group and multi-angle and includes anisotropic scattering, frequency(energy)-dependent scattering and absorption, complete velocity dependence to order v/c, rotation, and energy redistribution due to inelastic scattering. Hence, the "2D" realization is actually "6 1/2"-dimensional. The effects of radiation viscosity are automatically incorporated. Moreover, we develop Accelerated-Lambda-Iteration, Krylov subspace (GMRES), Discontinuous-Finite-Element, and Feautrier numerical methods for solving the equations and present the results of one-dimensional numerical tests of the new formalism. The virtues of the mixed-frame approach include simple velocity dependence with no velocity derivatives, straight characteristics, simple physical interpretation, and clear generalization to higher dimensions. Our treatment can be used for both photon and neutrino transport, but we focus on neutrino transport and applications to core-collapse supernova theory in the discussions and examples.