Computational Methods in Transport

Hydrodynamics codes are used to study nearly every astrophysical phenomena observed. Although coupling radiation and hydrodynamics is much less common, radiation hydrodynamics codes are being used in a growing number of astrophysics problems from energetic out flows of compact remnants such as core-collapse supernovae and active galactic nuclei to the formation of stars and planets to cosmological simulations of the first stars. In this paper, we review the current “state-of-the-art” radiation hydrodynamics techniques used in astrophysics.

Radiative transfer is particularly important in astrophysics. One reason is quite understandable: radiation is in most cases the only information we have (and will ever have) about distant objects (exceptions are detected neutrinos from the Sun and supernova SN1987a, and in a near future the gravitational waves). However, there is an even more compelling reason for the a need to deal with detailed radiation transport in astrophysics: In many astronomical objects the radiation is so strong that it significantly contributes to the energy and momentum budget of the medium. Therefore, radiation is not only a probe of the physical state, but is in fact an important constituent. In other words, radiation in fact determines the structure of the medium, yet the medium is probed only by this radiation.

Stars more massive than ~10 M_⊙ evolve to an onion-like conguration (Fig. 1), with an iron core surrounded by successive layers of silicon, oxygen, carbon, helium, and nally hydrogen. In addition to iron group nuclei, the core is composed of electrons, positrons, photons, and a small fraction of protons and neutrons. The pressure in the core, which supports it against the inward pull of gravity, is dominated at this stage by the electrons, and the balance between the electron pressure and gravity is only marginally stable. As a result of electron capture on the free protons and nuclei in the core and as a result of nuclear dissociation under the extreme densities and temperatures, electron and thermal pressure support are reduced, and the core becomes unstable and collapses.

The purpose of this paper is to briefly describe the application of discreteordinates methods to radiative transfer in the non-relativistic stellar regime. We consider the non-relativistic regime to be characterized by material velocities less than or equal to one percent of the speed of light. In most applications, the radiative transfer equations are coupled to the hydrodynamics equations. We first describe a radiation-hydrodynamics model that consists of the Euler equations and the radiation transport equation together with approximations for material-motion effects based upon the assumption of non-relativistic material velocities. For simplicity, we next focus on a model of radiative transfer in a static medium that is adequate for the purpose of describing numerical discretization and solution techniques for the transfer equation that can also be applied in the more general context of radiationhydrodynamics.

It is argued that, to directly target the mean fluxes through a structured medium with spatial correlations over a significant range of scales that includes the mean-free-path, one can use an effective propagation kernel that will necessarily be sub-exponential. We come to this conclusion using both standard transport theory for variable media and a point-process approach developed recently by A. Kostinski. The ramifications of this finding for multiple scattering and effective medium theory are examined. Finally, we describe a novel one-dimensional transport theory with asymptotically power-law propagation kernels and use it to shed new light onto recent observations of solar photon pathlength in the Earth’s cloudy atmosphere.

The solar radiation transport through broken clouds can be treated as photon transport through stochastic media. The stochastic radiative transfer equation and a new statistically inhomogeneous Markovian model are used to derive analytical equations for the ensemble-averaged intensity. The computational method for solving these equations is introduced, and their accuracy and robustness are discussed. Validation tests show good predictive performance for the mean radiative properties.

A general introduction to the field of ocean optics is presented, including features that make optical oceanography problems similar to and different from other transport problems. Methods for solving time-independent, one-dimensional problems are discussed that are appropriate for passive illumination conditions.

It is well known that generally to simulate accurately radiative transfer through a realistic cloudy atmosphere one should use numerical approaches such as Monte Carlo [12], or SHDOM [3]. However, it is usually required too much time to make a simulation which is inconvenient when just we need an answer on a simple question like how significant the 3D effects are for a given problem. The perturbation method is what comes to mind first if we need to go further into modelling of the radiative transfer through cloud atmosphere starting from the simplest framework of one dimensional radiative transfer [2, 8].

Biophysical considerations for vegetation canopy reflectance modeling are presented. Included is a brief overview outlining strengths and weaknesses of four possible canopy reflectance models. The overview is followed by the description of the LCM2 coupled leaf/canopy turbid medium reflectance model based on natural averaging. The model follows conventional radiative transfer theory with modification for canopy architecture as characterized by leaf orientation. The presentation concludes with a demonstration of LCM2 in a multiple pixel mode to estimate the amount of ripe coffee cherries at harvest in the fields of the Kauai Coffee Company and to detect targets hidden beneath canopies.

Accurate knowledge of the spatial (and temporal) variability of the biosphere’s characteristics is useful not only to address critical scientific issues (climate change, environmental degradation, biodiversity preservation, etc.) but also to provide appropriate initial state and boundary conditions for general circulation or landscape succession models. In particular, the 3-D structure of vegetation emerged as a crucial player in processes affecting carbon sequestration, landscape dynamics and the exchanges of energy, water and trace gases with the atmosphere e.g., [BWG04]. The growth and development of plant architecture, in turn, are primarily conditioned by effective interception of solar radiation that provides the necessary energy for photosynthesis and other physiological processes [VB86].

Numerical simulation of multidimensional particle transport processes is among the most dificult problems in applied mathematics with high computational burden. Deterministic methods are widely used at present time for solving transport equations numerically. Further development of such methods opens a prospect for simulating various physical processes of particle and energy transport in more realistic assumptions and with more accurate and profound consideration of details and specific features of the particular problems.

The equations of radiation transport for thermal photons are notoriously difficult to solve in thick media without resorting to asymptotic approximations such as the diffusion limit. One source of this difficulty is that in thick, absorbing media thermal emission is almost completely balanced by strong absorption. In a previous publication [SB03], the photon transport equation was written in terms of the deviation of the specific intensity from the local equilibrium field. We called the new form of the equations the difference formulation. The difference formulation is rigorously equivalent to the original transport equation. It is particularly advantageous in thick media, where the radiation field approaches local equilibrium and the deviations from the Planck distribution are small. The difference formulation for photon transport also clariffes the diffusion limit. In this paper, the transport equation is solved by the Symbolic Implicit Monte Carlo (SIMC) method and a comparison is made between the standard formulation and the difference formulation. The SIMC method is easily adapted to the derivative source terms of the difference formulation, and a remarkable reduction in noise is obtained when the difference formulation is applied to problems involving thick media.

In this paper we extend the difference formulation for radiation transport to the case of a single atomic line. We examine the accuracy, performance and stability of the difference formulation within the framework of the Symbolic Implicit Monte Carlo method. The difference formulation, introduced for thermal radiation by some of the authors, has the unique property that the transport equation is written in terms that become small for thick systems. We find that the difference formulation has a significant advantage over the standard formulation for a thick system. The correct treatment of the line profile, however, requires that the difference formulation in the core of the line be mixed with the standard formulation in the wings, and this may limit the advantage of the method. We bypass this problem by using the gray approximation. We develop three Monte Carlo solution methods based on different degrees of implicitness for the treatment of the source terms, and we find only conditional stability unless the source terms are treated fully implicitly.

A primary goal of numerical radiation transport is obtaining a selfconsistent solution for both the radiation field and plasma properties, which requires consideration of the coupling between the radiation and the plasma. The different characteristics of this coupling for continuum and line radiation have resulted in two separate sub-disciplines of radiation transport with distinct emphases and computational techniques. LTE radiation transfer focuses on energy transport and exchange through broadband radiation, primarily affecting temperature and ionization balance. Non-LTE line transfer focuses on narrowband radiation and the response of individual level populations, primarily affecting spectral properties. Many high energy density applications, particularly those with high-Z materials, incorporate characteristics of both these regimes. Applications where the radiation fields play an important role in the energy balance and include strong line components require a non-LTE broadband treatment of energy transport and exchange.

Numerical simulation of multidimensional particle transport problems falls into the category of the most complex and labor-intensive application problems.

In this paper, we investigate performance of a fully implicit formulation and solution method of a diffusion-reaction system modeling radiation diffusion with material energy transfer and a fusion fuel source. In certain parameter regimes this system can lead to a rapid conversion of potential energy into material energy. Accuracy in time integration is essential for a good solution since a major fraction of the fuel can be depleted in a very short time. Such systems arise in a number of application areas including evolution of a star [1] and inertial confinement fusion [2].

This paper concerns the analysis of approximations of transport equations in diffusive media. Firstly, we consider a variational formulation for the firstorder transport equation that has the correct diffusive behavior in the limit of small mean free paths. The associated bilinear form is shown to be coercive on a classical Hilbert space in transport theory with a constant of coercivity independent of the mean free path. This allows us to obtain the diffusion approximation of transport as an orthogonal projection onto a subspace of functions that are independent of the angular variable. Similarly, projections onto functions that are independent of the angular variable only in subsets of the full domain can be interpreted as a transport-diffusion coupling method. Convergence results based on averaging lemmas and error estimates are presented. Secondly, we address the problem of extended non-scattering layers or filaments surrounded by highly scattering media and derive generalized diffusion equations to model transport in such geometries.

We apply the nonlinear WENO (Weighted Essentially Nonoscillatory) scheme to the spatial discretization of the Boltzmann Transport Equation modeling linear particle transport. The method is a finite volume scheme which ensures not only conservation, but also provides for a more natural handling of boundary conditions, material properties and source terms, as well as an easier parallel implementation and post processing. It is nonlinear in the sense that the stencil depends on the solution at each time step or iteration level. By biasing the gradient calculation towards the stencil with smaller derivatives, the scheme eliminates the Gibb’s phenomenon with oscillations of size O(1) and recudes them to O(hr), where h is the mesh size and r is the order of accuracy. Our current implementation is three-dimensional, generalized for unequally spaced meshes, fully parallelized, and up to fifth order accurate (“WENO5”) in space. For unsteady problems, the resulting nonlinear spatial discretization yields a set of ODE’s in time, which in turn is solved via high order implicit time-stepping with error control. For the steady-state case, we need to solve the non-linear system, typically by Newton-Krylov iterations. There are several numerical examples presented to demonstrate the accuracy, nonoscillatory nature and efficiency of these high order methods, in comparison with other fixed-stencil schemes.

Domain decomposed Monte Carlo codes, like other domain-decomposed codes, are difficult to debug. Domain decomposition is prone to error, and interactions between the domain decomposition code and the rest of the algorithm often produces subtle bugs. These bugs are particularly difficult to find in a Monte Carlo algorithm, in which the results have statistical noise. Variations in the results due to statistical noise can mask errors when comparing the results to other simulations or analytic results.

One of the main difficulties during finite-difference simulation of spectral 2D problems of particle transport and interaction with medium is to find cost-efficient solutions to rather large systems of interconnected difference equations.

Extremely short collision mean free paths and near-singular elastic and inelastic differential cross sections (DCS) make analog Monte Carlo and deterministic computational approaches impractical for charged particle transport. The widely used alternative, the condensed history method, while efficient, also suffers from several limitations arising from the use of precomputed infinite medium distributions for sampling particle directions and energies. Accordingly, considerable attention has recently focused on the development of computationally efficient algorithms that implement the correct transport mechanics. Fokker-Planck [JEM81] and Boltzmann Fokker-Planck [CL83] approximations have historically proved very useful in handling highly peaked scattering in certain classes of problems but these approaches are limited in the accuracy they can ultimately deliver. A more general methodology that allows accuracy to be systematically increased with practically no enhancement of algorithmic complexity has become possible with the advent of recently proposed higher order Fokker-Planck expansions [GCP96] and their implementation in so-called Generalized Fokker-Planck models [LL01,PP01,PKH02]. The goal of these newer approaches is to approximate the analog transport problem by one which is characterized by longer or stretched mean free paths and nonsingular collision operators but which can be solved numerically with considerably less effort than the analog problem and whose accuracy and efficiency can be readily adapted to a broad class of problems. One such implementation that has proved particularly efficient uses purely discrete scattering angle and hybrid discrete-continuous scattering angle representations [FPKL1,FPKL2]. Moreover, generalizations of these methodologies to describe energy-loss straggling have been successfully demonstrated [PKH02].

The spherical harmonics (P _n) approximation to the transport equation for time dependent problems has previously been treated using Riemann solvers and explicit time integration. Here we present an implicit time integration method for the P _n equations using Riemann solvers. Both first-order and high-resolution spatial discretization schemes are detailed. One facet of the high-resolution scheme is that a system of nonlinear equations must be solved at each time step. This nonlinearity is the result of slope reconstruction techniques necessary to avoid the introduction of artifical extrema in the numerical solution. Results are presented that show auspicious agreement with analytical solutions using time steps well beyond the CFL limit.

The rapid growth of computing power, in the form of parallel architectures, over the last decade has provided the unprecedented capability for computational scientists and engineers to carry out large scale simulations of radiation transport and radiation-hydrodynamic phenomena. The development of massively parallel architectures on the scale of tens of thousands of processors provides, in principle, the rate of floating point operations needed to carry out multidimensional deterministic transport simulations involving multiple physical timescales. However, this new technological advance presents a tremendous challenge to the transport simulation developer in implementing a method for the parallel solution of the time-dependent discrete-ordinates Boltzmann equation on such platforms. Traditional iterative methods, such as source iteration, that have been developed in many research communities have undesirable features that present obstacles to efficient parallelization. In this paper we present an alternative approach, the full linear system solution via Krylov subspace algorithms, that is more readily amenable to implementation on massively parallel architectures.

Numerical solution of a non-stationary kinetic equation at a multi-group setting in the 2D spatial approximation demands much computer operating memory and calendar execution time. One of the ways of solving such problems is using algorithms of parallelization.

AMTRAN, a one, two and three dimensional Sn neutron transport code with adaptive mesh refinement (AMR) has been parallelized with MPI over spatial domains and energy groups and with threads over angles. Block refined AMR is used with linear finite element representations for the fluxes, which are node centered. AMR requirements are determined by minimum mean free path calculations throughout the problem and can provide an order of magnitude or more reduction in zoning requirements for the same level of accuracy, compared to a uniformly zoned problem.

We briefly summarize (i) the general characteristics of neutron (and photon) transport processes relevant to nuclear reactor problems, (ii) the nature of calculation techniques for these problems, and (iii) current areas of research aimed at improving the accuracy and efficiency of these techniques.