Radiation Diffusion Equations Research Articles

On the density of metastable and resonance atoms in a stratifiedpositive column in neon

The densities of metastable and resonance states 33P2 (S5) and 31P1(S2) are measured by a method of laser absorption in S and P striations in neon at low(1.4 Torr) and intermediate (12.8 Torr) pressures. The theoretical description of excitedatoms in striations requires the solution of the nonlocal kinetic equation in a spatiallyperiodic field to obtain the excitation rates, as well as the analysis of both theresonance radiation transport and diffusion equations for excited atoms. The solution ofthe radiation transport equation is obtained for the case of both the radial and axial(in particular, periodic) inhomogeneities. The densities of excited atoms are calculatedin different striation phases taking into account the transport processes, directintermixing by atomic and electron impact, and intermixing by excitation and radiation ofthe system 2p53p. Comparison of the theoretical and experimental resultsis performed for weakly nonlinear striations near a lower bound of existence by pressureand striations at low and intermediate pressures.

A Finite Volume Method for the Approximation of Diffusion Operators on Distorted Meshes

A new finite volume method is presented for discretizing general linear or nonlinear elliptic second-order partial-differential equations with mixed boundary conditions. The advantage of this method is that arbitrary distorted meshes can be used without the numerical results being altered. The resulting algorithm has more unknowns than standard methods like finite difference or finite element methods. However, the matrices that need to be inverted are positive definite, so the most powerful linear solvers can be applied. The method has been tested on a few elliptic and parabolic equations, either linear, as in the case of the standard heat diffusion equation, or nonlinear, as in the case of the radiation diffusion equation and the resistive diffusion equation with Hall term.

Iterative Linear Solvers in a 2D Radiation–Hydrodynamics Code: Methods and Performance

Computer codes containing both hydrodynamics and radiation play a central role in simulating both astrophysical and inertial confinement fusion (ICF) phenomena. A crucial aspect of these codes is that they require an implicit solution of the radiation diffusion equations. We present in this paper the results of a comparison of five different linear solvers on a range of complex radiation and radiation–hydrodynamics problems. The linear solvers used are diagonally scaled conjugate gradient, GMRES with incomplete LU preconditioning, conjugate gradient with incomplete Cholesky preconditioning, multigrid, and multigrid-preconditioned conjugate gradient. These problems involve shock propagation, opacities varying over 5–6 orders of magnitude, tabular equations of state, and dynamic ALE (Arbitrary Lagrangian Eulerian) meshes. We perform a problem size scalability study by comparing linear solver performance over a wide range of problem sizes from 1000 to 100,000 zones. The fundamental question we address in this paper is: Is it more efficient to invert the matrix in many inexpensive steps (like diagonally scaled conjugate gradient) or in fewer expensive steps (like multigrid)? In addition, what is the answer to this question as a function of problem size and is the answer problem dependent? We find that the diagonally scaled conjugate gradient method performs poorly with the growth of problem size, increasing in both iteration count and overall CPU time with the size of the problem and also increasing for larger time steps. For all problems considered, the multigrid algorithms scale almost perfectly (i.e., the iteration count is approximately independent of problem size and problem time step). For pure radiation flow problems (i.e., no hydrodynamics), we see speedups in CPU time of factors of ≈15–30 for the largest problems, when comparing the multigrid solvers relative to diagonal scaled conjugate gradient. For the incomplete factorization preconditioners, we see a weak dependence of iteration count on problem size. The speedups observed for pure radiation flow are typically on the order of 10 relative to diagonal scaled conjugate gradient. For radiation–hydrodynamics problems, we again see multigrid scaling perfectly. However, for the problems considered, we see speedups relative to diagonal scaled conjugate gradient of no more than ≈10, with incomplete Cholesky in fact either equaling or outperforming multigrid. We trace these observations to the time step control and the feature of ALE to relax distorted zones.

Time Step Size Selection for Radiation Diffusion Calculations

This work presents efficient solution techniques for radiative transfer in the smoothed particle hydrodynamics discretization. Two choices that impact efficiency are how the material and radiation energy are coupled, which determines the number of iterations needed to converge the emission source, and how the radiation diffusion equation is solved, which must be done in each iteration. The coupled material and radiation energy equations are solved using an inexact Newton iteration scheme based on nonlinear elimination, which reduces the number of Newton iterations needed to converge within each time step. During each Newton iteration, the radiation diffusion equation is solved using Krylov iterative methods with a multigrid preconditioner, which abstracts and optimizes much of the communication when running in parallel. The code is verified for an infinite medium problem, a one-dimensional Marshak wave, and a two and three-dimensional manufactured problem, and exhibits first-order convergence in time and second-order convergence in space. For these problems, the number of iterations needed to converge the inexact Newton scheme and the diffusion equation is independent of the number of spatial points and the number of processors.

Boundary layers and domain decomposition for radiative heat transfer and diffusion equations: applications to glass manufacturing process

In this paper domain decomposition methods for radiative transfer problems including conductive heat transfer are treated. The paper focuses on semi-transparent materials, like glass, and the associated conditions at the interface between the materials. Using asymptotic analysis we derive conditions for the coupling of the radiative transfer equations and a diffusion approximation. Several test casts are treated and a problem appearing in glass manufacturing processes is computed. The results clearly show the advantages of a domain decomposition approach. Accuracy equivalent to the solution of the global radiative transfer solution is achieved, whereas computation time is strongly reduced.

The calculation of anisotropic extinction coefficients for radiation diffusion in rigid fibrous ceramic insulations

The specific anisotropic extinction coefficient ρ ∗ couples the effective radiative properties of a fibrous insulation into the radiation diffusion equation. This coefficient can be calculated using electromagnetic scattering theory if fiber diameters, refractive indices and fiber orientation distributions are known. In general, fiber orientation distributions are not readily accessible and past calculations have considered fibers as either randomly distributed or normal to the heat flow direction. In certain rigid fibrous ceramic insulations neither of these cases apply well, and a simple procedure is described for approximating ρ ∗ from values calculated for the random and normal orientation cases. The intrinsic error associated with this scaling procedure is investigated. Numerical computations for several test structures and fiber materials show the average error to be less than 5% for net heat flux and radiation conductivity calculations.

Effects of nonuniform reflective boundaries and line competition on radiation trapping.

We develop a radiation diffusion equation for an infinite slab geometry where the slab may have different reflectivities on either surface, and for conditions where complete frequency redistribution is valid. Additionally we consider the effect of two competing trapped transitions. Specifically we consider the emission from the thallium 7 $^{2}$${\mathit{S}}_{1/2}$ state to either the 6 $^{2}$${\mathit{P}}_{3/2}$ (metastable state) at 535 nm or 6 $^{2}$${\mathit{P}}_{1/2}$ (ground state) at 378 nm. High densities of the long-lived metastable level can be produced in a number of ways. The escape factors (which are simply related to the decay rates) and excited-atom spatial profiles are calculated for a variety of conditions for the first several natural modes of the system. The spatial modes for this system are asymmetric. The asymmetry can vary dramatically with only small changes in either of the two lower-state densities. The correct description of the spatial modes is critical in order to determine the overall excited-atom profile accurately. For the simpler systems (i.e., a single transition and nonreflective boundaries) usually considered the mode profiles do not change appreciably as the optical depth is changed (given conditions of high optical depth and a single-line broadening mechanism). The fundamental mode decay rate is shown for a fixed metastable-state density as a function of the ground-state density.The dependence of the fundamental mode escape factor for the first resonance transition on the optical depth (or ground-state density) is in general more complicated than for the case of a single trapped transition. For conditions considered here a simple power rule cannot be used to scale the decay rates as a function of line-center optical depth. Also shown is the fundamental mode escape factor as a function of buffer gas pressure for fixed ground- and metastable-state densities. The calculations presented here allow us to model the narrow-bandwidth thallium fluorescence filter. This filter will be discussed in more detail in a separate paper. A thorough review of atomic resonance filters was done by Gelbwachs [IEEE J. Quantum Electron. 24, 1266 (1988)]. We have considered a much broader range of ground- and metastable-state densities than would be practical for efficient filter operation. This is done to illustrate several aspects of the radiation trapping processes.

Error estimates for use of the generalized radiation diffusion equation near boundaries

Using the two-flux formulation, the diffusion law has been derived for an optically-thick medium. The diffusion law is not valid near surfaces or boundaries. The errors near a boundary have been calculated for three cases: (i) in isothermal porous layer, (ii) a linear profile for T 4 s, and (iii) the general case.

Test Problems in Radiative Transfer Calculations

Several test problems are presented for evaluating the radiation diffusion equations. For spatial transport schemes, one-dimensional problems with known analytic solutions are tested on two-dimensi...

Solution of the diffusion equation by finite elements in Lagrangian hydrodynamic Codes

The radiation diffusion equation is solved by the finite element method. The energy densities are point centered. These are integrated into a program architecture which requires zonally averaged quantities. Numerical results are presented which compare the scheme with existing finite difference techniques for zonal variables. Finite elements give better results for transport dominated problems on non-orthogonal meshes such as might be generated by Lagrangian hydrodynamic distortions.

A synthetic acceleration scheme for radiative diffusion calculations

A synthetic method has been developed to accelerate the iterative convergence of the multigroup radiative diffusion equations with temporally implicit material-radiation coupling. The performance of the method is characterized by means of a Fourier analysis. Computational results are given and are found to be consistent with the analysis. Overall, the method appears to be quite effective.

Theoretical Study of Cepheid Light Curves

Starting with the initial understanding that pulsation in variable stars is caused by the heat engine of Hydrogen and Helium ionization in their atmospheres (A.S. Eddington in Cox 1980) it was soon realized that non-linear effects were responsible for the detailed features on their light and velocity curves. With the advent of the computer we were able to solve the coupled set of hydrodynamics and radiation diffusion equations to model these non-linear features (Christy 1968, Cox et. al. 1966). Calculations including the effects of multi-frequency radiative transfer (Davis 1975) showed that grey diffusion was adequate for modeling Cepheids but not for the RR Lyrae or W Virginis type variables.

A method of solving implicit difference equations approximating systems of radiation transport and diffusion equations

A new iterative method is described for the simultaneous solution of the implicit difference equations which approximate the energy equation and the non-stationary spectral equation of radiation transport or diffusion. It is shown that, in the “grey” linear case, the method converges at the rate of a geometric progression.

An Investigation of Coupled D. C.-Induction Arc Discharges

The object of this study is to analytically determine the properties of an arc discharge generated by a combination of induction and d. c. electromagnetic fields. Induction power is supplied to the discharge by a solenoidal coil whose axis of symmetry is aligned with the applied d. c. electric field. A numerical technique is used to simultaneously solve a set of linear differential equations which include Maxwell's field equations, the single fluid energy equation, the momentum balance equation, Ohm's law, and the radiation diffusion equation. It is necessary to make a number of reasonable assumptions to obtain the final set of equations for programming. These assumptions restrict the calculations to azimuthally symmetric plasmas with axial gradients only in static pressure. In addition, the radiant energy flux is assumed to be isotropic within the plasma.