We discuss several moment closure models for linear kinetic equations that have been developed over the past few years as alternatives to classical spectral and collocation methods. We then present numerical simulation results for a challenging benchmark problem known as the line source and observe the relative strengths and weaknesses of each closure. The results should prove useful for application scientists in making decisions about which methods are best suited to a given problem. They also provide a guide to help numerical experts who are interested in developing new, improved methods.

The Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) formalism is revisited in the framework of plasma spectroscopy. We address the issue of Stark line shape modeling by using kinetic transport equations. In the most simplified treatment of these equations, triple correlations between an emitter and the perturbing charged particles are neglected and a collisional description of Stark effect is obtained. Here we relax this assumption and retain triple correlations using a generalization of the Kirkwood truncature hypothesis to quantum operator. An application to hydrogen lines is done in the context of plasma diagnostic. It is shown that the neglect of triple correlations can lead to a significant overestimate of the line width.

The critical eigenvalues of the transport equation play an important role in the description of the dynamics of transport problems both in nuclear physics as well as in radiative transport theory. This article treats the problem of calculating numerically the critical spectrum of the transport equation with semireflecting boundary conditions. The eigenvalue problem is solved using spectral methods and numerical results are presented. The scattering kernel is considered to be one of three types, namely, isotropic, linearly anisotropic, or Rayleigh scattering, even although more general kernels could be considered.

We discuss heat conductance and diffusion, two remarkable transport processes characterized by instantaneous actions. We show that the assumption of local thermal equilibrium sets a limit to the speed of change in the distribution function of a statistical system . Using Onsager’s approximation, we show that the balance equations of the extensive parameters also have solutions with finite velocities involved. At the same time the infinite speed is obtainable when second order terms are neglected. We show how the presented technique is applied in heat transfer, diffusion, and plasma physics to determine the speeds of the physical processes.

This article presents a review of the special features exhibited by the transport of photons in dense vegetation canopies. The conservation of photons for particles moving in canopy media yields a linear Boltzmann Equation similar to conventional participating media. However, because of the turbid medium approximation where the atomic leaves are modeled as a gas cloud of infinitesimal plates with specified orientation, the overall system is not rotationally invariant. The latter affects the mathematical nature of the canopy equation and requires special numerical treatment to generate accurate numerical solutions. Here, the most common solution techniques are reviewed with special emphasis on two highly accurate methodologies, namely the Singular Eigenfunction Expansion, which is the basis of the F N method for canopy transport and the Analytical Discrete Ordinate (ADO) method. It is shown that the conventional ADO method can be easily extended to canopy transport problems by taking advantage of special symmetries exhibited by the intercept function and the area scattering function employed to describe the canopy optical properties.

This article describes the 3D discrete ordinates–based coupled code system TORT-TD/ATTICA3D that aims at steady state and transient analyses of pebble-bed high-temperature gas cooled reactors. In view of increasing computing power, the application of time-dependent neutron transport methods becomes feasible for best estimate evaluations of safety margins. The calculation capabilities of TORT-TD/ATTICA3D are presented along with the coupling approach, with focus on the time-dependent neutron transport features of TORT-TD. Results obtained for the OECD/NEA/NSC PBMR-400 benchmark demonstrate the transient capabilities of TORT-TD/ATTICA3D.

It is well known that the diffusion synthetic acceleration (DSA) methods for the Sn equations become ineffective in the Fokker-Planck forward-peaked scattering limit. In response to this deficiency, Morel and Manteuffel (1991) developed an angular multigrid method for the 1-D Sn equations. This method is very effective, costing roughly twice as much as DSA per source iteration, and yielding a maximum spectral radius of approximately 0.6 in the Fokker-Planck limit. Pautz, Adams, and Morel (PAM) (1999) later generalized the angular multigrid to 2-D, but it was found that the method was unstable with sufficiently forward-peaked mappings between the angular grids. The method was stabilized via a filtering technique based on diffusion operators, but this filtering also degraded the effectiveness of the overall scheme. The spectral radius was not bounded away from unity in the Fokker-Planck limit, although the method remained more effective than DSA. The purpose of this article is to recast the multidimensional PAM angular multigrid method without the filtering as an Sn preconditioner and use it in conjunction with the Generalized Minimal RESidual (GMRES) Krylov method. The approach ensures stability and our computational results demonstrate that it is also significantly more efficient than an analogous DSA-preconditioned Krylov method.

Proteinaceous aggregation occurs through self-assembly—a process not entirely understood. In a recent article, Knowles and colleagues (2009) presented an analytical theory for amyloid fibril growth via secondary rather than primary nucleation. Remarkably, with only a single kinetic parameter, the authors were able to unify growth characteristics for a variety of experimental data. In essence, they seem to have uncovered the underlying allometric law governing the evolution of filament elongation simply from two coupled nonlinear ordinary differential equations originally obtained from a master equation. While this work adds significantly to our understanding of filament self-assembly, it required an “approximate” analytical solution representation for the moments of the chain length distribution. If this were always true, the discovery of such scaling laws would be infrequent. Here, we show that the same results are found by purely numerical means. In addition, the numerical method used features a highly accurate solution strategy for the coupled Ordinary Differential Equations (ODEs) based only on a fundamental finite difference scheme and convergence acceleration. Once a reliable numerical solution has been established, a dimensional analysis then provides the scaling laws.

Transition rates in continuously driven steady states (relevant to sheared complex fluids) were derived in Evans (2004, 2005) by demanding that no information other than the microscopic laws of motion and the macroscopic observables of the system be used to describe it. This implies that the (nonequilibrium) reservoir, to which the system is weakly coupled, is fully characterized by its mean energy and mean flux. While we expect the resulting prescription for the rates in continuous- and discretized-time models to be equivalent, it is not trivial to see this from the expression for the rates derived in Evans (2005). We demonstrate this equivalence for a model of activated processes solved previously for continuous time (Evans 2005), thus demonstrating consistency of the theory.