The propagation of infinitesimal disturbances in an ultrarelativistic gas according to the method of elementary solutions

It has recently been shown that a linearized relativistic BGK model can be reduced, in the ultrarelativistic limit, to a system of three uncoupled transport equations for thermal, sound, and shear waves. The equation describing the propagation of thermal waves is the well-known one-speed neutron transport with isotropic scattering in the conservative case. In this paper the solution of the half-space problem for the equation describing the propagation of shear and sound waves is given according to Case's elementary solutions method.     

The solution of the neutron transport equation in a slab with anisotropic scattering

The neutron transport equation for a slab geometry with the extremely anisotropic scattering kernel is considered. The albedo and transmission factors are calculated using the variation method. The effect of the extremely anisotropic parameter on the variation of the slab albedo and transmission factor is calculated. The obtained results are compared with the published data.     

一维球几何菱形差分中子输运方程的扩散综合加速解法

本文导出了一维球几何定态中子输运方程菱形格式的扩散综合加速方程,并给出了差分公式。所给出的加速方法可以加速菱形格式的输运方程的迭代求解。并给出了部分模型的数值计算结果。     

The FN solution of the time-dependent neutron transport equation for a sphere with forward scattering

The one-speed time-dependent and stationary neutron transport equation in spherical geometry with forward scattering is considered. A formal equivalence between the transport equations for a critical and for a decaying system is established. By considering the pseudo-slab problem the scaled transport equation is solved using the F_N method. Numerical values of radii for a critical and time-dependent systems are tabulated as a function of the scattering parameters and the fundamental decay constant. Some of the results are discussed and compared with others obtained using various methods. The results agree for four or five significant figures with the published results. It is shown that the F_N method yields good numerical results for the problem considered. Finally, a few remarks about the effect of the forward anisotropy on the radius is also given.     

Equations of the surface harmonics method for solving time-dependent neutron transport problems and their verification

Time-dependent equations of the surface harmonics method (SHM) are obtained for planar one-dimensional geometry. The equations are verified by calculations of test problems from Benchmark Problem Book ANL-7416, and the capabilities and efficiency of applying the SHM for solving the time-dependent neutron transport equation in the diffusion approximation are demonstrated. The results of the work show that the implementation of the SHG for full-scale computations will make possible substantial progress in the efficient solution of time-dependent problems of neutron transport in nuclear reactors.     

解二维中子输运问题的三角网格配置法

本文研究了二维轴对称中子输运(包括各向同性散射和各向异性散射)问题的一种有限元数值方法。给出了一系列的数值计算结果,并与国内外的SN、间断有限元方法的数值结果及实验结果进行比较,取得了令人满意的结果。     

A probabilistic study of the influence of parameter uncertainty on solutions of the radiative transfer equation

The influence of uncertainty in the absorption and scattering coefficients on the solution and associated parameters of the radiative transfer equation is studied using polynomial chaos theory. The uncertainty is defined by means of uniform and log-uniform probability distributions. By expanding the radiation intensity in a series of polynomial chaos functions we may reduce the stochastic transfer equation to a set of coupled deterministic equations, analogous to those that arise in multigroup neutron transport theory, with the effective multigroup transfer scattering coefficients containing information about the uncertainty. This procedure enables existing transport theory computer codes to be used, with little modification, to solve the problem. Applications are made to a transmission problem and a constant source problem in a slab. In addition, we also study the rod model for which exact analytical solutions are readily available. In all cases, numerical results in the form of mean, variance and sensitivity are given that illustrate how absorption and scattering coefficient uncertainty influences the solution of the radiative transfer equation.     

Diffusion approximations for the scattering of resonance-line photons: Interior and boundary layer solutions

Resonance-line scattering in static low density media with large optical thickness has a diffusive behavior in both space and frequency because photons belonging to the Lorentzian wings of the line may be scattered almost monochromatically a very large number of times. This diffusive behavior holds on frequency scales and spatial scales, χ_c and τ_c, much larger than the scales associated with one elementary scattering of a wing-photon.A method developed for diffusion approximations in neutron transport theory, suitably generalized to handle diffusion in frequency space, is applied to the case of conservative scattering in a bounded medium with interior sources and zero incoming radiation. The method is to separate the line radiation field into an interior part and a boundary layer part which goes to zero in the interior. Each part is expanded in terms of a small parameter ?, which is the ratio of the mean free-path at frequency χ_c to the characteristic spatial scale τ_c.It is shown that the leading term in the interior asymptotic expansion is isotropic, zero on the boundary, and obeys a space and frequency diffusion equation. In the boundary-layer expansion, the leading term is of order ? and is a solution to a monochromatic transfer equation in a semi-infinite, plane-parallel medium. The emergent radiation field is shown to be of order ? and proportional to the gradient of the interior solution at the boundary. Its angular dependence, in the case of isotropic scattering in the atom frame, is given by the Ambartsoumian H-function. A comparison is presented between numerical solutions of the full transfer equation and asymptotic solutions. Non-conservative scattering and time-dependent problems are briefly discussed.     

解二维柱几何输运问题的离散——配置有限元方法

本文研究了求解多群多种介质的二维柱几何中子输运问题的离散一配置方法(TCC)。该方法对时间t用Crank-Nicholson中心差分,源项采用离散纵标近似和源迭代技巧,对空间变量采用配置有限元方法。本文讨论了算法。证明了守恒性。并且证明了本方法在K=1时等价于DSN^[2]方法,因此本方法是DSN方法的高阶推广。本文还给出了数值计算结果。我们对精确解以及多个二维柱几何的临介物理问题进行了数值计算,由分析比较可见,取得了较好的结果。     

Adomian decomposition method for solving the telegraph equation in charged particle transport

In this paper, the analysis for the telegraph equation in case of isotropic small angle scattering from the Boltzmann transport equation for charged particle is presented. The Adomian decomposition is used to solve the telegraph equation. By means of MAPLE the Adomian polynomials of obtained series (ADM) solution have been calculated. The behaviour of the distribution function are shown graphically. The results reported in this article provide further evidence of the usefulness of Adomain decomposition for obtaining solution of linear and nonlinear problems.     

Variational method for radiation transfer in an isotropically scattering, two infinite medium slab lattice

The monoenergetic transport equation with an isotropic scattering is applied to an infinite, two-medium slab lattice. Two coupled integral equations are obtained for the two regions. Numerical results are obtained by using the variational method and compared with the published results.     

Transport and Diffusion in a Random Medium

We consider particle transport in a spatially random medium, the transport governed by the traditional, linear, time- and space-dependent transport equation for “host and guest.” The scattering is elastic and isotropic; there is no absorption. If the host medium has uniform density we know that an initial burst will, in time, approach the solution to the time-dependent diffusion equation. In the case of random medium we find that for a large class of such media the asymptotic behavior is unchanged by the stochasticity; there is neither renormalization of the equation nor the diffusion co-efficient.The nature of the correlation between fluctuations of density at large separation plays an important role in the analysis.     

TN approximation to neutron transport equation and application to critical slab problem

The critical slab problem has been studied in one-speed neutron transport equation with isotropic scattering by using the T_N method. T_N moment criticality solutions are obtained for the uniform finite slab using Mark and Marshak type vacuum boundary conditions. Results obtained by T_N method, using the two type boundary conditions mentioned above, were presented in the Tables and also the Tables included the results obtained by P_N method for the comparisons.     

Calculation of Prompt Fission Neutron Lifetimes by the Monte Carlo Method

An algorithm for calculation of prompt fission neutron lifetimes in a nuclear reactor by the Monte Carlo method is described. Evaluation of the importance function is carried out with solution of the neutron transport equation without solving the adjoint equation. The results of the prompt neutron lifetime calculations performed within some critical experiments are presented and compared with the experimental results.     

Half-range moment method for solution of the transport equation in a spherically symmetric geometry

A half-range moment method is presented for solving, in various orders of approximation, a multi-group transport equation subject to generalized boundary conditions in a spherically symmetric geometry. The results for the plane-parallel geometry are obtainable from the present analysis as a special case. The equations and the boundary conditions considered are sufficiently general to characterize a variety of problems in radiative transfer, neutron transport and phonon transport if various coefficients appearing in the equations are properly specified.     

On pseudo-continuities in the eigenvalue continuum of one-speed neutron transport theory

We report on a new closed-form approximant to the singular eigenfunction transient solution of the one-speed, one-dimensional neutron transport equation with an anisotropic scattering kernel. It is proved that the associated eigenvalue continuum must be pointwise perforated.     

An application of the K-integrals for solving the radiative transfer equation with Fresnel boundary conditions

We introduce the reader to an approximate method of solving the transport equation which was developed in the context of neutron thermalisation by Kladnik and Kuscer in 1962 [Kladnik R, Kuscer I. Velocity dependent Milne's problem. Nucl Sci Eng 1962;13:149]. Essentially the method is based upon two special weighted integrals of the one-dimensional transport equation which are valid regardless of the boundary conditions, and any solution must satisfy these integral relationships which are called the K-integrals. To obtain an approximate solution to the transport equation we turn the argument around and insist that any approximate solution must also satisfy the K-integrals. These integrals are particularly useful when the problem under consideration cannot be solved easily by analytic methods. It also has the marked advantage of being applicable to problems where there is energy exchange in a collision and anisotropy of scattering. To establish the feasibility of the method we obtain a number of approximate solutions using the K-integral method for problems to which we have exact analytical solutions. This enables us to validate the method. It is then applied to a new problem that has not yet been solved; namely the calculation of the discontinuity in the scalar intensity at the boundary between two optically dissimilar materials.     

Radiation transport with anisotropic scattering

Reflection and transmission of radiation with anisotropic scattering in considered, using an invariant-imbedding formulation of the transport equation. Extensive numerical calculations show that the results of the transport approximation, in which the scattering kernel is represented by an isotropic part and a forward scattered component, are useful for an isotropic incident source but they are not satisfactory for a monodirectional source, especially in the case of reflection for large and small incident angles. A modified transport approximation is proposed in which singly-scattered radiation is taken into account exactly and higher-order scatterings are obtained from the transport approximation. The results derived from the modified transport approximation are in good agreement with other calculations for the cases considered.     

二维柱几何各向异性输运问题的数值计算方法

本文研究了求解二维输运问题的各向异性三角网间断有限元数值方法。详细讨论了求解过程,给出了解的预估公式,证明了数值解的存在唯一性和数值方法的稳定性。给出了各向异性问题的二维数值计算结果及钨——碳临界装置TCCA^[11]的二维数值计算结果。并与国内外S_N数值计算及实验结果进行比较,取得了令人满意的结果。     

Pure-triplet scattering for radiative transfer in binary discrete random finite media with reflective boundaries and internal source of radiation

The stochastic solution of the monoenergetic radiative transfer equation in a finite slab random medium with pure-triplet anisotropic scattering is considered. The random medium is assumed to consist of two randomly mixed immiscible fluids labelled by 1 and 2. The extinction function, the scattering kernel, and the internal source of radiation are treated as discrete random variables, which obey the same statistics. The theoretical model used here for stochastic media transport assumes Markovian processes and exponential chord length statistics. The boundaries of the medium under consideration are considered to have specular and diffuse reflectivities with an internal source of radiation inside the medium. The ensemble-average partial heat fluxes are obtained in terms of the average albedos of the corresponding source-free problem, whose solution is obtained by using the Pomraning-Eddington approximation. Numerical results are calculated for the average forward and backward partial heat fluxes for different values of the single scattering albedo with variation of the parameters that characterize the random medium. Compared to the results obtained by Adams et al. in the case of isotropic scattering based on the Monte Carlo technique, it can be demonstrated that we have good comparable data.