Asymptotic Diffusion Limits of the Multigroup Neutron Transport Equations

This work proposes to implement the cell calculation considering the fuel burning using the WIMSD-5B code. The cell calculation procedure allows determining the nuclear parameters present in the multi-group neutron diffusion equation and for this purpose the neutron transport theory is used in a problem with dimensional reduction, but in contrast is considered a large number of groups associated with the neutron spectrum. There are a variety of reactor physics codes that determine the nuclear parameters by solving the neutron transport equation applied to an equivalent cell representing a fuel element. The WIMSD-5B code is a deterministic code that solves the transport equation using collision probability method. The simulation of fuel burning in the cell calculation took into account different nuclear data libraries. The WIMSD-5B code supports several nuclear data libraries and in the present work the following libraries were used: IAEA, ENDFB-VII.1, JENDL3.2, JEFF3.1 and JEF2.2, all formatted for 69 energy groups.

SummaryThe multigroup neutron diffusion equations (an approximation of the neutron transport equation) are widely used for studying the motion of neutrons and their interactions with stationary background materials. Solving the multigroup neutron diffusion equations is challenging because the unknowns are tightly coupled through scattering and fission events, and solutions with high spatial resolution of full reactor cores in multiphysics environments are frequently required. In this paper, we focus on the development of a scalable, parallel preconditioner for solving the system of equations arising from the finite‐element discretization of the multigroup neutron diffusion equations in space and an implicit finite‐difference scheme in time. The parallel preconditioner (here referred to as the “fully coupled Schwarz preconditioner”) is constructed by monolithically applying the overlapping domain decomposition method together with a smoothed aggregation‐based coarse space to the coupled system. Our approach is different from the traditional block Gauss–Seidel sweep method that applies the preconditioner from the fast group to the thermal group sequentially, and we demonstrate that it provides significant improvements in terms of both the number of iterations required and the total compute time for a system of equations with millions of unknowns on a large supercomputer.

In this paper, we developed a new approach of analytic function expansion nodal (AFEN) method to solve the multi-group and multi-dimensional neutron diffusion equation in reactor cores with hexagonal fuel assembly. This method represents a multidimensional intra nodal flux distribution in terms of analytic basis functions at any points in the node. New types of boundary conditions have been considered that constrain the intranodal flux distributions in the hexagonal-z node, which include twelve radial surface-averaged partial currents and two axial surface-averaged partial currents. We utilized the coarse group rebalancing (CGR) method to increase the speed of code calculations. The computer code takes a few-groups cross sections produced by a lattice code and calculates the effective multiplication factor (keff ), flux in multi-group energy, reactivity, and the relative power density at each fuel assembly. Finally, the solution accuracy is tested for two well-known benchmark problems. The numerical results demonstrate that the new AFEN method is an accurate method for calculating keff and power density distribution in hexagonal-z geometries.

In the present work, we discuss the multi-group neutron diffusion equation solution in a homogeneous cylinder geometry case. We construct two kinds of solutions for the related problem considering two energy groups and a steady-state case. Upon applying the finite Hankel transform to the diffusion equation, one obtains a generalized system for the both neutron fluxes (fast and thermal), which may be solved in a closed form solution and depends on the specific source terms. As an equivalent approach we solve the problem by the infinite Hankel transform. Comparison of the two results shows that both methods provide acceptable solutions which are comparable to the ones in the literature. Applications for four parameter sets allow to conject that the infinite Hankel transform is the simpler approach although approximate, but providing solutions comparable to the exact solution by the finite Hankel transform.

Polynomial relaxation in the quasi-static approach and its implementation in nonlinear reactor transient analyses

Nodal Integral method for multi-group neutron diffusion equation in three dimensional cylindrical coordinate system

We show that diffusion approximations, including modified diffusion approximations, can be problematic since the proper choice of local boundary conditions (if any exist) is not obvious. For a class of Markov processes in one dimension, we show that to leading order it is proper to use a diffusion (Fokker-Planck) approximation to compute mean exit times with a simple absorbing boundary condition. However, this is only true for the leading term in the asymptotic expansion of the mean exit time. Higher order correction terms do not, in general, satisfy simple absorbing boundary conditions. In addition, the diffusion approximation for the calculation of mean exit times is shown to break down as the initial point approaches the boundary, and leads to an increasing relative error. By introducing a boundary layer, we show how to correct the diffusion approximation to obtain a uniform approximation of the mean exit time. We illustrate these considerations with a number of examples, including a jump process which leads to Kramers' diffusion model. This example represents an extension to a multivariate process.

We describe a procedure to obtain an approximate solution to the cosmic ray transport equation, which, contrary to the familiar diffusion approximation, is valid also for large anisotropies. Using some moments of the distribution function an approximation is constructed, in accordance with Jaynes’ principle of maximum entropy. We apply the procedure to the case of the one-dimensional transport equation and compare the resulting maximum entropy approximation and the diffusion approximation with the numerical solution. We find, that there is a qualitative agreement between the m.e.-approximation and the numerical solution, particularly also close to the particle source where the diffusion approximation breaks down.

We study the modeling and simulation of steady-state measurements of light scattered by a turbid medium taken at the boundary. In particular, we implement the recently introduced corrected diffusion approximation in two spatial dimensions to model these boundary measurements. This implementation uses expansions in plane wave solutions to compute boundary conditions and the additive boundary layer correction, and a finite element method to solve the diffusion equation. We show that this corrected diffusion approximation models boundary measurements substantially better than the standard diffusion approximation in comparison to numerical solutions of the radiative transport equation.

On asymptotic convergence and boundedness of stochastic systems with time-delay

On an improved Direct Discrete Method and its application in two dimensional multi-group neutron diffusion equation

PARCS code multi-group neutron diffusion constants generation using Monte Carlo method

Strong interplanetary coronal mass ejection driven shocks are often accompanied by high levels of low frequency compressive wave turbulence. This might require a non‐linear kinetic theory approach to properly describe energetic particle transport in their vicinity. We present a non‐linear diffusive kinetic theory for suprathermal particle transport and stochastic acceleration along the background magnetic field in strong compressive dynamic wave turbulence to which small‐scale Alfvén waves are coupled. Our theory shows that the standard cosmic‐ray transport equation must be revised for low suprathermal particle energies to accommodate fundamental changes in spatial diffusion (standard diffusion becomes turbulent diffusion) as well as modifications to particle convection, and adiabatic energy changes. In addition, a momentum diffusion term, which generates accelerated suprathermal particle spectra with a hard power law, must be added. Such effective first stage acceleration possibly leads to efficient injection of particles into second stage diffusive shock acceleration as described by standard theory.

The behavior of scalar wave propagation in a wide class of asymptotically conservative, dispersive, weakly inhomogeneous and weakly nonstationary, anisotropic, random media is investigated on the basis of a stochastic, collisionless, Liouville-type equation governing the temporal evolution of a phase-space Wigner distribution density function. Within the framework of the first-order smoothing approximation, a general diffusion–convolution-type kinetic or transport equation is derived for the mean phase-space distribution function containing generalized (nonloral, with memory) diffusion, friction, and absorption operators in phase space. Various levels of simplification are achieved by introducing additional constraints. In the long-time, Markovian, diffusion approximation, a general set of Fokker–Planck equations is derived. Finally, special cases of these equations are examined for spatially homogeneous systems and isotropic media.

In vivo bioluminescence imaging (BLI) has played a more and more important role in biomedical research of small animals. Bioluminescence tomography (BLT) further translates the BLI optical information into three-dimensional bioluminescent source distribution, which could greatly facilitate applications in related studies. Although the diffusion approximation (DA) is one of the most widely-used forward models, higher-order approximations are still needed for in vivo small animal imaging. In this work, as a relatively accurate and higher-order approximation theory, the performance of the simplified spherical harmonics approximation (SPN) in BLT is evaluated detailedly in heterogeneous small animals. In the numerical validations, the SPN based results demonstrate better imaging quality compared with diffusion approximation heterogeneously under various source locations over wide optical domain. Although the evaluation for the effects of the optical property mismatch indicates the sensitivity of SPN is similar with DA model in the source localization, it may offer improved performance with much less artifacts. In what follows, heterogeneous experimental BLT reconstructions using in vivo mouse further evaluate the capability of the higher-order method for practical biomedical applications.