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Title: Behavior of the Diamond Difference and Low-Order Nodal Numerical Transport Methods in the Thick Diffusion Limit for Slab Geometry

Abstract

The objective of this work is to investigate the thick diffusion limit of various spatial discretizations of the one-dimensional, steady-state, monoenergetic, discrete ordinates neutron transport equation. This work specifically addresses the two lowest order nodal methods, AHOT-N0 and AHOT-N1, as well as reconsiders the asymptotic limit of the Diamond Difference method. The asymptotic analyses of the AHOT-N0 and AHOT-N1 nodal methods show that AHOT-N0 does not possess the thick diffusion limit for cell edge or cell average fluxes except under very limiting conditions, which is to be expected considering the AHOT-N0 method limits to the Step method in the thick diffusion limit. The AHOT-N1 method, which uses a linear in-cell representation of the flux, was shown to possess the thick diffusion limit for both cell average and cell edge fluxes. The thick diffusion limit of the DD method, including the boundary conditions, was derived entirely in terms of cell average scalar fluxes. It was shown that, for vacuum boundaries, only when σ t, h, and Q are constant and σ a = 0 is the asymptotic limit of the DD method close to the finite-differenced diffusion equation in the system interior, and that the boundary conditions between the systems willmore » only agree in the absence of an external source. For a homogeneous medium an effective diffusion coefficient was shown to be present, which was responsible for causing numeric diffusion in certain cases. A technique was presented to correct the numeric diffusion in the interior by altering certain problem parameters. Numerical errors introduced by the boundary conditions and material interfaces were also explored for a two-region problem using the Diamond Difference method. A discrete diffusion solution which exactly solves the one-dimensional diffusion equation in a homogeneous region with constant cross sections and a uniform external source was also developed and shown to be equal to the finite-differenced diffusion discretization for c = 1 in the system interior, where again the boundary conditions again only agree in the absence of an external source. It was also shown that for c < 1 the exact discrete diffusion solution is written in terms of hyperbolic functions, with expressions which limit to the exact solution for the c = 1 case as c → 1. Finally, a transport discretization is developed which reproduces the exact S2 solution for the case of a purely scattering homogeneous region with vacuum boundary conditions, and an extension to the discretization for the case of c < 1 is found by considering a Taylor series expansion of the exact answer centered at c = 0.« less

Authors:
 [1]
  1. Pennsylvania State Univ., University Park, PA (United States)
Publication Date:
Research Org.:
Knolls Atomic Power Laboratory (KAPL), Niskayuna, NY (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
903208
Report Number(s):
LM-07K035
TRN: US200719%%642
DOE Contract Number:
AC12-00SN39357
Resource Type:
Thesis/Dissertation
Country of Publication:
United States
Language:
English
Subject:
42 ENGINEERING; BOUNDARY CONDITIONS; CROSS SECTIONS; DIAMONDS; DIFFUSION; DIFFUSION EQUATIONS; DISCRETE ORDINATE METHOD; EXACT SOLUTIONS; GEOMETRY; NEUTRON TRANSPORT; SCALARS; SCATTERING; SERIES EXPANSION; TRANSPORT

Citation Formats

Gill, Daniel Fury. Behavior of the Diamond Difference and Low-Order Nodal Numerical Transport Methods in the Thick Diffusion Limit for Slab Geometry. United States: N. p., 2007. Web. doi:10.2172/903208.
Gill, Daniel Fury. Behavior of the Diamond Difference and Low-Order Nodal Numerical Transport Methods in the Thick Diffusion Limit for Slab Geometry. United States. doi:10.2172/903208.
Gill, Daniel Fury. Tue . "Behavior of the Diamond Difference and Low-Order Nodal Numerical Transport Methods in the Thick Diffusion Limit for Slab Geometry". United States. doi:10.2172/903208. https://www.osti.gov/servlets/purl/903208.
@article{osti_903208,
title = {Behavior of the Diamond Difference and Low-Order Nodal Numerical Transport Methods in the Thick Diffusion Limit for Slab Geometry},
author = {Gill, Daniel Fury},
abstractNote = {The objective of this work is to investigate the thick diffusion limit of various spatial discretizations of the one-dimensional, steady-state, monoenergetic, discrete ordinates neutron transport equation. This work specifically addresses the two lowest order nodal methods, AHOT-N0 and AHOT-N1, as well as reconsiders the asymptotic limit of the Diamond Difference method. The asymptotic analyses of the AHOT-N0 and AHOT-N1 nodal methods show that AHOT-N0 does not possess the thick diffusion limit for cell edge or cell average fluxes except under very limiting conditions, which is to be expected considering the AHOT-N0 method limits to the Step method in the thick diffusion limit. The AHOT-N1 method, which uses a linear in-cell representation of the flux, was shown to possess the thick diffusion limit for both cell average and cell edge fluxes. The thick diffusion limit of the DD method, including the boundary conditions, was derived entirely in terms of cell average scalar fluxes. It was shown that, for vacuum boundaries, only when σt, h, and Q are constant and σa = 0 is the asymptotic limit of the DD method close to the finite-differenced diffusion equation in the system interior, and that the boundary conditions between the systems will only agree in the absence of an external source. For a homogeneous medium an effective diffusion coefficient was shown to be present, which was responsible for causing numeric diffusion in certain cases. A technique was presented to correct the numeric diffusion in the interior by altering certain problem parameters. Numerical errors introduced by the boundary conditions and material interfaces were also explored for a two-region problem using the Diamond Difference method. A discrete diffusion solution which exactly solves the one-dimensional diffusion equation in a homogeneous region with constant cross sections and a uniform external source was also developed and shown to be equal to the finite-differenced diffusion discretization for c = 1 in the system interior, where again the boundary conditions again only agree in the absence of an external source. It was also shown that for c < 1 the exact discrete diffusion solution is written in terms of hyperbolic functions, with expressions which limit to the exact solution for the c = 1 case as c → 1. Finally, a transport discretization is developed which reproduces the exact S2 solution for the case of a purely scattering homogeneous region with vacuum boundary conditions, and an extension to the discretization for the case of c < 1 is found by considering a Taylor series expansion of the exact answer centered at c = 0.},
doi = {10.2172/903208},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Tue May 01 00:00:00 EDT 2007},
month = {Tue May 01 00:00:00 EDT 2007}
}

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  • The neutron flux across the nuclear reactor core is of interest to reactor designers and others. The diffusion equation, an integro-differential equation in space and energy, is commonly used to determine the flux level. However, the solution of a simplified version of this equation when automated is very time consuming. Since the flux level changes with time, in general, this calculation must be made repeatedly. Therefore solution techniques that speed the calculation while maintaining accuracy are desirable. One factor that contributes to the solution time is the spatial flux shape approximation used. It is common practice to use the samemore » order flux shape approximation in each energy group even though this method may not be the most efficient. The one-dimensional, two-energy group diffusion equation was solved, for the node average flux and core k-effective, using two sets of spatial shape approximations for each of three reactor types. A fourth-order approximation in both energy groups forms the first set of approximations used. The second set used combines a second-order approximation with a fourth-order approximation in energy group two. Comparison of the results from the two approximation sets show that the use of a different order spatial flux shape approximation results in considerable loss in accuracy for the pressurized water reactor modeled. However, the loss in accuracy is small for the heavy water and graphite reactors modeled. The use of different order approximations in each energy group produces mixed results. Further investigation into the accuracy and computing time is required before any quantitative advantage of the use of the second-order approximation in energy group one and the fourth-order approximation in energy group two can be determined.« less
  • A family of highly efficient multidimensional multigroup advanced neutron-diffusion nodal methods, ILLICO, were implemented on sequential, vector, and vector-concurrent computers. Three-dimensional realistic benchmark problems can be solved in vectorized mode in less than 0.73 s (33.86 Mflops) on a Cray X-MP/48. Vector-concurrent implementations yield speedups as high as 9.19 on an Alliant FX/8. These results show that the ILLICO method preserves essentially all of its speed advantage over finite-difference methods. A self-consistent higher-order nodal diffusion method was developed and implemented. Nodal methods for global nuclear reactor multigroup diffusion calculations which account explicitly for heterogeneities in the assembly nuclear properties weremore » developed and evaluated. A systematic analysis of the zero-order variable cross section nodal method was conducted. Analyzing the KWU PWR depletion benchmark problem, it is shown that when burnup heterogeneities arise, ordinary nodal methods, which do not explicitly treat the heterogeneities, suffer a significant systematic error that accumulates. A nodal method that treats explicitly the space dependence of diffusion coefficients was developed and implemented. A consistent burnup-correction method for nodal microscopic depletion analysis was developed.« less
  • A new high-accuracy, coarse-mesh, nodal integral approach is developed for the efficient numerical solution of linear partial differential equations. It is shown that various special cases of this general nodal integral approach correspond to several high efficiency nodal methods developed recently for the numerical solution of neutron diffusion and neutron transport problems. The new approach is extended to the nonlinear Navier-Stokes equations of fluid mechanics; its extension to these equations leads to a new computational method, the nodal integral method which is implemented for the numerical solution of these equations. Application to several test problems demonstrates the superior computational efficiencymore » of this new method over previously developed methods. The solutions obtained for several driven cavity problems are compared with the available experimental data and are shown to be in very good agreement with experiment. Additional comparisons also show that the coarse-mesh, nodal integral method results agree very well with the results of definitive ultra-fine-mesh, finite-difference calculations for the driven cavity problem up to fairly high Reynolds numbers.« less