Improved Interpolation Schemes in Anisotropic Source Flux Iteration Techniques

Gopinath, Deepak; Natarajan, A.; Sundararaman, V.

doi:10.13182/nse80-a21307

Cited by 21 publications

(4 citation statements)

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“…A quadratic representation of the source distribution has also been developed for onedimensional Cartesian problems. 31 In this Quadratic method (QM), the spatial variation of the source term Q is approximated by a second order polynomial. For a given computational cell, the cell average flux in the cell and in the two adjacent cells are used to determine the coefficients of the polynomial.…”

Section: Characteristics Methods In Reactor Applicationsmentioning

confidence: 99%

A discrete ordinates approximation to the neutron transport equation applied to generalized geometries

DeHart¹

1992

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A method for applying the discrete ordinates method for solution of the neutron transport equation in arbitrary two-dimensional meshes has been developed. The finite difference approach normally used to approximate spatial derivatives in extrapolating angular fluxes across a cell is replaced by direct solution of the characteristic form of the transport equation for each discrete direction. Thus, computational cells are not restricted to the traditional shape of a mesh element within a given coordinate system. However, in terms of the treatment of energy and angular dependencies, this method resembles traditional discrete ordinates techniques. Using the method developed here, a general two-dimensional space can be approximated by an irregular mesh comprised of arbitrary polygons.The method of characteristics, originally developed to eliminate negative fluxes encountered in many finite difference approximations, had been previously applied to regular cell structures in multiple dimensions and various coordinate systems. In such geometries, the geometrical relationships between sides were determined analytically and incorporated directly into the numerical model. However, the present work makes no assumptions about the orientations or the number of sides in a given cell, and computes all geometric relationships between each set of sides in each cell for each discrete direction. A set of nonreentrant polygons can therefore be used to represent any given two-dimensional space.iv Results for a number of test problems have been compared to solutions obtained from traditional methods, with good agreement. Comparisons include benchmarks against analytical results for problems with simple geometry, as well numerical results obtained from traditional discrete ordinates methods by applying the ANISN and TWOTRAN computer programs. Numerical results were obtained for problems ranging from simple onedimensional geometry to complicated multidimensional configurations. These results have demonstrated the ability of the developed method to closely approximate complex geometrical configurations and to obtain accurate results for problems that are extremely difficult to model using traditional methods. v DEDICATION This work is dedicated to my wife, Leigh. Her patience, understanding and love over the long road leading to this point made the path seem much less rocky; her encouragement and willingness to listen to my problems helped to sustain my efforts; and the daughter she has given us has taught me that miracles are possible. vi ACKNOWLEDGEMENTS I wish to express my sincere and profound gratitude to Ron Pevey and Ted Parish, both of whom provided invaluable support and insight in my endeavors. Their guidance, encouragement, technical assistance and senses of humor were invaluable assets. I also wish to acknowledge the support given to me by the United States Department of Energy, the Westinghouse Savannah River Company, and most importantly, by several members of my direct management and by my colleagues within the Savannah Riv...

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Section: Characteristics Methods In Reactor Applicationsmentioning

confidence: 99%

A discrete ordinates approximation to the neutron transport equation applied to generalized geometries

DeHart¹

1992

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“…Remarks. As discussed in [6,4], Lemmas 3 and 4 together will prove the superconvergence of the cell-average fluxes, i.e., that (2.18)…”

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confidence: 91%

“…The specific difference methods considered by Larsen and Nelson are the step characteristic, diamond difference, linear discontinuous, linear characteristic, linear moments, and quadratic methods described in [1], [5]. One of the salient results in the work by Larsen and Nelson is that the discretization errors for the cell-edge and cell-average flux approximates to the linear characteristic and quadratic methods of Gopinath et al., and of Vaidyanathan [4], [9], [10], are of order four, whereas the discretization errors for the corresponding approximates to the widely used linear discontinuous method are of order three. These theoretical results had been confirmed numerically by Larsen and Miller in [5], who found that the asymptotic convergence rates for the cell-edge and cell-average fluxes to the linear characteristic and quadratic methods can be observed on coarser meshes than for the lesser rate of order three fgr the corresponding fluxes of the linear discontinuous method.…”

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confidence: 99%

A New Fourth-Order Finite-Difference Method for Solving Discrete-Ordinates Slab Transport Equations

Neta¹,

Victory²

1983

SIAM J. Numer. Anal.

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This work is concerned with a theoretical study of a new fourth-order finite-difference scheme for spatially discretizing the discrete-ordinates equations for solving numerically the slab transport (Boltzmann) equation. This analysis considers the quadratic continuous method, whose derivation parallels that of the commonly used diamond difference and linear discontinuous schemes from balance equations for particle conservation across a spatial cell. We provide a convergence analysis of this method and prove that superconvergence phenomena are present for cell-edge and cell-average fluxes. We also present results from an S2-test problem to show that the asymptotic convergence rates are observed on rather coarse spatial meshes.

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“…Refs. 1,2,[5][6][7][8]17,18) in analyzing the consistency and order of spatial finite-difference methods as employed in transport calculations. In effect our aim here is to show that such results in fact carry over to problems with scattering (i.e.…”

Section: Introductionmentioning

confidence: 99%

Closed linear one-cell functional spatial approximations: Consistency implies convergence and stability

Keller

Nelson

1988

Transport Theory and Statistical Physics

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The subject of this paper is spatial finite-difference approximations to the discrete-ordinate equations for azimuthally symmetric transport of monoenergetic particles in plane-parallel geometry. In previous work it has been shown that almost all closed linear one-cell junctional (CLOF) methods are equivalent to a linear finite-difference method. Here it is shown that any such method that is consistent, in a certain natural sense, is equivalent to a consistent one-step finite-difference approximation to a linear two-point boundary-value problem, in the sense of earlier work of Keller and White. It follows, from observations in this same work, that for CLOF methods consistency implies both convergence and stability.

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Improved Interpolation Schemes in Anisotropic Source Flux Iteration Techniques

Cited by 21 publications

References 0 publications

A discrete ordinates approximation to the neutron transport equation applied to generalized geometries

A discrete ordinates approximation to the neutron transport equation applied to generalized geometries

A New Fourth-Order Finite-Difference Method for Solving Discrete-Ordinates Slab Transport Equations

Closed linear one-cell functional spatial approximations: Consistency implies convergence and stability

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