An Extended Linear Discontinuous Method for One-group Fixed Source Discrete Ordinates Problems with Isotropic Scattering in Slab Geometry

Iram Barbaro Rivas-Ortiz, Dany Sanchez Dominguez, Carlos Rafael Garcia Hernandez, Susana Marrero Iglesias, Alberto Escrivá


Nowadays, the obtainment of an accurate numerical solution of fixed source discrete ordinates problems is relevant in many areas of engineering and science. In this work, we extend the hybrid Finite Element Spectral Green's Function method (FEM-SGF), originally developed to solve eigenvalue diffusion problems, for fixed source problems using as a mathematical model, the discrete ordinates formulation in one energy group with isotropic scattering in slab geometry. This new method, Extended Linear Discontinuous Discrete Ordinates (ELD-SN), is based on the use of neutron balance equations and the construction of a hybrid auxiliary equation. This auxiliary equation combines a linear discontinuous approximation and spectral parameters to approximate the neutron angular flux inside the cell. Numerical results for benchmark problems are presented to illustrate the accuracy and computational performance of our methodology. ELD-SN method is free from spatial truncation errors in S2 quadrature, and generate good results in the other quadrature sets. This method is more accurate than the conventional Diamond Difference (DD) and Linear Discontinuous (LD) methods, but surpassed by the Spectral Green's Function (SGF) method, for quadrature order greater than two.


fixed source problems; discrete ordinates formulation; hybrid method; linear-discontinuous; spectral parameters

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E. E. Lewis and J. W. F. Miller, Computational Methods of Neutron Transport. New York: Jhon Wiley & Sons, 1 ed., 1984.

G. F. Knoll, Radiation detection and measurement. John Wiley, 4 ed., 2010.

B. G. Carlson and K. D. Lathrop, "Transport theory - the method of discrete ordinates," in Computing Methods in Reactor Physics (H. Greenspan, C. N. Kelber, and O. D., eds.), ch. 3, New York: Gordon & Breach, 1968.

R. C. Barros and E. Larsen, "A Numerical Method for Multigroup Slab-

Geometry Discrete Ordinates Problems with No Spatial Truncation Error,"

Transport Theory and Statistical Physics, vol. 20, pp. 441-462, 1991.

R. C. Barros and E. Larsen, "A spectral nodal method for one-group x,y-geometry discrete ordinates problems," Nuclear Science and Engineering, vol. 111, pp. 34-45, 1992.

J. A. M. de Mello and R. C. Barros, "An exponential spectral nodal method for one-speed X,Y-geometry deep penetration discrete ordinates problems," Annals of Nuclear Energy, vol. 29, pp. 1855-1869, oct 2002.

D. S. Dominguez and R. C. Barros, "The spectral Green's function linear-nodal method for one-speed X,Y-geometry discrete ordinates deep penetration problems," Annals of Nuclear Energy, vol. 34, pp. 958-966, dec 2007.

M. P. de Abreu, H. A. Filho, and R. C. de Barros, "A numerical method

for multigroup slab-geometry eigenvalue problems in transport theory with no spatial truncation error," Transport Theory and Statistical Physics, vol. 25, pp. 61-83, aug 1996.

R. C. Barros, H. A. Filho, E. T. Valero Orellana, F. C. da Silva, N. do Couto, D. S. Dominguez, and C. R. Hernández, "The application of spectral nodal methods to discrete ordinates and diffusion problems in Cartesian geometry for neutron multiplying systems," Progress in Nuclear Energy, vol. 42, pp. 385-426, jan 2003.

R. C. Barros, C. R. Garcia, D. S. Dominguez, O. D. Garcia, and V. M. Tame, "Recent Advances in Spectral Nodal Methods for Numerically Solving Neutron - Diffusion Eigenvalue Problems," Transport Theory and Statistical Physics, vol. 33, no. 3-4, pp. 331-346, 2004.

D. S. Dominguez, F. B. S. Oliveira, H. Alves, and R. C. Barros, "Composite spatial grid spectral nodal method for one-speed discrete ordinates deep penetration problems in X , Y geometry," Progress in Nuclear Energy, vol. 52, pp. 298-303, 2010.

R. V. M. Rocha, D. S. Dominguez, S. M. Iglesias, and R. C. de Barros, "Finite Element Method with Spectral Green's Function in Slab Geometry for Neutron Diffusion in Multiplying Media and One Energy Group," TEMA: Tendências em Matemática Aplicada e Computacional, vol. 17, no. 2, pp. 173-186, 2016.

P. G. Maginot, J. C. Ragusa, and J. E. Morel, "Lumping Techniques for DFEM SN Transport in Slab Geometry," Nuclear Science and Engineering, vol. 179, pp. 148-163, feb 2015.

C. Ceolin, M. Schramm, B. E. J. Bodmann, M. T. Vilhena, and S. B. Leite, "On an analytical evaluation of the flux and dominant eigenvalue problem for the steady state multi-group multi-layer neutron diffusion equation," Kerntechnik, vol. 79, pp. 430-435, 2014.

E. W. Larsen and B. W. Kelley, "The Relationship between the Coarse-Mesh Finite Difference and the Coarse-Mesh Diffusion Synthetic Acceleration Methods," Nuclear Science and Engineering, vol. 178, pp. 1-15, sep 2014.

E. Sauter, F. S. Azevedo, M. Thompson, and M. T. M. B. Vilhena, "Solution of the one-dimensional transport equation by the vector Green function method: Error bounds and simulation," Applied Mathematics and Computation, vol. 219, pp. 11291-11301, 2013.

M. T. Vilhena, L. B. Barichello, J. Zabadal, C. F. Segatto, and A. V. Cardona, "General solution of one-dimensional approximations to the transport equation," Progress in Nuclear Energy, vol. 33, no. 1, pp. 99-115, 1998. Reviews from the X ENFIR/III ENAN Brazilian Joint Nuclear Conference.

C. F. Segatto, M. T. Vilhena, and M. G. Gomes, "The One-Dimensional LTSN Solution In a Slab With High Degree of Quadrature," PERGAMON Annals of Nuclear Energy, vol. 26, pp. 925-934, 1999.

T. R. Hill, "ONETRAN: A Discrete Ordinates Finite Element Code for the Solution of the One-Dimensional Muitigroup Transport Equation," tech. rep., Los Alamos National Laboratory, New Mexico, 1974. LA-5990-MS.

R. C. Barros, "On the Equivalence of Discontinuous Finite Element Methods and Discrete Ordinates Methods for the Angular Discretization of the Linearized Boltzmann Equation in Slab Geometry," Pergamon Ann. Nucl. Energy, vol. 24, no. 13, pp. 1013-1026, 1997.

K. M. Case and P. F. Zweifel, Linear Transport Theory. Reading, Massachussetts: Addison - Wesley, 1967.

W. A. Menezes, H. A. Filho, and R. C. de Barros, "Spectral Green's function nodal method for multigroup SN problems with anisotropic scattering in slab-geometry non-multiplying media," Annals of Nuclear Energy, vol. 64, pp. 270-275, feb 2014.

E. W. Larsen, "Spectral analysis of numerical methods for discrete-ordinates problems. I," Transport Theory and Statistical Physics, vol. 15, pp. 93-116, feb 1986.

R. C. Barros, A Spectral Nodal Method for The Solution of Discrete Ordinates Problems in One and Two Dimensional Cartesian Geometry. PhD thesis, University of Michigan, 1990.

R. C. Barros and E. W. Larsen, "A Numerical Method for One-Group Slab-Geometry Discrete Ordinates Problems with No Spatial Truncation Error," Nuclear Science and Engineering, vol. 104, no. 3, pp. 199-208, 1990.

Y. S. Jung and W. S. Yang, "A Consistent CMFD Formulation for the Acceleration of Neutron Transport Calculations Based on the Finite Element Method," Nuclear Science and Engineering, vol. 185, pp. 307-324, feb 2017.

D. M. Silva, E. J. Lydia, M. R. Guida, J. H. Zani, H. A. Filho, and R. C. Barros, "Analytical methods for computational modeling of fixed-source slab-geometry discrete ordinates transport problems: Response matrix and hybrid sn," Progress in Nuclear Energy, vol. 69, pp. 77-84, 2013.

R. S. Mansur, F. P. Santos, H. A. Filho, and R. C. Barros, "Diffusion synthetic methods for computational modeling of one-speed slab-geometry transport problems with linearly anisotropic scattering," Progress in Nuclear Energy, vol. 73, pp. 179-187, 2014.


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