scholarly journals The modified spectral deterministic method applied to fixed–source discrete ordinates problems in X,Y–geometry

2021 ◽  
Vol 8 (3A) ◽  
Author(s):  
Jesús Pérez Curbelo ◽  
Rafael Barbosa Libotte ◽  
Amaury Muñoz Oliva ◽  
Ricardo Carvalho Barros ◽  
Hermes Alves Filho
Keyword(s):  
Neutron Transport ◽  
Iteration Scheme ◽  
Discrete Ordinates ◽  
Coarse Mesh ◽  
Neutron Transport Equation ◽  
Balance Equations ◽  
Deterministic Method ◽  
New Approach ◽  
One Dimensional ◽  
Typical Model

A new approach for the application of the coarse–mesh Modified Spectral Deterministic method to numerically solve the two–dimensional neutron transport equation in the discrete ordinates (Sn) formulation is presented in this work. The method is based on within node general solution of the conventional one–dimensional Sn transverse integrated equations considering constant approximations for the transverse leakage terms and obtaining the Sn spatial balance equations. The discretized equations are solved by using a modified Source Iteration scheme without additional approximations since the average angular fluxes are computed analytically in each iteration. The numerical algorithm of the method presented here is algebraically simpler than other spectral nodal methods in the literature for the type of problems we have considered. Numerical results to two typical model problems are presented to test the accuracy of the offered method.

Variational Approach in Transport Theory

2004 ◽  
Author(s):  
Rube´n Panta Pazos ◽  
Marco Tu´llio de Vilhena
Keyword(s):  
Variational Approach ◽  
Transport Theory ◽  
Neutron Transport ◽  
Specular Reflection ◽  
Adjoint Equations ◽  
Discrete Ordinates ◽  
Neutron Transport Equation ◽  
Bounded Set ◽  
Transport Problems ◽  
Complementary Subset

In this work we present a variational approach to some methods to solve transport problems of neutral particles. We consider a convex domain X (for example the geometry of slab, or a convex set in the plane, or a convex bounded set in the space) and we use discrete ordinates quadrature to get a system of differential equations derived from the neutron transport equation. The boundary conditions are vacuum for a subset of the boundary, and of specular reflection for the complementary subset of the boundary. Recently some different approximation methods have been presented to solve these transport problems. We introduce in this work the adjoint equations and the conjugate functions obtained by means of the variational approach. First we consider the general formulation, and then some numerical methods such as spherical harmonics and spectral collocation method.

THE DISCRETE ORDINATES METHOD FOR THE NEUTRON TRANSPORT EQUATION IN AN INFINITE CYLINDRICAL DOMAIN

1992 ◽  
Vol 02 (03) ◽  
pp. 317-338 ◽  
Author(s):  
MOHAMMAD ASADZADEH ◽  
PETER KUMLIN ◽  
STIG LARSSON
Keyword(s):  
Transport Equation ◽  
Neutron Transport ◽  
Regularity Result ◽  
Discrete Ordinates ◽  
Cylindrical Domain ◽  
Neutron Transport Equation ◽  
Scalar Flux ◽  
Weakly Singular Kernel ◽  
Discrete Ordinates Method ◽  
Weakly Singular

We prove a regularity result for a Fredholm integral equation with weakly singular kernel, arising in connection with the neutron transport equation in an infinite cylindrical domain. The theorem states that the solution has almost two derivatives in L1, and is proved using Besov space techniques. This result is applied in the error analysis of the discrete ordinates method for the numerical solution of the neutron transport equation. We derive an error estimate in the L1-norm for the scalar flux, and as a consequence, we obtain an error bound for the critical eigenvalue.

A Closed Form Solution for a One-Dimensional Multi-Layered Neutron Transport Problem by Analytical Discrete Ordinates Method

2017 ◽  
Vol 372 ◽  
pp. 50-59
Author(s):  
João Francisco Prolo Filho ◽  
Marco Paulsen Rodrigues
Keyword(s):  
Closed Form ◽  
Heterogeneous Media ◽  
Neutron Transport ◽  
Closed Form Solution ◽  
Homogeneous Solution ◽  
Form Solution ◽  
Discrete Ordinates ◽  
Test Problems ◽  
Discrete Ordinates Method ◽  
One Dimensional

In this work, the Analytical Discrete Ordinates Method (ADO method) is used to provide a closed form solution for a class of one-dimensional neutron transport problems in Cartesian geometry, considering heterogeneous media with linearly anisotropic scattering effects. In this context, the mathematical model will describe a steady-state phenomenon, with neutron sources located inside and on the boundaries of the domain of interest. In the process, the integro-differential transport equation is transformed into an ODE system by the SN angular discretization, which homogeneous solution is obtained with a quadratic eigenvalues problem with reduced order. A particular solution in terms of constants is used. To validate the code, the method and provide benchmark results, test problems will be treated and results will be discussed.

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