An error bound estimate and convergence of the Nodal-LTSN solution in a rectangle

Hauser, Eliete Biasotto; Pazos, Rubén Panta; Vilhena, Marco T.

doi:10.1016/j.anucene.2005.02.007

Cited by 12 publications

(10 citation statements)

References 3 publications

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“…At this point, based on the physics of shielding problems, we assume that the neutron flux attenuates exponentially with increasing distance from the source, a hypothesis also assumed in [BaVi91], [HaViBa09], [HaEtAl08], [HaPaVi05], where the only approximation involved is in the transverse leakage terms. For instance, in (13.4) we use…”

Section: The Multigroup Nodal Lts N Formulation In a Rectanglementioning

confidence: 99%

“…However, to the best of our knowledge, a convergence analysis of the nodal methods is not found in the literature, except for the isotropic scattering and a monoenergetic LT S N nodal approach. Therefore, we extend the convergence analysis for this nodal technique, [HaPaVi05], now assuming the multigroup model, specializing the study for the two-dimensional problem. The basic idea relies on the definition of an error with a proper norm both for the LT S N nodal solution and for the Gaussian quadrature approximation of the integral scattering term appearing in the neutron transport equation.…”

Section: Introductionmentioning

confidence: 99%

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Integral Methods in Science and Engineering

2013

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Section: The Multigroup Nodal Lts N Formulation In a Rectanglementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Integral Methods in Science and Engineering

2013

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“…Let us consider the two dimensional S N nodal problem assuming Klein-Nishina scattering kernel and multigroup model [8]:…”

Section: The Lts N Nodal Solution In a Rectangular Domainmentioning

confidence: 99%

“…The main idea comprehends the following steps: application of the Laplace transform technique to the set of the S N equations, solution of the resulting algebraic equation by the matrix diagonalization approach and inversion of the transformed angular flux by standard results of the Laplace transform theory. In earlier works [6,8] the LTS N method was applied in the solution of two dimensional transport equation assuming neutrons and photons in cartesian geometry.…”

Section: Introductionmentioning

confidence: 99%

An overview of the Boltzmann transport equation solution for neutrons, photons and electrons in Cartesian geometry

Rodriguez

Vilhena

Bodmann

2011

Progress in Nuclear Energy

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Questions regarding accuracy and efficiency of deterministic transport methods are still on our mind today, even with modern supercomputers. The most versatile and widely used deterministic methods are the P N approximation, the S N method (discrete ordinates method) and their variants. In the discrete ordinates (S N) formulations of the transport equation, it is assumed that the linearized Boltzmann equation only holds for a set of distinct numerical values of the direction-of-motion variables. In this work, looking forward to confirm the capabilities of deterministic methods in obtaining accurate results, we present a general overview of deterministic methods to solve the Boltzmann transport equation for neutral and charged particles. First, we describe a review in the Laplace transform technique applied to S N two dimensional transport equation in a rectangular domain considering Compton scattering. Next, we solved the Fokker-Planck (FP) equation, an alternative approach for the Boltzmann transport equation, assuming a monoenergetic electron beam in a rectangular domain. The main idea relies on applying the P N approximation, a recent advance in the class of deterministic methods, in the angular variable, to the two dimensional Fokker-Planck equation and then applying the Laplace Transform in the spatial x-variable. Numerical results are given to illustrate the accuracy of deterministic methods presented.

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“…Note, that the explicit form of the leakage angular flux is necessary in order to determine a closed-form solution for the transport equation, but the specific choice maid in this work does not restrict generality of the approach. A mathematical proof of convergence of the LTS N nodal solution is discussed in Hauser et al (2005).…”

Section: Introductionmentioning

confidence: 99%

A closed-form solution for the two-dimensional transport equation by the LTSN nodal method in the energy range of Compton effect

Rodriguez

Vilhena

Hoff

et al. 2011

Annals of Nuclear Energy

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In the present work we report on a closed-form solution for the two-dimensional Compton transport equation by the LTS N nodal method in the energy range of Compton effect. The solution is determined using the LTS N nodal approach for homogeneous and heterogeneous rectangular domains, assuming the Klein-Nishina scattering kernel and a multi-group model. The solution is obtained by two one-dimensional S N equation systems resulting from integrating out one of the orthogonal variables of the S N equations in the rectangular domain. The leakage angular fluxes are approximated by exponential forms, which allows to determine a closed-form solution for the photons transport equation. The angular flux and the parameters of the medium are used for the calculation of the absorbed energy in rectangular domains with different dimensions and compositions. In this study, only the absorbed energy by Compton effect is considered. We present numerical simulations and comparisons with results obtained by using the simulation platform GEANT4 (version 9.1) with its low energy libraries.

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An error bound estimate and convergence of the Nodal-LTSN solution in a rectangle

Cited by 12 publications

References 3 publications

Integral Methods in Science and Engineering

Integral Methods in Science and Engineering

An overview of the Boltzmann transport equation solution for neutrons, photons and electrons in Cartesian geometry

A closed-form solution for the two-dimensional transport equation by the LTSN nodal method in the energy range of Compton effect

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