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

2019 ◽  
Vol 20 (1) ◽  
pp. 61
Author(s):  
Iram Barbaro Rivas-Ortiz ◽  
Dany Sanchez Dominguez ◽  
Carlos Rafael Garcia Hernandez ◽  
Susana Marrero Iglesias ◽  
Alberto Escrivá
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Isotropic Scattering ◽  
Benchmark Problems ◽  
Discrete Ordinates ◽  
Auxiliary Equation ◽  
Slab Geometry ◽  
Spectral Parameters ◽  
Computational Performance ◽  
Discontinuous Approximation

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.

A Sparse Reformulation of the Green’s Function Formalism Allows Efficient Simulations of Morphological Neuron Models

2015 ◽  
Vol 27 (12) ◽  
pp. 2587-2622 ◽  
Author(s):  
Willem A. M. Wybo ◽  
Daniele Boccalini ◽  
Benjamin Torben-Nielsen ◽  
Marc-Oliver Gewaltig
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Linear Scaling ◽  
Cable Equation ◽  
Neuron Models ◽  
Tree Graphs ◽  
Computational Performance ◽  
Sums Of Exponentials ◽  
Simulation Paradigm ◽  
Set Of Points

We prove that when a class of partial differential equations, generalized from the cable equation, is defined on tree graphs and the inputs are restricted to a spatially discrete, well chosen set of points, the Green’s function (GF) formalism can be rewritten to scale as [Formula: see text] with the number n of inputs locations, contrary to the previously reported [Formula: see text] scaling. We show that the linear scaling can be combined with an expansion of the remaining kernels as sums of exponentials to allow efficient simulations of equations from the aforementioned class. We furthermore validate this simulation paradigm on models of nerve cells and explore its relation with more traditional finite difference approaches. Situations in which a gain in computational performance is expected are discussed.

Monte Carlo Simulation Compared With Classic Deterministic Solutions for Neutron Transport and Diffusion

2010 ◽  
Author(s):  
Zafar Ullah Koreshi ◽  
Sadaf Siddiq
Keyword(s):  
Monte Carlo ◽  
Green’S Function ◽  
Green's Function ◽  
Transport Theory ◽  
Fourier Transforms ◽  
Neutron Transport ◽  
Isotropic Scattering ◽  
Finite Slab ◽  
Deterministic Methods ◽  
Mc Simulation

The Monte Carlo (MC) simulation method, known to handle complex problems which may be formidable for deterministic methods, will always require validation with classic problems that have evolved historically from deterministic methods [1–5] based on Chandrasekhar’s method in radiative transfer, Fourier transforms, Green’s functions, Weiner-Hopf method etc which are restricted to simple geometries, such as infinite or semiinfinite media, and simple scattering laws too for practical application. This work compares deterministic results with MC simulation results for neutron flux in a slab. We consider mono-energetic transport problem in an infinite medium and in a 1-D finite slab with isotropic scattering. The transport theory solutions used in infinite geometry are the Green’s function solution and the spherical harmonics (P1, P3) solutions, while for the 1-D finite slab, we refer to a transport benchmark for which an exact solution is available. For diffusion theory, we consider the Green’s function infinite geometry solution, and the exact and eigen-function numerical solution for finite geometry (1-D slab). The objective of this work is to illustrate the results from all the methods considered especially near the source and boundaries, and as a function of the scattering probability. The results are plotted for six elements that include a strong absorber, such as gadolinium, and a strong “scaterrer” such as aluminium. The present work is didactic and focuses on problems which are simple enough, yet important, to illustrate the conceptual difference and computational complexity of the deterministic and stochastic approaches.

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