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Krylov Techniques for 3D Problems in Transport Theory

[+] Author Affiliations
Rubén Panta Pazos

UNISC – Universidade de Santa Cruz do Sul, Santa Cruz do Sul, RS, Brazil

Paper No. ICONE14-89562, pp. 447-454; 8 pages
doi:10.1115/ICONE14-89562
From:
  • 14th International Conference on Nuclear Engineering
  • Volume 3: Structural Integrity; Nuclear Engineering Advances; Next Generation Systems; Near Term Deployment and Promotion of Nuclear Energy
  • Miami, Florida, USA, July 17–20, 2006
  • Conference Sponsors: Nuclear Engineering Division
  • ISBN: 0-7918-4244-4 | eISBN: 0-7918-3783-1
  • Copyright © 2006 by ASME

abstract

When solving integro-differential equations by means of numerical methods one has to deal with large systems of linear equations, such as happens in transport theory [10]. Many iterative techniques are now used in Transport Theory in order to solve problems of 2D and 3D dimensions. In this paper, we choose two problems to solve the following transport equation,

μ∂ψ∂x(x,μ) + h(x,μ)ψ(x,μ) =       = −11 k(x,μ,μ')ψ(x,μ')dμ' + q(x,μ)   (1)
where x: represents the spatial variable, μ: the cosine of the angle, ψ: the angular flux, h(x, μ): is the collision frequency, k(x, μ, μ'): the scattering kernel, q(x, μ): the source. The aim of this work is the straightforward application of the Krylov spaces technique [2] to the governing equation or to its discretizations derived of the discrete ordinates method (choosing a finite number of directions and then approximating the integral term by means of a proper sum). The equation (1) can be written in functional form as
A ψ = q   (2)
with ψ in the Hilbert space L2 ([0,a] × [-1,1])., and q is the source function. The operator à derived from a discrete ordinates scheme that approximates the operator
A ψ = μ∂ψ∂x(x,μ) +           + h(x,μ)ψ(x,μ) − −11 k(x,μ,μ')ψ(x,μ')dμ'   (3)
generates the following subspace
K'''(Ã,q) = Span{q,Ãq,Ã2q,...,Ãm−1q}   (4)
i.e. the subspace generated by the iterations of order 0, 1, 2, ... , m−1 of the source function q. Two methods are specially outstanding, the Lanczos method to solve the problem given by equation (2) with certain boundary conditions, and the conjugate gradient method to solve the same problem with identical boundary conditions. We discuss and accelerate the basic iterative method [8]. An important conclusion is the generation of these methods to solve linear systems in Hilbert spaces, if verify the convergence conditions, which are outlined in this work. The first problem is a cubic domain with two regions, one with a source near the vertex at the origin and the shield region. In this case, the Cartesian planes (specifically 0<x<L, 0<y<L, 0<z<L) are reflexive boundaries and the rest faces of the cube are vacuum boundaries. The second problem has a geometry with three regions, one of the source, other is the void region and the third one is a shield domain.

Copyright © 2006 by ASME

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