Iterative Methods for Solving the Neutron Transport Equation in 1-D Spherical Geometry

Tizaoui, Abdelkader

doi:10.1007/s10894-009-9197-y

Cited by 3 publications

(19 citation statements)

References 4 publications

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“…Note This is precisely the weak form of the equation discussed in (12) and (14) coupled with the boundary substitutions in (15)- (20). ; r iþ 1 2 Â ½l dÀ 1 2 ; l dþ 1 2 showing a few normal vectors n as solid arrows, a couple of characteristic direction vectors jðr; lÞ ¼ ðl; 1Àl 2 r Þ as short dotted lines with arrows , and a few characteristic lines as long dashed lines with arrows.…”

Section: Transport Operator Discretizationmentioning

confidence: 78%

“…The preconditioner matrix M can have a variety of forms, and is often highly problem dependent (see [20] for a few examples). In the cases considered here:…”

Section: Comparisons and Discussionmentioning

confidence: 99%

“…In 2002, the otherwise comprehensive study in [1] included only an investigation of the matter in the absence of any spatial or angular discretization. Tizaoui's piece [20] is the only recent theoretical result specific to the scattering 1-D spherical case known to this author.…”

Section: Brief Overview Of Previous Effortsmentioning

confidence: 96%

“…There have been a few investigations into the theoretical convergence properties of the various iterative schemes used in resolving the scattering component of the operator [3,6,20,21] but again, few focused on the 1-D spherical case. In 2002, the otherwise comprehensive study in [1] included only an investigation of the matter in the absence of any spatial or angular discretization.…”

Section: Brief Overview Of Previous Effortsmentioning

confidence: 99%

“…Since the mean free (uncollided) path length for a neutron is the reciprocal of r t , the asymptotic diffusive (or thick) regime has path lengths of OðÞ which has shown to be problematic for mesh-size spacing as decreases [1,[17][18][19][20][21][22][23] in various other geometric settings.…”

Section: Highly-scattering Media and Dsamentioning

confidence: 99%

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Discontinuous 1-D Spherical Transport Solvers using Diffusion Preconditioner and Iterative Methods

Machorro

2011

J Fusion Energ

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Using discontinuous Galerkin finite-element methods to solve the hyperbolic components of the transport operator in the 1-D spherically symmetric case, the convergence behavior of various iterative methods are compared and evaluated. Diffusion synthetic accelerator, a preconditioner based on the diffusion approximation to the transport equation, is presented formally for this geometry for the first time. Compared with classical, finite-difference like methods (diamond difference methods), it is found that DG diffusion based preconditioners performed extremely well in resolving problems with strong scattering effects and material discontinuities.Keywords Neutron transport equation Á Iterative methods Á Integro-differential operator Discontinuous Galerkin Methods with Isotropic ScatteringThe Boltzmann operator considered here iswhere r t C r s C 0. The notation S½w r s ðrÞ Z þ1 À1 wðr;lÞ dl 2 ¼ r s ðrÞuðrÞ ð 2Þ may be used on occasion where the scalar flux is defined as uðrÞ ¼ Z þ1 À1 wðr;lÞ dl 2 :The streaming-collision operator T ¼ B À S can be written as T ¼ j Á r þ r t where j ¼ ðl; 1Àl 2 r Þ: The Boltzmann transport equation (BTE) problem properly stated is as follows: Given a domain D ¼ ½0; 1 Â ½À1; 1 in (r, l)-space, and source term q(r), find a solution w(r, l) such thatNote that in many of the examples it will be commonly assumed that r t , q, and r s are all isotropic (independent of l), but in all cases these quantities are non-negative. Brief Overview of Previous EffortsIn 1973, Lesaint and Ravaint [12] produced one of the seminal studies of the convergence properties of the discontinuous Galerkin (DG) method as applied to the nonscattering constant coefficient 2-D Cartesian transport equation

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Section: Transport Operator Discretizationmentioning

confidence: 78%

“…The preconditioner matrix M can have a variety of forms, and is often highly problem dependent (see [20] for a few examples). In the cases considered here:…”

Section: Comparisons and Discussionmentioning

confidence: 99%

Section: Brief Overview Of Previous Effortsmentioning

confidence: 96%

Section: Brief Overview Of Previous Effortsmentioning

confidence: 99%

Section: Highly-scattering Media and Dsamentioning

confidence: 99%

See 3 more Smart Citations

Discontinuous 1-D Spherical Transport Solvers using Diffusion Preconditioner and Iterative Methods

Machorro

2011

J Fusion Energ

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Theoretical and numerical analysis of the Minimal Residual Method for solving the neutron transport equation in spherical geometry

Tizaoui¹

2023

Preprint

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This paper focuses on an infinite dimensional adaptation of the Minimal Residual method for solving the neutron transport equation in spherical geometry. This method is based on a splitting of the collision operator strategy taking into account the characteristics of the transport operator. The theoretical and numerical analysis of this algorithm are given and compared to the methods given in [1, 2, 3].

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A Splitting Iterative Method for Solving the Neutron Transport Equation

Awono

Tagoudjeu

2009

Mathematical Modelling and Analysis

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This paper presents an iterative method based on a self‐adjoint and m‐accretive splitting for the numerical treatment of the steady state neutron transport equation. Theoretical analysis shows that this method converges unconditionally to the unique solution of the transport equation. The convergence of the method is numerically illustrated and compared with the standard Source Iteration method and multigrid method on sample problems in slab geometry and in two dimensional space.

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Iterative Methods for Solving the Neutron Transport Equation in 1-D Spherical Geometry

Cited by 3 publications

References 4 publications

Discontinuous 1-D Spherical Transport Solvers using Diffusion Preconditioner and Iterative Methods

Discontinuous 1-D Spherical Transport Solvers using Diffusion Preconditioner and Iterative Methods

Theoretical and numerical analysis of the Minimal Residual Method for solving the neutron transport equation in spherical geometry

A Splitting Iterative Method for Solving the Neutron Transport Equation

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