A Fast Multigrid Algorithm for Isotropic Transport Problems I: Pure Scattering

Manteuffel, Thomas A.; McCormick, Steve; Morel, Jim E.; Oliveira, Suely; Yang, Gang

doi:10.1137/0916038

Cited by 25 publications

(12 citation statements)

References 11 publications

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“…Our approach may be useful with this type of iterative scheme, as well. Manteuffel et al (1995Manteuffel et al ( , 1996 have also developed a multigrid method based on cellwise block iteration. This method was later extended to heterogeneous media (Lansrud and Adams, 2005a) as well as multiple spatial dimensions (Lansrud and Adams, 2005b;Sheehan, 2007;Chang et al, 2007).…”

Section: Introductionmentioning

confidence: 99%

“…However, there are two significant differences between our approach and this previous work. First, the particular form of cellwise block iteration employed by Manteuffel et al (1995Manteuffel et al ( , 1996 involves solving for the angular flux in two adjacent spatial cells simultaneously, as implemented in one-dimensional planar geometry. Lansrud and Adams (2005b), Sheehan (2007), and Chang et al (2007) all adapted this iteration to two-dimensional Cartesian geometry by solving in four adjacent cells simultaneously.…”

Section: Introductionmentioning

confidence: 99%

“…Instead, we solve within a single cell, which is computationally less expensive. Second, Manteuffel et al (1995Manteuffel et al ( , 1996 and Lansrud and Adams (2005a, b) use rather complicated procedures for transferring quantities between spatial grids. While the resulting multigrid method converges very quickly for homogeneous media in one-dimensional planar geometry, performance can degrade significantly in the presence of material heterogeneities and is not nearly as impressive in multiple spatial dimensions.…”

Section: Introductionmentioning

confidence: 99%

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Cellwise Block Iteration as a Multigrid Smoother for Discrete-Ordinates Radiation-Transport Calculations

Densmore¹,

Gill²,

Pounders

2016

Journal of Computational and Theoretical Transport

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We improve the convergence properties of cellwise block iteration for discrete-ordinates radiation-transport calculations by adapting it for use as a smoother within a multigrid method. Cellwise block iteration by itself converges very slowly for optically thin spatial cells. However, multigrid methods involve a sequence of increasingly coarser grids such that cells on the coarsest grid should be optically thick, for which cellwise block iteration converges quickly. This fast convergence on the coarsest grid should enable fast convergence overall. A novel aspect of this paper is that, along with the usual first-order form of the transport equation, we also consider the Self-Adjoint Angular Flux (SAAF) form. We present numerical results generated using several multigrid methods based on cellwise block iteration for smoothing that demonstrate our approach yields robust convergence regardless of cell optical thickness as specified by the finest grid as well as for heterogeneous media. In addition, we find that the multigrid methods for the SAAF form of the transport equation have superior convergence properties as compared to those for the firstorder form.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Cellwise Block Iteration as a Multigrid Smoother for Discrete-Ordinates Radiation-Transport Calculations

Densmore¹,

Gill²,

Pounders

2016

Journal of Computational and Theoretical Transport

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“…The algorithm described here is an extension to two spatial dimensions of the work presented by Manteuffel et al in [13] and [14]. The partial differential equation (PDE) to be solved is…”

Section: Introductionmentioning

confidence: 99%

“…Other limitations of DSA stem from source iteration, the relaxation step on which it is based [12]. This relaxation is not ideally suited to parallel computing and does not allow for acceleration by spatial multigrid [13,8,15]. In contrast, the method outlined below employs the corner balance discretization, acceleration by spatial multigrid, and is well suited to parallel computing.…”

Section: Introductionmentioning

confidence: 99%

Spatial Multigrid for Isotropic Neutron Transport

Chang¹,

Manteuffel²,

McCormick³

et al. 2007

SIAM J. Sci. Comput.

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A spatial multigrid algorithm for isotropic neutron transport is presented in x-y geometry. The linear system is obtained using discrete ordinates in angle and corner balance finite differencing in space. Spatial smoothing is accomplished by a four-color block-Jacobi relaxation, where the diagonal blocks correspond to 4-cell blocks on the spatial grid. A bilinear interpolation operator and its transpose are used for the grid transfer operators. Good convergence factors were observed for homogeneous material properties. Heterogeneous material properties prove more difficult, especially the case of a vacuum region surrounded by a thick, diffusive region. In this case, a small amount of absorption, or "effective absorption" in a time-dependent problem, restores good convergence. Numerical results are presented.

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Numerical Methods for Radiative Heat Transfer in Diffusive Regimes and Applications to Glass Manufacturing

Klar

Thömmes

2004

Transport in Transition Regimes

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A Fast Multigrid Algorithm for Isotropic Transport Problems I: Pure Scattering

Cited by 25 publications

References 11 publications

Cellwise Block Iteration as a Multigrid Smoother for Discrete-Ordinates Radiation-Transport Calculations

Cellwise Block Iteration as a Multigrid Smoother for Discrete-Ordinates Radiation-Transport Calculations

Spatial Multigrid for Isotropic Neutron Transport

Numerical Methods for Radiative Heat Transfer in Diffusive Regimes and Applications to Glass Manufacturing

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