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Title: Iterative convergence acceleration of neutral particle transport methods via adjacent-cell preconditioners

Abstract

The author proposes preconditioning as a viable acceleration scheme for the inner iterations of transport calculations in slab geometry. In particular he develops Adjacent-Cell Preconditioners (AP) that have the same coupling stencil as cell-centered diffusion schemes. For lowest order methods, e.g., Diamond Difference, Step, and 0-order Nodal Integral Method (ONIM), cast in a Weighted Diamond Difference (WDD) form, he derives AP for thick (KAP) and thin (NAP) cells that for model problems are unconditionally stable and efficient. For the First-Order Nodal Integral Method (INIM) he derives a NAP that possesses similarly excellent spectral properties for model problems. The two most attractive features of the new technique are:(1) its cell-centered coupling stencil, which makes it more adequate for extension to multidimensional, higher order situations than the standard edge-centered or point-centered Diffusion Synthetic Acceleration (DSA) methods; and (2) its decreasing spectral radius with increasing cell thickness to the extent that immediate pointwise convergence, i.e., in one iteration, can be achieved for problems with sufficiently thick cells. He implemented these methods, augmented with appropriate boundary conditions and mixing formulas for material heterogeneities, in the test code APID that he uses to successfully verify the analytical spectral properties for homogeneous problems. Furthermore, he conductsmore » numerical tests to demonstrate the robustness of the KAP and NAP in the presence of sharp mesh or material discontinuities. He shows that the AP for WDD is highly resilient to such discontinuities, but for INIM a few cases occur in which the scheme does not converge; however, when it converges, AP greatly reduces the number of iterations required to achieve convergence.« less

Authors:
 [1]
  1. Oak Ridge National Lab., TN (United States)
Publication Date:
OSTI Identifier:
361773
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Computational Physics; Journal Volume: 152; Journal Issue: 1; Other Information: PBD: 10 Jun 1999
Country of Publication:
United States
Language:
English
Subject:
66 PHYSICS; ITERATIVE METHODS; CONVERGENCE; NEUTRAL-PARTICLE TRANSPORT; BOUNDARY-VALUE PROBLEMS; SPECTRA UNFOLDING

Citation Formats

Azmy, Y.Y. Iterative convergence acceleration of neutral particle transport methods via adjacent-cell preconditioners. United States: N. p., 1999. Web. doi:10.1006/jcph.1999.6251.
Azmy, Y.Y. Iterative convergence acceleration of neutral particle transport methods via adjacent-cell preconditioners. United States. doi:10.1006/jcph.1999.6251.
Azmy, Y.Y. Thu . "Iterative convergence acceleration of neutral particle transport methods via adjacent-cell preconditioners". United States. doi:10.1006/jcph.1999.6251.
@article{osti_361773,
title = {Iterative convergence acceleration of neutral particle transport methods via adjacent-cell preconditioners},
author = {Azmy, Y.Y.},
abstractNote = {The author proposes preconditioning as a viable acceleration scheme for the inner iterations of transport calculations in slab geometry. In particular he develops Adjacent-Cell Preconditioners (AP) that have the same coupling stencil as cell-centered diffusion schemes. For lowest order methods, e.g., Diamond Difference, Step, and 0-order Nodal Integral Method (ONIM), cast in a Weighted Diamond Difference (WDD) form, he derives AP for thick (KAP) and thin (NAP) cells that for model problems are unconditionally stable and efficient. For the First-Order Nodal Integral Method (INIM) he derives a NAP that possesses similarly excellent spectral properties for model problems. The two most attractive features of the new technique are:(1) its cell-centered coupling stencil, which makes it more adequate for extension to multidimensional, higher order situations than the standard edge-centered or point-centered Diffusion Synthetic Acceleration (DSA) methods; and (2) its decreasing spectral radius with increasing cell thickness to the extent that immediate pointwise convergence, i.e., in one iteration, can be achieved for problems with sufficiently thick cells. He implemented these methods, augmented with appropriate boundary conditions and mixing formulas for material heterogeneities, in the test code APID that he uses to successfully verify the analytical spectral properties for homogeneous problems. Furthermore, he conducts numerical tests to demonstrate the robustness of the KAP and NAP in the presence of sharp mesh or material discontinuities. He shows that the AP for WDD is highly resilient to such discontinuities, but for INIM a few cases occur in which the scheme does not converge; however, when it converges, AP greatly reduces the number of iterations required to achieve convergence.},
doi = {10.1006/jcph.1999.6251},
journal = {Journal of Computational Physics},
number = 1,
volume = 152,
place = {United States},
year = {Thu Jun 10 00:00:00 EDT 1999},
month = {Thu Jun 10 00:00:00 EDT 1999}
}
  • The adjacent-cell preconditioner (AP) formalism originally derived in slab geometry is extended to multidimensional Cartesian geometry for generic fixed-weight, weighted diamond difference neutron transport methods. This is accomplished for the thick-cell regime (KAP) and thin-cell regime (NAP). A spectral analysis of the resulting acceleration schemes demonstrates their excellent spectral properties for model problem configurations, characterized by a uniform mesh of infinite extent and homogeneous material composition, each in its own cell-size regime. Thus, the spectral radius of KAP vanishes as the computational cell size approaches infinity, but it exceeds unity for very thin cells, thereby implying instability. In contrast, NAPmore » is stable and robust for all cell sizes, but its spectral radius vanishes more slowly as the cell size increases. For this reason, and to avoid potential complication in the case of cells that are thin in one dimension and thick in another, NAP is adopted in the remainder of this work. The most important feature of AP for practical implementation in production level codes is that it is cell centered, reducing the size of the algebraic system comprising the acceleration stage compared to face-centered schemes. Boundary conditions for finite extent problems and a mixing formula across material and cell-size discontinuity are derived and used to implement NAP in a test code, AHOT, and a production code, TORT. Numerical testing for algebraically linear iterative schemes for the cases embodied in Burre's Suite of Test Problems demonstrates the high efficiency of the new method in reducing the number of iterations required to achieve convergence, especially for optically thick cells where acceleration is most needed. Also, for algebraically nonlinear (adaptive) methods, AP generally performs better than the partial current rebalance method in TORT and the diffusion synthetic acceleration method in TWODANT. Finally, application of the AP formalism to a simplified linear nodal (SLN) method similar, but not identical, to TORT's linear nodal option is shown to possess two eigenvalues that approach either one or infinity with increasing cell size regardless of the preconditioner parameters. This implies impossibility of unconditionally robust acceleration of SLN-type methods with cell-centered preconditioners that have a block-diffusion coupling stencil. Edge-centered acceleration methods, or methods that do not require the linear moments of the flux to converge, might have an advantage in this regard but at a significant penalty to computational efficiency due to the larger system solved or the inability to utilize the computed linear moments.« less
  • In this note we demonstrate that using Anderson Acceleration (AA) in place of a standard Picard iteration can not only increase the convergence rate but also make the iteration more robust for two transport applications. We also compare the convergence acceleration provided by AA to that provided by moment-based acceleration methods. Additionally, we demonstrate that those two acceleration methods can be used together in a nested fashion. We begin by describing the AA algorithm. At this point, we will describe two application problems, one from neutronics and one from plasma physics, on which we will apply AA. We provide computationalmore » results which highlight the benefits of using AA, namely that we can compute solutions using fewer function evaluations, larger time-steps, and achieve a more robust iteration.« less
  • The Adjacent-cell Preconditioner (AP) is derived for accelerating generic fixed-weight, Weighted Diamond Difference (WDD) neutron transport methods in multidimensional Cartesian geometry. The AP is determined by requiring: (a) the eigenvalue of the combined mesh sweep-AP iterations to vanish in the vicinity of the origin in Fourier space; and (b) the diagonal and off-diagonal elements of the preconditioner to satisfy a diffusion-like condition. The spectra of the resulting iterations for a wide range of problem parameters exhibit a spectral radius smaller than .25, that vanishes implying immediate convergence for very large computational cells. More importantly, unlike other unconditionally stable acceleration schemes,more » the AP is cell-centered and its spectral radius remains small when the cell aspect ratio approaches 0 or {infinity}. Testing of the AP and comparison of its rate of convergence to the standard Source Iterations (SI) for Burre`s Suite of Test Problems (BSTeP) demonstrates its high efficiency in reducing the number of iterations required to achieve convergence, especially for optically thick cells where acceleration is most needed.« less
  • In particle transport applications there are numerous physical constructs in which heterogeneities are randomly distributed. The quantity of interest in these problems is the ensemble average of the flux, or the average of the flux over all possible material 'realizations.' The Levermore-Pomraning closure assumes Markovian mixing statistics and allows a closed, coupled system of equations to be written for the ensemble averages of the flux in each material. Generally, binary statistical mixtures are considered in which there are two (homogeneous) materials and corresponding coupled equations. The solution process is iterative, but convergence may be slow as either or both materialsmore » approach the diffusion and/or atomic mix limits. A three-part acceleration scheme is devised to expedite convergence, particularly in the atomic mix-diffusion limit where computation is extremely slow. The iteration is first divided into a series of 'inner' material and source iterations to attenuate the diffusion and atomic mix error modes separately. Secondly, atomic mix synthetic acceleration is applied to the inner material iteration and S{sup 2} synthetic acceleration to the inner source iterations to offset the cost of doing several inner iterations per outer iteration. Finally, a Krylov iterative solver is wrapped around each iteration, inner and outer, to further expedite convergence. A spectral analysis is conducted and iteration counts and computing cost for the new two-step scheme are compared against those for a simple one-step iteration, to which a Krylov iterative method can also be applied.« less