Diffusion-synthetic Acceleration Methods for the Discrete-ordinates Equations PDF Download

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Languages : en
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The diffusion-synthetic acceleration (DSA) method is an iterative procedure for obtaining numerical solutions of discrete-ordinates problems. The DSA method is operationally more complicated than the standard source-iteration (SI) method, but if encoded properly it converges much more rapidly, especially for problems with diffusion-like regions. In this article we describe the basic ideas beind the DSA method and give a (roughly chronological) review of its long development. We conclude with a discussion which covers additional topics, including some remaining open problems and the status of current efforts aimed at solving these problems.

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Languages : en
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A class of acceleration schemes is investigated which resembles the conventional synthetic method in that they utilize the diffusion operator in the transport iteration schemes. The accelerated iteration involves alternate diffusion and transport solutions where coupling between the equations is achieved by using a correction term applied to either the diffusion coefficient, the removal cross section, or the source of the diffusion equation. The methods involving the modification of the diffusion coefficient and of the removal term yield nonlinear acceleration schemes and are used in k/sub eff/ calculations, while the source term modification approach is linear at least before discretization, and is used for inhomogeneous source problems. A careful analysis shows that there is a preferred differencing method which eliminates the previously observed instability of the conventional synthetic method. Use of this preferred difference scheme results in an acceleration method which is at the same time stable and efficient. This preferred difference approach renders the source correction scheme, which is linear in its continuous form, nonlinear in its differenced form. An additional feature of these approaches is that they may be used as schemes for obtaining improved diffusion solutions for approximately twice the cost of a diffusion calculation. Numerical experimentation on a wide range of problems in one and two dimensions indicates that improvement from a factor of two to ten over rebalance or Chebyshev acceleration is obtained. The improvement is most pronounced in problems with large regions of scattering material where the unaccelerated transport solutions converge very slowly.

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Languages : en
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The standard iterative procedure for solving fixed-source discrete-ordinates problems converges very slowly for problems in optically large regions with scattering ratios c near unity. The diffusion-synthetic acceleration method has been proposed to make use of the fact that for this class of problems the diffusion equation is often an accurate approximation to the transport equation. However, stability difficulties have historically hampered the implementation of this method for general transport differencing schemes. In this article we discuss a recently developed procedure for obtaining unconditionally stable diffusion-synthetic acceleration methods for various transport differencing schemes. We motivate the analysis by first discussing the exact transport equation; then we illustrate the procedure by deriving a new stable acceleration method for the linear discontinuous transport differencing scheme. We also provide some numerical results.

Author: Greenberg
Publisher: Birkhäuser
ISBN: 303485675X
Category : Science
Languages : en
Pages : 339

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The Eleventh International Transport Theory Conference and Symposium in honor of the sixty-fifth birthday of Kenneth Case and the sixtieth birthday of Paul Zweifel was held in Blacksburg, Virginia, during May 22-26, 1989, on the campus of Virginia Polytechnic Institute and State University (Virginia Tech). This volume consists of a selection of the invited papers delivered at the Conference, and represents a cross section of the research currently being carried out in the field of transport theory. The volume is divided into two sections. The Symposium lectures are intended each to summarize an important aspect of transport theory, as well as to present timely new results of the author's research interest. The Conference lectures are contributions of each author on his current research. As has been the custom in this series of conferences, each lecturer was invited to participate by the organizing committee of the Conference: W. Greenberg, Virginia Tech, chairman; V. Boffi, Universita di Firenze; N. Corngold, California Institute of Technology; B. Ganapol, University of Arizona; N. McCormick, University of Washington; P. Nelson, Texas Tech; G. Pomraning, University of California, Los Angeles. The Eleventh International Transport Theory Conference was funded by generous con tributions from Science Applications International Corporation, R. Beyster, president, and from Virginia Polytechnic Institute and State University. Conference participants, and, we believe, researchers in this and related areas, are indebted to these organizations. We would like to thank Lamberto Rondoni, in the graduate program at Virginia Tech, for proofreading manuscripts of all the Italian contributors.

Author: Bruno Roger Fernand Turcksin
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Languages : en
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In this dissertation, advanced numerical methods for highly forward peaked scattering deterministic calculations are devised, implemented, and assessed. Since electrons interact with the surrounding environment through Coulomb interactions, the scattering kernel is highly forward-peaked. This bears the consequence that, with standard preconditioning, the standard Legendre expansion of the scattering kernel requires too many terms for the discretized equation to be solved efficiently using a deterministic method. The Diffusion Synthetic Acceleration (DSA), usually used to speed up the calculation when the scattering is weakly anisotropic, is inefficient for electron transport. This led Morel and Manteuffel to develop a one-dimensional angular multigrid (ANMG) which has proved to be very effective when the scattering is highly anisotropic. Later, Pautz et al. generalized this scheme to multidimensional geometries, but this method had to be stabilized by a diffusive filter that degrades the overall convergence of the iterative scheme. In this dissertation, we recast the multidimensional angular multigrid method without the filter as a preconditioner for a Krylov solver. This new method is stable independently of the anisotropy of the scattering and is increasingly more effective and efficient as the anisotropy increases compared to DSA preconditioning wrapped inside a Krylov solver. At the coarsest level of ANMG, a DSA step is needed. In this research, we use the Modified Interior Penalty (MIP) DSA. This DSA was shown to be always stable on triangular cells with isotropic scattering. Because this DSA discretization leads to symmetric definite-positive matrices, it is usually solved using a conjugate gradient preconditioned (CG) by SSOR but here, we show that algebraic multigrid methods are vastly superior than more common CG preconditioners such as SSOR. Another important part of this dissertation is dedicated to transport equation and diffusion solves on arbitrary polygonal meshes. The advantages of polygonal cells are that the number of unknowns needed to mesh a domain can be decreased and that adaptive mesh refinement implementation is simplified: rather than handling hanging nodes, the adapted computational mesh includes different types of polygons. Numerical examples are presented for arbitrary quadrilateral and polygonal grids. The electronic version of this dissertation is accessible from http://hdl.handle.net/1969.1/148243

Author: Jeffery Lewins
Publisher: Springer Science & Business Media
ISBN: 9780306436147
Category : Science
Languages : en
Pages : 322

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This twenty-fifth volume in a distinguished series addresses a range of topics including: the difficult matter of questioning scientific hypotheses in court the use of Monte Carlo simulation to evaluate time-dependent development and to study system reliability in nuclear reactors of considerable complexity the genetic optimization algorith wavelet analysis ergonomic design of safer and more efficient plant control rooms.

Author: MUSA YAVUZ
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Category :
Languages : en
Pages : 530

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performance of the new multigrid method degrades with an increasing number of processors, not because of the degradation in the convergence rate, but because of the domination of communication overhead and unemployment of some processors on the coarse grids.

Author: Jeremiah U. Brackbill
Publisher: Academic Press
ISBN: 1483257568
Category : Mathematics
Languages : en
Pages : 457

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Multiple Time Scales presents various numerical methods for solving multiple-time-scale problems. The selection first elaborates on considerations on solving problems with multiple scales; problems with different time scales; and nonlinear normal-mode initialization of numerical weather prediction models. Discussions focus on analysis of observations, nonlinear analysis, systems of ordinary differential equations, and numerical methods for problems with multiple scales. The text then examines the diffusion-synthetic acceleration of transport iterations, with application to a radiation hydrodynamics problem and implicit methods in combustion and chemical kinetics modeling. The publication ponders on molecular dynamics and Monte Carlo simulations of rare events; direct implicit plasma simulation; orbit averaging and subcycling in particle simulation of plasmas; and hybrid and collisional implicit plasma simulation models. Topics include basic moment method, electron subcycling, gyroaveraged particle simulation, and the electromagnetic direct implicit method. The selection is a valuable reference for researchers interested in pursuing further research on the use of numerical methods in solving multiple-time-scale problems.

Author: Susana Gomez
Publisher: SIAM
ISBN: 9780898712698
Category : Mathematics
Languages : en
Pages : 388

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The papers in this volume emphasize the numerical aspects of three main areas: optimization, linear algebra and partial differential equations. Held in January, 1989, in Yucatan, Mexico, the workshop was organized by the Institute for Research in Applied Mathematics of the National University of Mexico in collaboration with the mathematical Sciences Department at Rice University.

Author: Nicholas J. Prins
Publisher:
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Category : Molecular dynamics
Languages : en
Pages : 388

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The discrete ordinates method is widely used to solve the Boltzmann transport equation for neutral particle transport for many engineering applications. Source iteration is used to solve the discrete ordinates system of equations, but can be slow to converge in highly scattering problems. Synthetic acceleration techniques have been developed to address this shortcoming; however, recent research has shown synthetic acceleration to lose effectiveness or diverge for certain problems. LTC Wager introduced an alternative to source iteration and demonstrated it in slab geometry. Here the method is further developed, enhancing efficiency in various ways, and demonstrated in XY-geometry as well as slab geometry. It is shown to be efficient even for those problems for which diffusion-synthetic and transport-synthetic accelerations fail or are ineffective. The method has significant advantages for massively-parallel implementations.