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Transport Synthetic Acceleration Methods For One Dimensional Deterministic Transport Problems
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Book Synopsis Transport Synthetic Acceleration Methods for One-dimensional Deterministic Transport Problems by : William Daryl Hawkins
Download or read book Transport Synthetic Acceleration Methods for One-dimensional Deterministic Transport Problems written by William Daryl Hawkins and published by . This book was released on 2003 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis On the Use of Diffusion Synthetic Acceleration in Parallel 3D Neutral Particle Transport Calculations by :
Download or read book On the Use of Diffusion Synthetic Acceleration in Parallel 3D Neutral Particle Transport Calculations written by and published by . This book was released on 1998 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The linear Boltzmann transport equation (BTE) is an integro-differential equation arising in deterministic models of neutral and charged particle transport. In slab (one-dimensional Cartesian) geometry and certain higher-dimensional cases, Diffusion Synthetic Acceleration (DSA) is known to be an effective algorithm for the iterative solution of the discretized BTE. Fourier and asymptotic analyses have been applied to various idealizations (e.g., problems on infinite domains with constant coefficients) to obtain sharp bounds on the convergence rate of DSA in such cases. While DSA has been shown to be a highly effective acceleration (or preconditioning) technique in one-dimensional problems, it has been observed to be less effective in higher dimensions. This is due in part to the expense of solving the related diffusion linear system. We investigate here the effectiveness of a parallel semicoarsening multigrid (SMG) solution approach to DSA preconditioning in several three dimensional problems. In particular, we consider the algorithmic and implementation scalability of a parallel SMG-DSA preconditioner on several types of test problems.
Book Synopsis Transport Theory by : Richard Bellman
Download or read book Transport Theory written by Richard Bellman and published by American Mathematical Soc.. This book was released on 1969 with total page 340 pages. Available in PDF, EPUB and Kindle. Book excerpt: The industrial and military applications of atomic energy have stimulated much mathematical research in neutron transport theory. The possibility of controlled thermonuclear processes has similarly focussed attention upon plasmas, sometimes called the "fourth state of matter". Independently, many classical aspects of kinetic theory and radiative transfer theory have been studied both because of their basic mathematical interest and of their physical applications to areas such as upper-atmosphere meteorology - introduction.
Book Synopsis A Transport Synthetic Acceleration Method for Transport Iterations by : Gilles Lionel Ramone
Download or read book A Transport Synthetic Acceleration Method for Transport Iterations written by Gilles Lionel Ramone and published by . This book was released on 1996 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Acceleration Techniques for Discrete-Ordinates Transport Methods with Highly Forward-Peaked Scattering by : Bruno Roger Fernand Turcksin
Download or read book Acceleration Techniques for Discrete-Ordinates Transport Methods with Highly Forward-Peaked Scattering written by Bruno Roger Fernand Turcksin and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Book Synopsis Diffusion Synthetic Acceleration Methods for the Diamond-differenced Discrete-ordinates Equations by :
Download or read book Diffusion Synthetic Acceleration Methods for the Diamond-differenced Discrete-ordinates Equations written by and published by . This book was released on 1976 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis ASYMPTOTIC DIFFUSION ACCELERATED DISCONTINUOUS FINITE ELEMENT METHODS FOR TRANSPORT PROBLEMS (DIFFUSION SYNTHETIC ACCELERATION). by : TODD ARLIN WAREING
Download or read book ASYMPTOTIC DIFFUSION ACCELERATED DISCONTINUOUS FINITE ELEMENT METHODS FOR TRANSPORT PROBLEMS (DIFFUSION SYNTHETIC ACCELERATION). written by TODD ARLIN WAREING and published by . This book was released on 1991 with total page 490 pages. Available in PDF, EPUB and Kindle. Book excerpt: system of equations that cannot be collapsed into an efficiently solvable discrete diffusion equation.
Book Synopsis Transport Equation and Tomography by : D. S. Anikonov
Download or read book Transport Equation and Tomography written by D. S. Anikonov and published by VSP. This book was released on 2002 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book deals mainly with the results of the authors' research devoted to both the study of the transport equation (the linear Boltzmann equation) and its applications in X-ray tomography. The introduction gives an outline of the book and deals with certain aspects of the methodology. The first part of the book is devoted to the investigation of known and new problems for the stationary transport equation of a general form. New problems are treated as problems of tomography. The second part deals with the monoenergetic transport equation. This book will be of interest to specialists in transport theory and tomography.
Book Synopsis Acceleration Methods for Massively Parallel Deterministic Transport by : Rachel N. Slaybaugh
Download or read book Acceleration Methods for Massively Parallel Deterministic Transport written by Rachel N. Slaybaugh and published by . This book was released on 2011 with total page 167 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Unconditionally Stable Diffusion-acceleration of the Transport Equation by :
Download or read book Unconditionally Stable Diffusion-acceleration of the Transport Equation written by and published by . This book was released on 1982 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Numerical Transport Theory ; Final Report by :
Download or read book Numerical Transport Theory ; Final Report written by and published by . This book was released on 1994 with total page 12 pages. Available in PDF, EPUB and Kindle. Book excerpt: The basic problem addressed in the project was that of accelerating the iterative convergence of Discrete Ordinates (S{sub N}) problems. Important previous work on this problem, much of which was done at LANL, has shown that the Diffusion Synthetic Acceleration (DSA) method can be a very effective acceleration procedure. However, in two-dimensional geometries, only the diamond differenced S{sub N} equations have been efficiently solved using DSA. This is because, for the 2-D diamond-differenced S{sub N} equations, the standard DSA procedure leads to a relatively simple discretized low-order diffusion equation that for many problems can be efficiently solved by a multigrid method. For other discretized versions of the S{sub N} equations, the standard DSA procedure leads to much more complicated discretizations of the low-order diffusion equation that have not been efficiently solved by multigrid (or other) methods. In this project, we have developed a new procedure to obtain discretized diffusion equations for DSA-accelerating the convergence of the S{sub N} equations using certain lumped discontinuous finite element spatial differencing methods. The idea is to use an asymptotic analysis for the derivation of the discretized diffusion equation. This is based on the fact that diffusion theory is an asymptotic limit of transport theory. The asymptotic analysis also shows that the schemes considered in this project are highly accurate for diffusive problems with spatial meshes that are optically thick. Specifically, we apply this DSA procedure to a lumped Linear Discontinuous (LD) scheme for slab geometry and a lumped Bilinear Discontinuous (BLD) scheme for x, y-geometry. Our theoretical and numerical results indicate that these schemes are very accurate and can be solved efficiently using the new method. We describe the concept that underlies the DSA method. We describe the basic asymptotic relationship between transport and diffusion theory.
Book Synopsis Computational Methods in Transport by : Frank Graziani
Download or read book Computational Methods in Transport written by Frank Graziani and published by Springer Science & Business Media. This book was released on 2006-02-17 with total page 539 pages. Available in PDF, EPUB and Kindle. Book excerpt: Thereexistawiderangeofapplicationswhereasigni?cantfractionofthe- mentum and energy present in a physical problem is carried by the transport of particles. Depending on the speci?capplication, the particles involved may be photons, neutrons, neutrinos, or charged particles. Regardless of which phenomena is being described, at the heart of each application is the fact that a Boltzmann like transport equation has to be solved. The complexity, and hence expense, involved in solving the transport problem can be understood by realizing that the general solution to the 3D Boltzmann transport equation is in fact really seven dimensional: 3 spatial coordinates, 2 angles, 1 time, and 1 for speed or energy. Low-order appro- mations to the transport equation are frequently used due in part to physical justi?cation but many in cases, simply because a solution to the full tra- port problem is too computationally expensive. An example is the di?usion equation, which e?ectively drops the two angles in phase space by assuming that a linear representation in angle is adequate. Another approximation is the grey approximation, which drops the energy variable by averaging over it. If the grey approximation is applied to the di?usion equation, the expense of solving what amounts to the simplest possible description of transport is roughly equal to the cost of implicit computational ?uid dynamics. It is clear therefore, that for those application areas needing some form of transport, fast, accurate and robust transport algorithms can lead to an increase in overall code performance and a decrease in time to solution.
Book Synopsis SYNTHETIC ACCELERATION METHODS FOR LINEAR TRANSPORT PROBLEMS WITH HIGHLY ANISOTROPIC SCATTERING. by : KASSEM MOHAMMAD KHATTAB
Download or read book SYNTHETIC ACCELERATION METHODS FOR LINEAR TRANSPORT PROBLEMS WITH HIGHLY ANISOTROPIC SCATTERING. written by KASSEM MOHAMMAD KHATTAB and published by . This book was released on 1989 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: Synthetic Acceleration (MDSA) method.
Book Synopsis Modern Mathematical Methods in Transport Theory by : William Greenberg
Download or read book Modern Mathematical Methods in Transport Theory written by William Greenberg and published by Birkhauser. This book was released on 1991 with total page 350 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Agreement Concerning the Adoption of Uniform Conditions of Approval and Reciprocal Recognition of Approval for Motor Vehicle Equipment and Parts. Done at Geneva on 20 March 1958. Addendum 77: Regulation No. 78 to be Annexed to the Agreement: Uniform Provisions Concerning the Approval of Vehicles of Category L with Regard to Braking by :
Download or read book Agreement Concerning the Adoption of Uniform Conditions of Approval and Reciprocal Recognition of Approval for Motor Vehicle Equipment and Parts. Done at Geneva on 20 March 1958. Addendum 77: Regulation No. 78 to be Annexed to the Agreement: Uniform Provisions Concerning the Approval of Vehicles of Category L with Regard to Braking written by and published by . This book was released on 1988 with total page 23 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Boundary-projection Acceleration by :
Download or read book Boundary-projection Acceleration written by and published by . This book was released on 1987 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We present a new class of synthetic acceleration methods which can be applied to transport calculations regardless of geometry, discretization scheme, or mesh shape. Unlike other synthetic acceleration methods which base their acceleration on P1 equations, these methods use acceleration equations obtained by projecting the transport solution onto a coarse angular mesh only on cell boundaries. We demonstrate, via Fourier analysis of a simple model problem as well as numerical calculations of various problems, that the simplest of these methods are unconditionally stable with spectral radius less than or equal toc/3 (c being the scattering ratio), for several different discretization schemes in slab geometry. 28 refs., 4 figs., 3 tabs.
Book Synopsis Transport Methods Conquering the Seven-Dimensional Mountain by :
Download or read book Transport Methods Conquering the Seven-Dimensional Mountain written by and published by . This book was released on 2003 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In a wide variety of applications, a significant fraction of the momentum and energy present in a physical problem is carried by the transport of particles. Depending on the circumstances, the types of particles might involve some or all of photons, neutrinos, charged particles, or neutrons. In application areas that use transport, the computational time is usually dominated by the transport calculation. Therefore, there is a potential for great synergy; progress in transport algorithms could help quicken the time to solution for many applications. The complexity, and hence expense, involved in solving the transport problem can be understood by realizing that the general solution to the Boltzmann transport equation is seven dimensional: 3 spatial coordinates, 2 angles, 1 time, and 1 for speed or energy. Low-order approximations to the transport equation are frequently used due in part to physical justification but many times simply because a solution to the full transport problem is too computationally expensive. An example is the diffusion equation, which effectively drops the two angles in phase space by assuming that a linear representation in angle is adequate. Another approximation is the grey approximation, which drops the energy variable by averaging over it. If the grey approximation is applied to the diffusion equation, the expense of solving what amounts to the simplest possible description of transport is roughly equal to the cost of implicit computational fluid dynamics. It is clear therefore, that for those application areas needing some form of transport, fast, accurate and robust transport algorithms can lead to an increase in overall code performance and a decrease in time to solution. The seven-dimensional nature of transport means that factors of 100 or 1000 improvement in computer speed or memory are quickly absorbed in slightly higher resolution in space, angle, and energy. Therefore, the biggest advances in the last few years and in the next several years will be driven by algorithms. Because transport is an implicit problem requiring iteration, the biggest gains are to be made in finding faster techniques for acceleration to convergence. Some of these acceleration methods are very application specific because they are physics based; others are very general because they address the mathematics of the transport equation. Funding more research in the latter area could have a large impact on many physics applications. Usually it is a collaboration of someone with a tough problem to solve and someone with a new idea that makes the big advances. More heads are needed to continue the progress of the last few years. Unfortunately, transport as a discipline is not taught in many graduate schools. Students and researchers too often pick up transport theory in pieces on an ad hoc basis. Therefore, they don't know the published literature and existing techniques. Knowledge of advances in one application area often takes years to propagate to other application areas.
