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Larsen, E.W.

Los Alamos National Lab., NM (USA)

Los Alamos National Lab., NM (USA)

AbstractAbstract

[en] 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

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1982; 22 p; Los Alamos/CEA conference; Paris (France); 19 - 23 Apr 1982; CONF-820429--11; Available from NTIS., PC A02/MF A01 as DE82014064

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Larsen, E.W.

Los Alamos National Lab., NM (USA)

Los Alamos National Lab., NM (USA)

AbstractAbstract

[en] 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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1983; 21 p; American Nuclear Society topical conference on computational methods; Salt Lake City, UT (USA); 28 - 31 Mar 1983; CONF-830304--10; Available from NTIS, PC A02/MF A01 as DE83004738

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[en] In recent year, substantial work has been performed on the problem of determining the cross sections of a system by introducing known steady-state neutron or photon flux distributions through the outer boundary of the system, measuring the exiting flux distributions, and then suitably processing this boundary flux information. In all of this work, the angular fluxes must be known at every point and every angle on the other boundary, and the resulting equations that relate the material cross sections to the incident and exiting boundary fluxes are ill conditioned. Here is proposed an entirely different solution having the following features: 1. The known incident fluxes must by pencil beams, i.e., delta functions in space, angle, and energy. 2. The detectors must be capable of measuring weak delta functions of angle, consisting of the uncollided flux generated by a line source. 3. Our solution is conceptually applicable in any convex geometry. 4. If the physical system is homogeneous, then our solution is explicit, requires only a few measurements, and is well conditioned

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American Nuclear Society annual meeting; San Diego, CA (USA); 12-16 Jun 1988; CONF-880601--

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Larsen, E.W.

University of Michigan, Ann Arbor, MI (United States)

University of Michigan, Ann Arbor, MI (United States)

AbstractAbstract

[en] It is standard practice to require an S

_{N}spatial discretization scheme to preserve the ''flat infinite-medium'' solution of the transport equation. This solution consists of a spatially independent source that gives rise to a spatially independent flux. However, there exist many other exact solutions of the transport equation that are typically not preserved by approximation schemes. Here, we discuss one of these: a source that is linear in space giving rise to an angular flux that is linear in space and angle. For one-group, planar-geometry S_{N}problems, we show that (a) among the class of weighted-diamond schemes, only one - the diamond-difference scheme - preserves this exact ''linear'' solution; (b) consequently, only the diamond scheme preserves the correct Fick's Law; and (c) as a further consequence, nondiamond schemes can produce significant errors (not observed in the diamond solution) for diffusive problems with spatial cells that are not optically thin. These results demonstrate that it is advantageous for S_{N}discretization schemes to preserve the ''flat'' and ''linear'' infinite-medium solutionsPrimary Subject

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17 Jun 2001; 2 p; 2001 Annual Meeting; Milwaukee, WI (United States); 17-21 Jun 2001; ISSN 0003-018X; ; CODEN TANSAO; Available from American Nuclear Society, P.O. Box 97781, Chicago, IL 60678 (US); Transactions of the American Nuclear Society, volume 84

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Miscellaneous

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Brantley, P.S.; Larsen, E.W.

Lawrence Livermore National Lab., CA (United States)

Lawrence Livermore National Lab., CA (United States)

AbstractAbstract

[en] The simplified P

_{3}(SP_{3}) approximation to the multigroup neutron transport equation in arbitrary geometries is derived using a variational analysis. This derivation yields the SP_{3}equations along with material interface and Marshak-like boundary conditions. The multigroup SP_{3}approximation is reformulated as a system of within-group problems that can be solved iteratively. An explicit iterative algorithm for solving the within-group problems that can be solved iteratively. An explicit iterative algorithm for solving the within-group problem is described. Fourier analyzed, and shown to be more efficient than the traditional FLIP implicit algorithm. Numerical results compare diffusion (P_{1}), simplified P_{2}(SP_{2}), and simplified P_{3}calculations of a mixed-oxide (MOX) fuel benchmark problem to a reference transport calculation. The SP_{3}approximation can eliminate much of the inaccuracy in the diffusion and SP_{2}calculations of MOX fuel problemsPrimary Subject

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AC05-76OR00033; W-7405-ENG-48

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[en] The Diffusion Synthetic Acceleration (DSA) method has become a widely used numerical algorithm for accelerating the iterative convergence of S

_{n}problems. The primary advantage of DSA is that it produces, for most problems, an iteration error that is reduced by at least two orders of magnitude for every three iterations. However, for problems in which the entire exterior boundary of the physical system is reflecting (such problems occur in modeling a single cell in an infinite lattice), DSA is much more slowly converging, and in some cases, unstable. Here, the authors explain why this occurs and we describe an easily implemented modification of DSA that yield the same rapid rate of convergence for this type of problem as current DSA now exhibits for other types of problemsPrimary Subject

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American Nuclear Society annual meeting; San Diego, CA (USA); 12-16 Jun 1988; CONF-880601--

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[en] One of the oldest and most commonly used algorithms for accelerating the iterative convergence of S

_{n}problems is the rebalance method. Although there are many problems for which rebalance accelerates effectively, there are also many problems (typically having scattering ratios c close to unity or optically thick spatial cells) for which it is unstable. Also, the fact that the rebalance method is nonlinear has inhibited previous theoretical attempts to quantify its stability properties. Here the authors describe a Fourier stability analysis of fine-mesh rebalance (FMR) for a special class of S_{n}problems; numerical experimentation, however, shows that this analysis accurately predicts the stability of FMR over a much larger class of problemsPrimary Subject

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American Nuclear Society annual meeting; San Diego, CA (USA); 12-16 Jun 1988; CONF-880601--

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Alcouffe, R.E.; Larsen, E.W.

Los Alamos Scientific Lab., NM (USA)

Los Alamos Scientific Lab., NM (USA)

AbstractAbstract

[en] Characteristic methods used to solve the linear transport equation are reviewed. Characteristic methods are based upon the solution of the transport equation written in the form psi(s) = psi(s

_{0})e/sup -sigma/sub T/(s-s_{0})/ + ∫/sub s_{0}//sup s/Q(t')e/sup -sigma/sub T/(s-t')/dt', where s is arc length along the characteristic. The methods of solution distinguish themselves in how the characteristics used for computation are selected and how the source term is approximated. Criteria upon which a production method should be based are recommended. 3 figuresPrimary Subject

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1981; 14 p; ANS/ENS joint topical meeting on mathematical methods in nuclear engineering; Munich, F.R. Germany; 27 - 29 Apr 1981; CONF-810415--4; Available from NTIS., PC A02/MF A01

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Larsen, E.W.; Alcouffe, R.E.

Los Alamos Scientific Lab., NM (USA)

Los Alamos Scientific Lab., NM (USA)

AbstractAbstract

[en] A new linear characteristic (LC) spatial differencing scheme for the discrete-ordinates equations in (x,y) geometry is described, and numerical comparisons are given with the diamond difference (DD) method. The LC method is more stable with mesh size and is generally much more accurate than the DD method on both fine and coarse meshes, for eigenvalue and deep-penetration problems. The LC method is based on computations involving the exact solution of a cell problem that has spatially linear boundary conditions and interior source. The LC method is coupled to the diffusion synthetic acceleration (DSA) algorithm in that the linear variations of the source are determined in part by the results of the DSA calculation from the previous inner iteration. An inexpensive negative-flux fixup is used which has very little effect on the accuracy of the solution. The storage requirements for LC are essentially the same as that for DD, while the computational times for LC are generally less than twice the DD computational times for the same mesh. This increase in computational cost is offset if one computes LC solutions on somewhat coarser meshes than DD; the resulting LC solutions are still generally much more accurate than the DD solutions. 4 figures, 6 tables

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1981; 15 p; ANS/ENS joint topical meeting on mathematical methods in nuclear engineering; Munich, F.R. Germany; 27 - 29 Apr 1981; CONF-810415--3; Available from NTIS., PC A02/MF A01

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AbstractAbstract

[en] An asymptotic solution of the neutron transport equation is constructed in a large heterogeneous medium using a multiscale method. The solution is asymptotic with respect to a small dimensionless parameter, epsilon, which is defined as the ratio of a mean-free-path to the diameter of the medium. The leading term of the solution is the product of two functions, one determined by a cell calculation and the other as the solution of a diffusion equation. The coefficients in the diffusion equation contain functions that are determined by cell calculations and are then averaged over the cell. The asymptotic diffusion coefficients are compared to other homogenized diffusion coefficients that have been proposed in the literature. A substantial numerical disagreement exists for a large class of problems. A physical interpretation is given to the asymptotic solution and to the numerical results concerning the asymptotic diffusion coefficients

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Journal Article

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Nuclear Science and Engineering; v. 60(4); p. 357-368

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