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Transport Acceleration Methods as Two-Level Multigrid Algorithms

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Modern Mathematical Methods in Transport Theory

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 51))

Abstract

In this paper, a unified framework is developed to describe acceleration methods for the iterative convergence of neutron transport problems. The framework is that of a two-level multigrid algorithm. All widely-known transport acceleration methods not based on direct matrix manipulations (i.e., SOR or Chebyschev acceleration) can be described within this framework. Our presentation provides a way of understanding older methods within a unified theory, and it easily allows for the development of new methods.

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References

  1. E.E. Lewis and W.F. Miller Jr., Computational Methods of Neutron Transport, Wiley, New York (1984).

    Google Scholar 

  2. A. Brandt, “Multilevel Adaptive Solutions to Boundary Value Problems,” Math. Comp. 31, 333 (1977).

    Article  Google Scholar 

  3. W. Hackbusch and U. Trottenberg, Multigrid Methods, Springer-Verlag, Berlin (1982).

    Book  Google Scholar 

  4. W.L. Briggs, A Multigrid Tutorial, SIAM, Philadelphia (1987).

    Google Scholar 

  5. S. McCormick, Multigrid Methods, SIAM, Philadelphia (1987).

    Book  Google Scholar 

  6. W.F. Miller Jr., “Generalized Rebalance: A Common Framework for Transport Acceleration Methods,” Nucl. Sci. Eng. 65, 226 (1978).

    Google Scholar 

  7. V.Ya. Gol’din, “A Quasi-Diffusion Method of Solving the Kinetic Equation,” Zh. Vych. Mat. 4, 1078 (1964). English translation published in U S S R Computational Mathematics and Mathematical Physics 4, 136 (1967).

    Google Scholar 

  8. K.D. Lathrop, “Transport Theory Numerical Methods,” in Proc. Conf. Mathematical Models and Computational Techniques for Analysis of Nuclear System, Ann Arbor, Michigan, April 1973, CONF-7304144, Vol. 1, U.S. Atomic Energy Commision (1974).

    Google Scholar 

  9. R.E. Alcouffe, “Diffusion Synthetic Acceleration Methods for the Diamond-Differenced Discrete-Ordinates Equations,” Nucl. Sci. Eng. 64, 344 (1977).

    Google Scholar 

  10. E.W. Larsen, “Unconditionally Stable Diffusion-Synthetic Acceleration Methods for the Slab Geometry Discrete-Ordinates Equations. Part I: Theory,” Nucl. Sci. Eng. 82, 47 (1982).

    Google Scholar 

  11. E.W. Larsen, “Diffusion-Synthetic Acceleration Methods for the Discrete-Ordinates Equations,” in Proc. ANS Topical Meeting, Advances in Reactor Computations, March 28–31, 1983, Salt Lake City, 2, 705 (1983). Also published in Transport Theory Statist. Phys. 13, 107 (1984).

    Google Scholar 

  12. W.A. Rhoades and R.R. Mynatt, “The DOT III Two-Dimensional Discrete Ordinates Transport Code,” ORNL-TM-42800, Oak Ridge National Laboratory (1973).

    Google Scholar 

  13. K.D. Lathrop and F.W. Brinkly, “TWOTRAN-II: An Interfaced, Exportable Version of the TWOTRAN Code for Two-Dimensional Transport,” LA-5990-MS, Los Alamos Scientific Laboratory (1973).

    Book  Google Scholar 

  14. G.R. Cefus and E.W. Larsen, “Stability Analysis of Rebalance,” in Proc. ANS Topical Meeting, Advances in Nuclear Engineering Computation and Radiation Shielding, April 10–13, 1989, Santa Fe, 1, 15:1–15:12. An expanded version of this paper has appeared in Nucl. Sci. Eng. 105, 31 (1990).

    Google Scholar 

  15. G. Cefus and E.W. Larsen, “Stability Analysis of the Quasidiffusion and Second Moment Methods for Iteratively Solving Discrete-Ordinates Problems,” Transport Theory Statist. Phys. 18, 493 (1990).

    Article  Google Scholar 

  16. L.J. Lorence Jr., J.E. Morel, and E.W. Larsen, “An 52-Synthetic Acceleration Scheme for the One-Dimensional S n Equations with Linear-Discontinuous Spatial Differencing,” Nucl. Sci. Eng. 101, 341 (1989).

    Google Scholar 

  17. M.L. Adams and W.R. Martin, “Boundary Projection Acceleration: A New Approach to Synthetic Acceleration of Transport Calculations,” Nucl. Sci. Eng. 100, 177 (1988).

    Google Scholar 

  18. E.W. Larsen, “A Grey Transport Aceleration Method for Thermal Radiative Transfer Problems,” J. Comp. Phys. 78, 459 (1988).

    Article  Google Scholar 

  19. W.H. Reed, “The Effectiveness of Acceleration Techniques for Iterative Methods in Transport Theory,” Nucl. Sci. Eng. 45, 245 (1971).

    Google Scholar 

  20. R.E. Alcouffe, F.W. Brinkley Jr., D.R. Marr, and R.D. O’Dell, “User’s Guide for TWODANT: A Code Package for Two-Dimensional, Diffusion-Accelerated, Neutral-Particle Iransport,” Los Alamos National Laboratory Report LA-10049-M, Rev. (October, 1984).

    Google Scholar 

  21. R.E. Alcouffe, A. Brandt, J.E. Dendy Jr., and J.W. Painter, “The Multigrid Method for the Diffusion Equation with Strongly Discontinuous Coefficients,” SIAM J. Sci. Stat. Comput. 2, 430 (1981).

    Article  Google Scholar 

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© 1991 Springer Basel AG

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Larsen, E.W. (1991). Transport Acceleration Methods as Two-Level Multigrid Algorithms. In: Greenberg, W., Polewczak, J. (eds) Modern Mathematical Methods in Transport Theory. Operator Theory: Advances and Applications, vol 51. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5675-1_4

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  • DOI: https://doi.org/10.1007/978-3-0348-5675-1_4

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-5677-5

  • Online ISBN: 978-3-0348-5675-1

  • eBook Packages: Springer Book Archive

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