Krylov iterative methods and the degraded effectiveness of diffusion synthetic acceleration for multidimensional SN calculations in problems with material discontinuities

Nuclear Science and Engineering

Volumn 147, Issue 3, 2004, Pages 218-248

  Warsa, James S ^a   Wareing, Todd A ^a   Morel, Jim E ^a  

^a LOS ALAMOS NATIONAL LABORATORY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL METHODS; ENCODING (SYMBOLS); FINITE ELEMENT METHOD; FOURIER TRANSFORMS; MATHEMATICAL MODELS; NUMERICAL ANALYSIS; PROBLEM SOLVING;

CARTESIAN GRIDS; DIFFUSION EQUATIONS; DIFFUSION SYNTHETIC ACCELERATION (DSA); KRYLOV METHODS;

ITERATIVE METHODS;

EID: 3242783342     PISSN: 00295639     EISSN: None     Source Type: Journal    
DOI: 10.13182/NSE02-14     Document Type: Article

References (41)

1. R. E. ALCOUFFE, "Diffusion Synthetic Acceleration Methods for the Diamond-Differenced Discrete-Ordinates Equations," Nucl. Sci. Eng., 64, 344 (1977).
2. 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).
3. D. R. McCOY and E. W. LARSEN, "Unconditionally Stable Diffusion-Synthetic Acceleration Methods for the Slab Geometry Discrete Ordinates Equations. Part II: Numerical Results," Nucl. Sci. Eng., 82, 64 (1982).
4. M. L. ADAMS and T. A. WAREING, "Diffusion-Synthetic Acceleration Given Anisotropic Scattering, General Quadratures, and Multidimensions," Trans. Am. Nucl. Soc., 68, 203 (1993).
5. N Transport Discretizations on Unstructured Tetrahedral Meshes," Nucl. Sci. Eng., 141, 236 (2002).
6. M. L. ADAMS and E. W. LARSEN, "Fast Iterative Methods for Discrete-Ordinates Particle Transport Calculations," Prog. Nucl. Energy, 40, 3 (2002).
7. M. L. ADAMS and W. R. MARTIN, "Diffusion Synthetic Acceleration of Discontinuous Finite Element Transport Iterations," Nucl. Sci. Eng., 111, 145 (1992).
8. N Equations with Corner Balance Spatial Differencing," Joint Int. Conf. Mathematical Methods and Supercomputing in Nuclear Applications, Karlsruhe, Germany, April 19-23, 1993, Vol. 2, p. 500 (1993).
9. 1 Equations in Two-Dimensional Cartesian Geometry with Two-Level Preconditioning," SIAM J. Sci. Comput., 24, 2093 (2003).
10. J. S. WARSA, M. BENZI, T. A. WAREING, and J. E. MOREL, "Two-Level Preconditioning of a Discontinuous Galerkin Method for Radiation Diffusion," Numerical Mathematics and Advanced Applications, Proceedings of ENUMATH 2001, p. 967, F. BREZZI, A. BUFFA, S. CORSARO, and A. MURLI, Eds., Springer-Verlag, New York and Berlin (2003).
11. Y. AZMY, "Impossibility of Unconditional Stability and Robustness of Diffusive Acceleration Schemes," Proc. Topl. Mtg. 1998 ANS Radiation Protection and Shielding Division, Nashville, Tennessee, April 19-23, 1998, Vol. 1, p. 480, American Nuclear Society (1998).
12. N Methods," Int. Conf. Mathematics and Computation, Reactor Physics, and Environmental Analysis in Nuclear Applications, Madrid, Spain, September 27-30, 1999, Vol. 1, p. 55 (1999).
13. Y. AZMY, "Unconditionally Stable and Robust Adjacent-Cell Diffusive Preconditioning of Weighted-Difference Particle Transport Methods is Impossible," J. Comp. Phys., 182, 213 (2002).
14. S. F. ASHBY, P. N. BROWN, M. R. DORR, and A. C. HINDMARSH, "Preconditioned Iterative Methods for Discretized Transport Equations," Proc. Int. Topl. Mtg. Advances in Mathematics, Computations, and Reactor Physics, Pittsburgh, Pennsylvania, April 28-May 2, 1991, Vol. 2, p. 6.1 2-1, American Nuclear Society (1991); see also UCRL-JC-104901, Lawrence Livermore National Laboratory (July 1990).
15. S. F. ASHBY, P. N. BROWN, M. R. DORR, and A. C. HINDMARSH, "Preconditioned Iterative Methods for Discretized Transport Equations," Proc. Int. Topl. Mtg. Advances in Mathematics, Computations, and Reactor Physics, Pittsburgh, Pennsylvania, April 28-May 2, 1991, Vol. 2, p. 6.1 2-1, American Nuclear Society (1991); see also UCRL-JC-104901, Lawrence Livermore National Laboratory (July 1990).
16. P. N. BROWN, "A Linear Algebraic Development of Diffusion Synthetic Acceleration for Three-Dimensional Transport Equations," SIAM J. Numer. Anal., 32, 179 (1995).
17. B. GUTHRIE, J. P. HOLLOWAY, and B. W. PATTON, "GMRES as a Multi-Step Transport Sweep Accelerator," Transp. Theory Stat. Phys., 28, 1, 83 (1999).
18. Y. SAAD, Iterative Methods for Sparse Linear Systems, PWS Publishing Company, Boston (1996).
19. n Methods on Three-Dimensional Unstructured Grids," Nucl. Sci. Eng., 138, 256 (2001).
20. J. S. WARSA, T. A. WAREING, and J. E. MOREL, "Fully Consistent Linear Discontinuous Diffusion Synthetic Acceleration on 3D Unstructured Meshes," Proc. Int. Mtg. Mathematical Methods for Nuclear Applications, Salt Lake City, Utah, September 9-13, 2001, American Nuclear Society (2001).
21. E. E. LEWIS and W. F. MILLER, Computational Methods of Neutron Transport, Wiley & Sons, New York (1984).
22. M. L. ADAMS, "Discontinuous Finite Element Transport Solutions in Thick Diffusive Problems," Nucl. Sci. Eng., 137, 298 (2001).
23. E. W. LARSEN, J. E. MOREL, and W. F. MILLER, Jr., "Asymptotic Solutions of Numerical Transport Problems in Optically Thick, Diffusive Regimes I," J. Comp. Phys., 69, 283 (1987).
24. E. W. LARSEN and J. E. MOREL, "Asymptotic Solutions of Numerical Transport Problems in Optically Thick, Diffusive Regimes II," J. Comp. Phys., 83, 212 (1989).
25. T. A. WAREING, J. M. MCGHEE, and J. E. MOREL, "ATTILA: A Three-Dimensional, Unstructured Tetrahedral Mesh Discrete Ordinates Transport Code," Trans. Am. Nucl. Soc., 75, 146 (1996).
26. V. FABER and T. A. MANTEUFFEL, "A Look at Transport Theory from the Point of View of Linear Algebra," Lecture Notes in Pure and Applied Mathematics, Vol. 115, p. 31, P. NELSON et al., Eds., Marcel Dekker, New York (1989).
27. C. T. KELLEY, "Multilevel Source Iteration Accelerators for the Linear Transport Equation in Slab Geometry," Transp. Theory Stat. Phys., 24, 4 & 5, 697 (1995).
28. C. T. KELLEY and Z. Q. XUE, "GMRES and Integral Operators," SIAM J. Sci. Comput., 17, 1, 217 (1996).
29. S. OLIVEIRA and Y. DENG, "Preconditioned Krylov Subspace Methods for Transport Equations," Prog. Nucl. Energy, 33, 155 (1998).
30. R. SANCHEZ and S. SANTANDREA, "Symmetrization of the Transport Operator and Lanzcos' Iterations," Proc. Int. Mtg. Mathematical Methods for Nuclear Applications, Salt Lake City, Utah, September 9-13, 2001, American Nuclear Society (2001).
31. B. W. PATTON and J. P. HOLLOWAY, "Application of Krylov Subspace Methods to the Slab Geometry Transport Equation," Topl. Mtg. Radiation Protection and Shielding, No. Falmouth, Massachusetts, April 21-25, 1996, Vol. 1, p. 384 (1996).
32. B. W. PATTON and J. P. HOLLOWAY, "Application of Preconditioned GMRES to the Numerical Solution of the Neutron Transport Equation," Ann. Nucl. Energy, 29, 2, 109 (2002).
33. A. GREENBAUM, Iterative Methods for Solving Linear Systems, SIAM, Philadelphia (1997).
34. G. L. RAMONE, M. L. ADAMS, and P. F. NOWAK, "A Transport Synthetic Acceleration Method for Transport Iterations," Nucl. Sci. Eng., 125, 257 (1997).
35. M. R. ZIKA and M. L. ADAMS, "Transport Synthetic Acceleration with Opposing Reflecting Boundary Conditions," Nucl. Sci. Eng., 134, 159 (2000).
36. S. F. ASHBY, P. N. BROWN, M. R. DORR, and A. C. HINDMARSH, "A Linear Algebraic Analysis of Diffusion Synthetic Acceleration for the Boltzmann Transport Equation," SIAM J. Numer. Anal., 32, 128 (1995).
37. S. L. CAMPBELL, I. C. F. IPSEN, C. T. KELLEY, and C. D. MEYER, "GMRES and the Minimal Polynomial," BIT, 36, 664 (1996).
38. N Discretization Method," Proc. Topl. Mtg. Mathematics and Computations in Nuclear Applications, Gatlinburg, Tennessee, April 6-10, 2003, American Nuclear Society (2003).
39. J. ZITKO, "Using Successive Approximations for Improving the Convergence of GMRES Method," Applic. Math., 43, 5, 321 (1998).
40. A. BOURAS and V. FRAYSSÉ, "A Relaxation Strategy for Inexact Matrix-Vector Products for Krylov Methods," CERFACS TR/PA/00/15, European Centre for Research and Advanced Training in Scientific Computation (2000).
41. V. SIMONCINI and D. SZYLD, "Theory of Inexact Krylov Subspace Methods and Applications to Scientific Computing," SIAM J. Sci. Comput., 25, 454 (2003).