A block Arnoldi method for the SP_N equations

References

• P. Arbenz, U.L. Hetmaniuk, R.B. Lehoucq, and R.S. Tuminaro, A comparison of eigensolvers for large-scale 3d modal analysis using amg-preconditioned iterative methods, Int. J. Numer. Methods Eng. 64 (2005), pp. 204–236.
   Web of Science ®Google Scholar
• J. Baglama, Augmented block householder arnoldi method, Linear Algebra Appl. 429 (2008), pp. 2315–2334.
   Web of Science ®Google Scholar
• W. Bangerth, R. Hartmann, and G. Kanschat, deal.II – A general-purpose object-oriented finite element library, ACM Trans. Math. Softw. 33 (2007), pp. 1–27.
   Web of Science ®Google Scholar
• P. Binding, B. Najman, and Q. Ye, A variational principle for eigenvalues of pencils of hermitian matrices, Integr. Equat. Operat. Theor. 35 (1999), pp. 398–422.
   Web of Science ®Google Scholar
• P.S. Brantley and E.W. Larsen, The simplified P3 approximation, Nucl. Sci. Technol. 134 (2000), pp. 1–21.
   Google Scholar
• M. Capilla, C. Talavera, D. Ginestar, and G. Verdú, Application of a nodal collocation approximation for the multidimensional PL equations to the 3D takeda benchmark problems, Ann. Nucl. Energy 40 (2012), pp. 1–13.
   Web of Science ®Google Scholar
• S. Carney, F. Brown, B. Kiedrowski, and W. Martin, Theory and applications of the fission matrix method for continuous-energy monte carlo, Ann. Nucl. Energy 73 (2014), pp. 423–431.
   Web of Science ®Google Scholar
• A. Carreño, A. Vidal-Ferràndiz, D. Ginestar, and G. Verdú, Block hybrid multilevel method to compute the dominant λ-modes of the neutron diffusion equation, Ann. Nucl. Energy 121 (2018), pp. 513–524.
   Web of Science ®Google Scholar
• E.M. Gelbard, Application of spherical harmonics methods to reactor problems, Tech. Rep. WAPD-BT-20, Bettis Atomic Power Laboratory, 1960.
   Google Scholar
• G.H. Golub and Q. Ye, An inverse free preconditioned krylov subspace method for symmetric generalized eigenvalue problems, SIAM J. Sci. Comput. 24 (2002), pp. 312–334.
   Web of Science ®Google Scholar
• S.P. Hamilton and T.M. Evans, Efficient solution of the simplified PN equations, J. Comput. Phys. 284 (2015), pp. 155–170.
   Web of Science ®Google Scholar
• S.P. Hamilton, T.M. Evans, G.G. Davidson, S.R. Johnson, T.M. Pandya, and A.T. Godfrey, Hot zero power reactor calculations using the insilico code, J. Comput. Phys. 314 (2016), pp. 700–711.
   Web of Science ®Google Scholar
• V. Hernandez, J.E. Roman, and V. Vidal, SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems, ACM Trans. Math. Softw. 31 (2005), pp. 351–362.
   Web of Science ®Google Scholar
• A.V. Knyazev, Preconditioned eigensolvers – an oxymoron? Electron. Trans. Numer. Anal 7 (1998), pp. 104–123.
   Google Scholar
• M. Kronbichler and K. Kormann, A generic interface for parallel cell-based finite element operator application, Comput. Fluids 63 (2012), pp. 135–147.
   Web of Science ®Google Scholar
• E.E. Lewis and W.F.J. Miller, Computational Methods of Neutron Transport, John Wiley & Sons, Ltd., New York, USA, 1984.
   Google Scholar
• R.B. Morgan and D.S. Scott, Generalizations of davidson's method for computing eigenvalues of sparse symmetric matrices, SIAM J. Sci. and Stat. Comput. 7 (1986), pp. 817–825.
   Web of Science ®Google Scholar
• P. Quillen and Q. Ye, A block inverse-free preconditioned krylov subspace method for symmetric generalized eigenvalue problems, J. Comput. Appl. Math. 233 (2010), pp. 1298–1313.
   Web of Science ®Google Scholar
• Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., Society for Industrial and Applied Mathematics, Philadelphia, USA, 2003.
   Google Scholar
• M.A. Smith, E.E. Lewis, and B.C. Na, Benchmark on deterministic transport calculations without spatial homogenisation – A 2-D/3-D MOX Fuel Assembly Benchmark (C5G7 MOX Benchmark), Tech. Rep. NEA/NSC/DOC(2003)16, OECD/NEA, 2003.
   Google Scholar
• W.M. Stacey, Nuclear Reactor Physics, 2nd ed., John Wiley & Sons, Weinheim, Germany, 2007.
   Google Scholar
• G.W. Stewart, A krylov–schur algorithm for large eigenproblems, SIAM J. Matrix Anal. Appl. 23 (2002), pp. 601–614.
   Web of Science ®Google Scholar
• G. Verdú, D. Ginestar, V. Vidal, and J. Muñoz-Cobo, 3D lambda-modes of the neutron-diffusion equation, Ann. Nucl. Energy 21 (1994), pp. 405–421.
   Web of Science ®Google Scholar
• A. Vidal-Ferràndiz, R. Fayez, D. Ginestar, and G. Verdú, Solution of the lambda modes problem of a nuclear power reactor using an h–p finite element method, Ann. Nucl. Energy 72 (2014), pp. 338–349.
   Web of Science ®Google Scholar
• A. Vidal-Ferràndiz, S. González-Pintor, D. Ginestar, G. Verdú, and C. Demazière, Schwarz type preconditioners for the neutron diffusion equation, J. Comput. Appl. Math. 309 (2017), pp. 563–574.
   Web of Science ®Google Scholar
• J.S. Warsa, T.A. Wareing, J.E. Morel, J.M. McGhee, and R.B. Lehoucq, Krylov subspace iterations for deterministic k-eigenvalue calculations, Nucl. Sci. Technol. 147 (2004), pp. 26–42.
   Google Scholar
• K. Wu, Y. Saad, and A. Stathopoulos, Inexact newton preconditioning techniques for large symmetric eigenvalue problems, Electron. Trans. Numer. Anal. 7 (1998), pp. 202–214.
   Google Scholar