Newton’s Method for Solving k-Eigenvalue Problems in Neutron Diffusion Theory

Abstract We present an approach to the k-eigenvalue problem in multigroup diffusion theory based on a nonlinear treatment of the generalized eigenvalue problem. A nonlinear function is posed whose roots are equal to solutions of the k-eigenvalue problem; a Newton-Krylov method is used to find these roots. The Jacobian-vector product is found exactly or by using the Jacobian-free Newton-Krylov (JFNK) approximation. Several preconditioners for the Krylov iteration are developed. These preconditioners are based on simple approximations to the Jacobian, with one special instance being the use of power iteration as a preconditioner. Using power iteration as a preconditioner allows for the Newton-Krylov approach to heavily leverage existing power method implementations in production codes. When applied as a left preconditioner, any existing power iteration can be used to form the kernel of a JFNK solution to the k-eigenvalue problem. Numerical results generated for a suite of two-dimensional reactor benchmarks show the feasibility and computational benefits of the Newton formulation as well as examine some of the numerical difficulties potentially encountered with Newton-Krylov methods. The performance of the method is also seen to be relatively insensitive to the dominance ratio for a one-dimensional slab problem.

[1]  Alain Hébert,et al.  Variational principles and convergence acceleration strategies for the neutron diffusion equation , 1985 .

[2]  Allan F. Henry,et al.  Nuclear Reactor Analysis , 1977, IEEE Transactions on Nuclear Science.

[3]  Gumersindo Verdú,et al.  The implicit restarted Arnoldi method, an efficient alternative to solve the neutron diffusion equation , 1999 .

[4]  Y. Y. Azmy,et al.  A JACOBIAN-FREE NEWTON-KRYLOV ITERATIVE SCHEME FOR CRITICALITY CALCULATIONS BASED ON THE NEUTRON DIFFUSION EQUATION , 2009 .

[5]  Homer F. Walker,et al.  Choosing the Forcing Terms in an Inexact Newton Method , 1996, SIAM J. Sci. Comput..

[6]  K. L. Derstine,et al.  Optimized iteration strategies and data management considerations for fast reactor finite difference diffusion theory codes , 1977 .

[7]  Richard B. Lehoucq,et al.  Krylov Subspace Iterations for Deterministic k-Eigenvalue Calculations , 2004 .

[8]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[9]  Trond Steihaug,et al.  Truncated-newtono algorithms for large-scale unconstrained optimization , 1983, Math. Program..

[10]  D. F. Gill,et al.  Jacobian-Free Newton-Krylov as an Alternative to Power Iterations for the k-Eigenvalue Transport Problem , 2009 .

[11]  R. Dembo,et al.  INEXACT NEWTON METHODS , 1982 .

[12]  D. F. Gill,et al.  Newton -Krylov methods for the solution of the k-eigenvalue problem in multigroup neutronics calculations , 2009 .

[13]  Alain Hébert,et al.  Application of the Hermite Method for Finite Element Reactor Calculations , 1985 .

[14]  Hengbin An,et al.  A choice of forcing terms in inexact Newton method , 2007 .

[15]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

[16]  L. B. Rall,et al.  The solution of characteristic value-vector problems by Newton's method , 1968 .

[17]  Jeffery D. Densmore,et al.  Newton’s Method for the Computation of k-Eigenvalues in SN Transport Applications , 2011 .

[18]  Kord Smith,et al.  APPLICATION OF THE JACOBIAN-FREE NEWTON-KRYLOV METHOD IN COMPUTATIONAL REACTOR PHYSICS , 2009 .

[19]  J. H. Wilkinson,et al.  Inverse Iteration, Ill-Conditioned Equations and Newton’s Method , 1979 .