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.