Newton-Krylov Methods for the Solution of the K-Eigenvalue Problem in Multigroup Neutronics Calculations

In this work we propose using Newton's method, specifically the inexact Newton-GMRES formulation, to solve the k-eigenvalue problem in both transport and diffusion neutronics problems. This is achieved by choosing a nonlinear function whose roots are the eigenpairs of the k-eigenvalue calculation and then using Newton's method to solve the nonlinear system. The exibility resulting from the use of a Krylov subspace method to solve the linear Newton step can be further extended via the use of the Jacobian-Free Newton-Krylov (JFNK) approximation, which requires no knowledge of the system's Jacobian; instead only the ability to evaluate the system residual is necessary. For the diffusion approximation, the nonlinear function is written in the form of the generalized eigenvalue problem and a set of preconditioners is developed and applied to the GMRES iterations that are used to solve the linearized Newton problem. Most of the developed methods can be implemented as either Newton-Krylov (NK) methods, where the Jacobian-vector product is formed using the explicitly constructed Jacobian, or via the JFNK approximation, where a finite-difference perturbation is used to approximate the Jacobian-vector product. One particularly effective preconditioning option comprises the use of the standard power iteration to precondition the GMRES iteration on either the right or the left. Pre-conditioning on the left, denoted JFNK(PI), results in a modified nonlinear system whose implementation only requires the ability to perform a single traditional outer iteration, making this approach relatively simple to wrap around an existing diffusion theory k-eigenvalue problem solver. Similar methods were developed for transport theory, cast using an operator notation that greatly simplifies their presentation. All of the nonlinear functions developed are written in terms of a generic fixed-point iteration, with a number of specific fixed-point formulations considered. Each fixed-point scheme represents a viable k-eigenvalue problem solution method, with two of the techniques corresponding to traditionally used iterative schemes. The new methods developed can also be wrapped around existing software in most instances, simplifying the implementation process. Ultimately it is seen that the most effective of the Newton formulations in transport theory is wrapped around a k-eigenvalue formulation that is a very special instance of traditional methods: no upscattering iterations are performed, only one inner iteration completed per outer, using source iteration with the previous outer iterate as the initial guess. In the Newton approach an extra degree of freedom is introduced by including the eigenvalue as an unknown, meaning an additional relation is necessary to close the system. In the diffusion theory case a normalization condition on the eigenvector was generally used, however in transport theory a number of so-called constraint relations were considered. These fall into two categories: normalization relations and eigenvalue update formulations. It was observed that the most effective of these constraint relations is the fission-rate eigenvalue update, derived directly from the eigenvalue update formula traditionally used to solve the k-eigenvalue problem. Numerical results, including measured performance quantified in number of iterations and execution time, were generated for suites of benchmark problems using the various Newton's Method formulations for the k-eigenvalue problem in both transport and diffusion theories. These results showed that the choice of the perturbation parameter in the JFNK approximation has very little impact on the calculation...