Newton's Method for the Computation of k-Eigenvalues in S_N Transport Applications

Recently, Jacobian-Free Newton-Krylov (JFNK) methods have been used to solve the k-eigenvalue problem in diffusion and transport theories. We propose an improvement to Newton's method (NM) for solving the k-eigenvalue problem in transport theory that avoids costly within-group iterations or iterations over energy groups. We present a formulation of the k-eigenvalue problem where a nonlinear function, whose roots are solutions of the k-eigenvalue problem, is written in terms of a generic fixed-point iteration (FPI). In this way any FPI that solves the k-eigenvalue problem can be accelerated using the Newton approach, including our improved formulation. Calculations with a one-dimensional multigroup S_N transport implementation in MATLAB provide a proof of principle and show that convergence to the fundamental mode is feasible. Results generated using a three-dimensional Fortran implementation of several formulations of NM for the well-known Takeda and C5G7-MOX benchmark problems confirm the efficiency of NM for realistic k-eigenvalue calculations and highlight numerous advantages over traditional FPI.