A comparison of acceleration methods for solving the neutron transport k-eigenvalue problem

Over the past several years a number of papers have been written describing modern techniques for numerically computing the dominant eigenvalue of the neutron transport criticality problem. These methods fall into two distinct categories. The first category of methods rewrite the multi-group k-eigenvalue problem as a nonlinear system of equations and solve the resulting system using either a Jacobian-Free Newton–Krylov (JFNK) method or Nonlinear Krylov Acceleration (NKA), a variant of Anderson Acceleration. These methods are generally successful in significantly reducing the number of transport sweeps required to compute the dominant eigenvalue. The second category of methods utilize Moment-Based Acceleration (or High-Order/Low-Order (HOLO) Acceleration). These methods solve a sequence of modified diffusion eigenvalue problems whose solutions converge to the solution of the original transport eigenvalue problem. This second class of methods is, in our experience, always superior to the first, as most of the computational work is eliminated by the acceleration from the LO diffusion system. In this paper, we review each of these methods. Our computational results support our claim that the choice of which nonlinear solver to use, JFNK or NKA, should be secondary. The primary computational savings result from the implementation of a HOLO algorithm. We display computational results for a series of challenging multi-dimensional test problems.