RAYLEIGH QUOTIENT ITERATION IN 3D, DETERMINISTIC NEUTRON TRANSPORT

Today’s “grand challenge” neutron transport problems require 3-D meshes with billions of cells,
hundreds of energy groups, and accurate quadratures and scattering expansions. Leadership-class
computers provide platforms on which high-fidelity fluxes can be calculated. However, appropriate
methods are needed that can use these machines effectively. Such methods must be able to to use
hundreds of thousands of cores and have good convergence properties. Taking advantage of the
multigroup Krylov solver that scales to hundreds of thousands of cores, Rayleigh quotient iteration
(RQI) is an eigenvalue solver that has been added to the S N code Denovo to address convergence.
Rayleigh quotient iteration is an optimal shifted inverse iteration method. RQI should converge in
fewer iterations than the more common power method and other shifted inverse iteration methods
for many problems of interest. Denovo’s RQI uses a new multigroup Krylov solver for the fixed
source solutions inside every iteration that allows parallelization in energy in addition to space and
angle. This Krylov solver has scaled successfully to 200,000 cores. This paper shows that RQI
works for some small problems. However, the Krylov method upon which it relies does not always
converge because RQI creates ill-conditioned systems. This result leads to the conclusion that
preconditioning is needed to allow this method to be applicable to a wider variety of problems.