  Three complementary methods have been implemented in the code Denovo that
accelerate neutral particle transport calculations with methods that use
leadership-class computers fully and effectively: a multigroup block (MG)
Krylov solver, a Rayleigh Quotient Iteration (RQI) eigenvalue solver, and a
multigrid in energy (MGE) preconditioner. The MG Krylov solver converges more
quickly than Gauss Seidel and enables energy decomposition such that Denovo can
scale to hundreds of thousands of cores. RQI should converge in fewer
iterations than power iteration (PI) for large and challenging problems. RQI
creates shifted systems that would not be tractable without the MG Krylov
solver. It also creates ill-conditioned matrices. The MGE preconditioner
reduces iteration count significantly when used with RQI and takes advantage of
the new energy decomposition such that it can scale efficiently. Each
individual method has been described before, but this is the first time they
have been demonstrated to work together effectively.
  The combination of solvers enables the RQI eigenvalue solver to work better
than the other available solvers for large reactors problems on leadership
class machines. Using these methods together, RQI converged in fewer iterations
and in less time than PI for a full pressurized water reactor core. These
solvers also performed better than an Arnoldi eigenvalue solver for a reactor
benchmark problem when energy decomposition is needed. The MG Krylov, MGE
preconditioner, and RQI solver combination also scales well in energy. This
solver set is a strong choice for very large and challenging problems.
