"DSA Preconditioning for the Sn Equations with Strictly Positive Spatial Discretization,"
M.S. Thesis, Nuclear Engineering, Texas A&M University, College Station, TX (2012).
Preconditioners based upon sweeps and diffusion-synthetic
acceleration (DSA) have been constructed and applied to the zeroth
and first spatial moments of the 1-D transport equation using SN
angular discretization and a strictly positive nonlinear spatial
closure (the CSZ method). The sweep preconditioner was applied
using the linear discontinuous Galerkin (LD) sweep operator and the
nonlinear CSZ sweep operator. DSA preconditioning was applied using
the linear LD S2 equations and the nonlinear CSZ S2 equations.
These preconditioners were applied in conjunction with a
Jacobian-free Newton Krylov (JFNK) method utilizing Flexible GMRES.
The action of the Jacobian on the Krylov vector was difficult to
evaluate numerically with a finite difference approximation because
the angular flux spanned many orders of magnitude. The evaluation
of the perturbed residual required constructing the nonlinear CSZ
operators based upon the angular flux plus some perturbation. For
cases in which the magnitude of the perturbation was comparable to
the local angular flux, these nonlinear operators were very
sensitive to the perturbation and were significantly different than
the unperturbed operators. To resolve this shortcoming in the
finite difference approximation, in these cases the residual
evaluation was performed using nonlinear operators "frozen" at the
unperturbed local psi. This was a Newton method with a perturbation
fixup. Alternatively, an entirely frozen method always performed
the Jacobian evaluation using the unperturbed nonlinear operators.
This frozen JFNK method was actually a Picard iteration scheme. The
perturbed Newton's method proved to be slightly less expensive than
the Picard iteration scheme. The CSZ sweep preconditioner was
significantly more effective than preconditioning with the LD
sweep. Furthermore, the LD sweep is always more expensive to apply
than the CSZ sweep. The CSZ sweep is superior to the LD sweep as a
preconditioner. The DSA preconditioners were applied in conjunction
with the CSZ sweep. The nonlinear CSZ DSA preconditioner did not
form a more effective preconditioner than the linear DSA
preconditioner in this 1-D analysis. As it is very difficult to
construct a CSZ diffusion equation in more than one dimension, it
will be very beneficial if the results regarding the effectiveness
of the LD DSA preconditioner are applicable to multi-dimensional
problems. See Document
Associated Project(s):SHIELD (Smuggled HEU Interdiction through Enhanced anaLysis and Detection): A Framework for Developing Novel Detection Systems Focused on Interdicting Shielded HEU