DSA Preconditioning for the S_N Equations with Strictly Positive Spatial Discretization

Abstract

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.