Erin D. Fichtl

Nonlinear Krylov acceleration applied to a discrete ordinates formulation of the k-eigenvalue problem

We compare a variant of Anderson Mixing with the Jacobian-Free Newton-Krylov and Broyden methods applied to an instance of the k-eigenvalue formulation of the linear Boltzmann transport equation. We present evidence that one variant of Anderson Mixing finds solutions in the fewest number of iterations. We examine and strengthen theoretical results of Anderson Mixing applied to linear problems.

The Newton-Krylov Method Applied to Negative-Flux Fixup in SN Transport Calculations

Abstract Under some circumstances, spatial discretizations of the SN transport equation will lead to negativity in the scalar flux; therefore, negative-flux fixup schemes are often employed to ensure that the flux is positive. The nonlinear nature of these schemes precludes the use of powerful linear iterative solvers such as Krylov methods; thus, solutions are generally computed using so-called source iteration (SI), which is a simple fixed-point iteration. In this paper, we use Newton’s method to solve fixed-source SN transport problems with negative-flux fixup, for which the analytic form of the Jacobian is shown to be nonsingular. It is necessary to invert the Jacobian at each Newton iteration. Generally, an exact inversion is prohibitively expensive and furthermore is not necessary for convergence of Newton’s method. In the inexact Newton-Krylov method, the Jacobian is inverted using a Krylov method, which completes at some prescribed tolerance. This tolerance may be quite large in the initial stages of the Newton iteration. In this paper, we compare the use of the exact Jacobian with two approximations of the Jacobian in the inexact Newton-Krylov method. The first approximation is a finite difference approximation. The second is that used in the Jacobian-free Newton-Krylov (JFNK) method, which performs a finite difference approximation without actually generating the Jacobian itself. Numerical results comparing standard SI with the three methods demonstrate that Newton-Krylov can outperform SI, particularly for diffusive materials. The results also show that the additional level of approximation introduced by the JFNK approach does not adversely affect convergence, indicating that JFNK will be robust and efficient in large-scale applications.

Atomic Mix Synthetic Acceleration of Dose Computations in Binary Statistical Media

Abstract An iterative solution of coupled standard model equations arising in electron transport in binary statistical mixtures is considered. Convergence degradation is observed in certain energy groups and is attributed to chunk sizes appearing optically thin in the higher energy groups. Fourier analysis shows that the spectral radius approaches unity for the zero wave-number error mode as the chunk sizes become vanishingly small. It is shown that the atomic mix model accurately approximates transport under these circumstances and moreover provides a suitable low-order approximation to the iteration error. Fourier analysis and numerical implementation confirm that atomic mix acceleration is unconditionally effective for the application considered here. Our computations also demonstrate the inaccuracy of the atomic mix model for electron dose, especially for materials with strongly contrasting physical properties.