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