Spatial Multigrid for Isotropic Neutron Transport

Abstract

A spatial multigrid algorithm for isotropic neutron transport is presented in x-y geometry. The linear system is obtained using discrete ordinates in angle and corner balance finite differencing in space. Spatial smoothing is accomplished by a four-color block-Jacobi relaxation, where the diagonal blocks correspond to 4-cell blocks on the spatial grid. A bilinear interpolation operator and its transpose are used for the grid transfer operators. Good convergence factors were observed for homogeneous material properties. Heterogeneous material properties prove more difficult, especially the case of a vacuum region surrounded by a thick, diffusive region. In this case, a small amount of absorption, or "effective absorption" in a time-dependent problem, restores good convergence. Numerical results are presented.

Cellwise Block Iteration as a Multigrid Smoother for Discrete-Ordinates Radiation-Transport Calculations

“…Manteuffel et al (1995Manteuffel et al ( , 1996 have also developed a multigrid method based on cellwise block iteration. This method was later extended to heterogeneous media (Lansrud and Adams, 2005a) as well as multiple spatial dimensions (Lansrud and Adams, 2005b;Sheehan, 2007;Chang et al, 2007). However, there are two significant differences between our approach and this previous work.…”

“…While the resulting multigrid method converges very quickly for homogeneous media in one-dimensional planar geometry, performance can degrade significantly in the presence of material heterogeneities and is not nearly as impressive in multiple spatial dimensions. Sheehan (2007) and Chang et al (2007) improved performance somewhat by instead employing multiple applications of cellwise block iteration, each with a different combination of cells. In addition, Kanschat and Ragusa (2014) have developed a similar multigrid method, but with the angular flux solved for in a single cell as in our approach.…”

“…First, the particular form of cellwise block iteration employed by Manteuffel et al (1995Manteuffel et al ( , 1996 involves solving for the angular flux in two adjacent spatial cells simultaneously, as implemented in one-dimensional planar geometry. Lansrud and Adams (2005b), Sheehan (2007), and Chang et al (2007) all adapted this iteration to two-dimensional Cartesian geometry by solving in four adjacent cells simultaneously. Presumably, this would become eight cells in three spatial dimensions.…”

Adaptive angle and parallel multigrid for deterministic shielding problems

“…This started with S n discretisations in angle, with multigrid applied both in angle [10,11] and space [12][13][14][15]. Spherical harmonics (P n ) have also been used, again with multigrid performed in both space [16][17][18][19] and angle [20]. Stepping away from purely spatial or angular multigrid, mixed space/angle [21,22] and space/angle/energy [23] multigrid schemes with S n have been explored recently.…”

Extending the applicability of multigrid methods

“…In Section 3, we present numerical experiments demonstrating that our adaptive smooth aggregation solver is applicable to the linear systems arising in quantum chromodynamics (QCD). Then, in Section 4, we discuss our recent success [15] in applying Multigrid to the systems arising in radiation transport. The success of our approach is due to a robust multi-cell block-Jacobi smoother that allows for standard MG coarsening.…”