A Fast Multigrid Algorithm for Isotropic Transport Problems. II: With Absorption

Abstract

A multigrid method for solving the 1-D slab-geometry S N equations with isotropic scattering and absorption is presented. The case with no absorption was treated in part I of this paper 10]. Relaxation is based on a two-cell inversion, which is very e cient because it takes advantage of the structure of the two-cell problem. For interpolation we use kinked linear elements. The kink is based on the amount of absorption present. The restriction operator is full weighting. Numerical results show this algorithm to be faster than DSA in all regimes. This scheme is also well-suited for massively parallel computer architectures.

Polynomial preconditioning and the Generalized Minimal Residual algorithm solver for the 2-D Boltzmann transport equation

“…Several acceleration methods of the convergence of this algorithm have been introduced and studied. In particular, the diffusion synthetic acceleration (DSA) methods developed in [5], and multigrid algorithms [7]. The main difficulties encountered while studying these methods lead the authors either to consider the discretized equation in the angular variable [5], or the continuous equation with a truncated expansion of h with respect to this angular variable [5,7].…”

“…In particular, the diffusion synthetic acceleration (DSA) methods developed in [5], and multigrid algorithms [7]. The main difficulties encountered while studying these methods lead the authors either to consider the discretized equation in the angular variable [5], or the continuous equation with a truncated expansion of h with respect to this angular variable [5,7]. The idea in [1,2] is to introduce and study various algorithms, relying on a splitting of the collision operator, and adapted from the methods of Jacobi, Gauss-Seidel, SOR and the Generalized Minimal Residual, in the infinite dimensional case.…”

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

“…Our approach may be useful with this type of iterative scheme, as well. 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. 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.…”

“…Instead, we solve within a single cell, which is computationally less expensive. Second, Manteuffel et al (1995Manteuffel et al ( , 1996 and Lansrud and Adams (2005a, b) use rather complicated procedures for transferring quantities between spatial grids. 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.…”