Chaotic Iterations for S_N Transport

Abstract

An iterative solution method is introduced for S_N transport calculations called “chaotic” iterations. For S_N sweeps on parallel-decomposed meshes, a full-parallel sweep can be employed, in which processors must wait to start a sweep until incoming boundary data are received from one or more neighboring processes. This causes delays in the computation that affects efficiency. The parallel block Jacobi (PBJ) method, by contrast, is a splitting method in which all processor-local sweeps are computed using incoming data from the previous iteration with no waiting. This eliminates the delay associated with full-parallel sweeps but adversely impacts the iterative convergence rate. The chaotic iteration is a hybrid of the two possibilities, using current incoming data from neighboring processors when available and previous iteration data otherwise. Whether the boundary data are available or not depends on the communication between processes. It can be viewed as a splitting that changes from one iteration to the next, making the iteration chaotic. In this article, we prove that several iteration schemes associated with the chaotic splitting converge. The analysis presumes some splitting has been imposed at any given iteration, and so the results also apply to fixed, as well as chaotic, splittings. We present numerical results showing the convergence rate of the chaotic iterations method is between the full sweep method and the PBJ method. The numerical results also compare timings between the methods. Notably, for most of the test problems in this article, the chaotic iterations method is at least as fast as the PBJ method.

Keywords:

• Radiation transport
• sweeps
• chaotic iterations
• discrete ordinates

Acknowledgments

This work was supported by the US Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Los Alamos National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract DEAC52-06NA25396).

Notes

1 Many times, q_m, σ_t and σ_s are approximations to Q_m, Σ_t, and Σ_s respectively. The cross sections are sometimes approximated to be constant in each cell or projections onto the space P while q_m is typically a projection onto the space P. However, for the proof in this paper, it does not matter what space q_m, σ_t and σ_s are in.

2 This unpredictability means from run to run, one cannot expect results to vary less than the tolerance of the iterative solver. Even the number of iterations for convergence can change from run to run. This is in contrast to fixed splittings like PBJ, where one typically gets close to machine precision differences between runs and the number of iterations for convergence remains constant.

3 For $a,b\in \mathbb{R},ab\le \frac{1}{2}{a}^2+\frac{1}{2}{b}^2$.

4 ${(v·w)}^2\le \left|\right|v\left|{|}_2^2\right|\left|w\right|{|}_2^2$