Chaotic Iterations for SN Transport

Abstract

An iterative solution method is introduced for SN transport calculations called “chaotic” iterations. For SN 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.