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.

引入用于S _N传输计算的迭代求解方法，称为“混沌”迭代。对于S _N在并行分解的网格上执行扫描，可以采用全并行扫描，其中处理器必须等待开始扫描，直到从一个或多个相邻进程接收到传入的边界数据为止。这会导致计算延迟，从而影响效率。相比之下，并行块Jacobi（PBJ）方法是一种拆分方法，其中所有处理器本地扫描都是使用前一次迭代的输入数据来计算的，而无需等待。这样可以消除与全并行扫描相关的延迟，但会对迭代收敛速度产生不利影响。混沌迭代是这两种可能性的混合，使用可用时来自相邻处理器的当前传入数据，否则使用先前迭代数据。边界数据是否可用取决于过程之间的通信。可以将其视为从一个迭代到下一个迭代的分裂，从而使迭代变得混乱。在本文中，我们证明了与混沌分裂相关的几种迭代方案收敛。分析假设在任何给定的迭代中都施加了一些分裂，因此结果也适用于固定分裂以及混沌分裂。我们提供的数值结果表明混沌迭代方法的收敛速度在全扫描方法和PBJ方法之间。数值结果还比较了两种方法之间的时序。值得注意的是，对于本文中的大多数测试问题，混沌迭代方法至少与PBJ方法一样快。