Abstract

We discuss the numerical solution of the nonlinear integro-differential equation for the probability of a divergent neutron chain in a stationary system (i.e., the probability of initiation (POI)). We follow the development described in Bell’s classic paper on the stochastic theory of neutron transport. As noted by Bell, the linearized form of this equation resembles the linear adjoint neutron transport equation. A matrix formalism for the discretized steady state (or forward) neutron equation in slab geometry is first developed, and is then used to derive the discrete adjoint equation. A main advantage of this discrete development is that the resulting discrete adjoint equation does not depend upon how the multigroup cross sections for the forward problem are obtained. That is, we derive the discrete adjoint directly from the discrete forward equations rather than discretizing directly the adjoint equation. This also guarantees that the discrete adjoint operator is consistent with the inner product used to define the adjoint operator. We discuss three approaches for the numerical solution of the POI equations, and present numerical results on several test problems. The three solution methods are a simple fixed point iteration, a second approach that is akin to a nonlinear Power iteration, and a third approach which uses a Newton-Krylov nonlinear solver. We also give sufficient conditions to guarantee the existence and uniqueness of nontrivial solutions to our discrete POI equations when the discrete system is supercritical, and that only the trivial solution exists when the discrete system is subcritical. Our approach is modeled after the analysis presented for the continuous POI equations by Mokhtar-Kharroubi and Jarmouni-Idrissi, and by Pazy and Rabinowitz.

摘要

我们讨论了稳态系统中发散中子链概率（即引发概率（POI））的非线性积分微分方程的数值解。我们遵循贝尔关于中子输运随机理论的经典论文中描述的发展。正如贝尔所指出的，该方程的线性化形式类似于线性伴随中子输运方程。首先开发了板几何中离散稳态（或正向）中子方程的矩阵形式，然后用于推导离散伴随方程。这种离散发展的主要优点是所得到的离散伴随方程不依赖于如何获得正向问题的多组横截面。那是，我们直接从离散前向方程导出离散伴随方程，而不是直接离散伴随方程。这也保证了离散伴随算子与用于定义伴随算子的内积一致。我们讨论了 POI 方程数值求解的三种方法，并给出了几个测试问题的数值结果。这三种求解方法是简单的定点迭代、类似于非线性幂迭代的第二种方法以及使用 Newton-Krylov 非线性求解器的第三种方法。我们还给出了足够的条件来保证当离散系统是超临界时离散 POI 方程非平凡解的存在和唯一性，并且当离散系统是亚临界时只有平凡解存在。