Monte Carlo Solution of k-Eigenvalue Problem Using Subspace Iteration Method

Abstract

Monte Carlo calculations for the evaluation of fundamental mode solution of k-eigenvalue problems generally make use of the Power Iteration (PI) method, which suffers from poor convergence, particularly in the case of large, loosely coupled systems. In the present paper, a method called Meyer’s Subspace Iteration (SSI) method, also called the Simultaneous vector iteration algorithm, is applied for the Monte Carlo solution of the k-eigenvalue problem. The SSI method is the block generalization of the single-vector PI method and has been found to work efficiently for solving the problem with the deterministic neutron transport setup. It is found that the convergence of the fundamental k-eigenvalue and corresponding fission source distribution improves substantially with the SSI-based Monte Carlo method as compared to the PI-based Monte Carlo method. To reduce the extra computational effort needed for simultaneous iterations with several vectors, a novel procedure is adopted in which it takes almost the same effort as with the single-vector PI-based Monte Carlo method. The algorithm is applied to several one-dimensional slab test cases of varying difficulty, and the results are compared with the standard PI method. It is observed that unlike the PI method, the SSI-based Monte Carlo method converges quickly and does not require many inactive generations before the mean and variance of eigenvalues could be estimated. It has been demonstrated that the SSI method can simultaneously find a set of the most dominant higher k-eigenmodes in addition to the fundamental mode solution.

Keywords:

• Monte Carlo method for neutron transport
• subspace iteration
• k-eigenvalue problem
• higher mode eigensolutions

Acknowledgments

The first author wishes to dedicate this work to the memory of his coauthor and teacher, Dr. R. S. Modak, who passed away in August 2018.