Stability of Monte Carlo k-eigenvalue simulations with CMFD feedback

In this paper we perform a Fourier stability analysis of MC-CMFD, a hybrid Monte Carlo k-eigenvalue method that utilizes coarse mesh finite difference (CMFD) feedback. The MC-CMFD method is nonlinear and contains random statistical errors; both of these features are inconsistent with the direct application of a Fourier stability analysis. To accomplish this analysis, we first formulate a non-random iteration method that approximates MC-CMFD, by assuming an infinite number of Monte Carlo particles per cycle. Then we (i) linearize this method, and (ii) Fourier-analyze the linearized method to theoretically predict its convergence properties. Finally, we demonstrate by direct numerical simulations that the Fourier analysis of the linearized non-random method accurately predicts the stability and fission source convergence rate (during the inactive cycles) of the original nonlinear MC-CMFD method. We do this by comparing the predictions of the Fourier analysis to simulations that utilize (i) a high-fidelity SN-CMFD code (which has no random statistical errors), and (ii) a MC-CMFD code (which has random statistical errors). The Fourier analysis and our two test codes confirm that the MC-CMFD method is stable if the optical thickness of the coarse grid (in the low-order CMFD calculation) is sufficiently small. However, the spectral radius increases monotonically with the coarse grid size, and if the latter exceeds a critical value, the MC-CMFD method becomes unstable. We discuss some implications of our results for practical MC-CMFD simulations.