Solving eigenvalue response matrix equations with nonlinear techniques

Abstract

This paper presents new algorithms for use in the eigenvalue response matrix method (ERMM) for reactor eigenvalue problems. ERMM spatially decomposes a domain into independent nodes linked via boundary conditions approximated as truncated orthogonal expansions, the coefficients of which are response functions. In its simplest form, ERMM consists of a two-level eigenproblem: an outer Picard iteration updates the k-eigenvalue via balance, while the inner λ-eigenproblem imposes neutron balance between nodes. Efficient methods are developed for solving the inner λ-eigenvalue problem within the outer Picard iteration. Based on results from several diffusion and transport benchmark models, it was found that the Krylov-Schur method applied to the λ-eigenvalue problem reduces Picard solver times (excluding response generation) by a factor of 2-5. Furthermore, alternative methods, including Picard acceleration schemes, Steffensen's method, and Newton's method, are developed in this paper. These approaches often yield faster k-convergence and a need for fewer k-dependent response function evaluations, which is important because response generation is often the primary cost for problems using responses computed online (i.e., not from a precomputed database). Accelerated Picard iteration was found to reduce total computational times by 2-3 compared to the unaccelerated case for problems dominated by response generation. In addition, Newton's method was found to provide nearly the same performance with improved robustness.

Multigroup diffusion preconditioners for multiplying fixed-source transport problems

“…All of our analysis is based on the discrete ordinates (S N ) method, though much of the analysis applies equally to the method of characteristics (MOC). Ultimately, this work supports our simultaneous effort to develop advanced response matrix methods, which aim to solve large reactor eigenvalue problems by spatially-decomposing a domain into independent nodes linked via approximate boundary conditions [6,7,8]. The boundary conditions are defined in terms of fixed source problems for each node, and because many such transport problems are required, methods to solve them efficiently are highly desirable.…”

A High-Order, Time-Dependent Response Matrix Method for Reactor Kinetics

“…Their work extends previous work on the steady-state COarsh MEsh Transport (COMET) method from the same group, although similar response matrix methods (RMMs) were developed in the 1970s 2,3 and continue to be studied by a variety of groups around the world. 4,5 A survey of the RMM literature can be found in Ref. angular, and temporal discretization; for example, the COMET approach is based on polynomial representations of all variables, while the approach of Sicilian and Pryor is limited to low-order representations, e.g., linear in angle and time, and segmented in space.…”