A Multilevel in Space and Energy Solver for 3-D Multigroup Diffusion and Coarse-Mesh Finite Difference Eigenvalue Problems

Abstract

The Multilevel in Space and Energy Diffusion (MSED) method accelerates the iterative convergence of multigroup diffusion eigenvalue problems by performing work on lower-order equations with only one group and/or coarser spatial grids. It consists of two primary components: (1) a grey (one-group) diffusion eigenvalue problem that is solved via Wielandt-shifted power iteration (PI) and (2) a multigrid-in-space linear solver. In previous work, the efficiency of MSED was verified using Fourier analysis and numerical results from a one-dimensional multigroup diffusion code. Since that work, MSED has been implemented as a solver for the coarse-mesh finite difference (CMFD) system in the three-dimensional Michigan Parallel Characteristics Transport (MPACT) code. In this paper, the results from the implementation of MSED in MPACT are presented, and the changes needed to make MSED more suitable for MPACT are described. For problems without feedback, the results in this paper show that MSED can reduce the CMFD run time by an order of magnitude and the overall run time by a factor of 2 to 3 compared to the default CMFD solver in MPACT [PI with the generalized minimal residual (GMRES) method]. For problems with feedback, the convergence of the outer Picard iteration scheme is worsened by the well-converged CMFD solutions produced by the standard MSED method. To overcome this unintuitive deficiency, MSED may be run with looser convergence criteria (a modified version of the MSED method called MSED-L) to circumvent the issue until the multiphysics iteration in MPACT is improved. Results show that MSED-L can reduce the CMFD run time in MPACT by an order of magnitude, without negatively impacting the outer Picard iteration scheme.

Keywords:

• CMFD
• multilevel
• eigenvalue
• three-dimensional, multigroup

View correction statement:

Correction

Acknowledgments

The work of the first author was supported by a U.S. Department of Energy (DOE) Nuclear Energy University Programs Graduate Fellowship from August 2013 to August 2016. This research was supported by the Consortium for Advanced Simulation of Light Water Reactors (www.casl.gov), an Energy Innovation Hub (http://www.energy.gov/hubs) for Modeling and Simulation of Nuclear Reactors under DOE contract number DE-AC05-00OR22725. Moreover, this research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the DOE Office of Science under contract number DE-AC05-00OR22725.

Lawrence Livermore National Laboratory (LLNL) is operated by Lawrence Livermore National Security, LLC, for the DOE National Nuclear Security Administration under contract number DE-AC52-07NA27344. The first author was supported by LLNL during the writing of this paper.

This document (IM release number LLNL-JRNL-759325) was prepared as an account of work sponsored by an agency of the U.S. government.

Notes

^a The only differences between odCMFD and CMFD are the values of the diffusion and correction coefficients. Although MPACT uses odCMFD, we will refer to it as simply CMFD from this point forward because the methods developed in this paper can be applied to both odCMFD and standard CMFD.

^b In MPACT, the role of the CMFD problem is to minimize the number of CMFD-accelerated transport sweeps (outer iterations). Fully converging CMFD in the early outer iterations is not necessary since the CMFD correction factors $\hat{D}$ from the transport problem are far from converged. Nonetheless, the results of the paper show that the default CMFD solver in MPACT struggles to converge even when the convergence criterion is loose.

^c Equation (8)(8) $P^{-1}A\mathit{x}=P^{-1}\mathit{b}$(8) is an example of left-preconditioning, but right-preconditioning is also commonly used.

^d The signs on $\hat{D}$ are erroneously swapped in Eq. 6.3d of Ref. 9. Equation (13d)(13d) $〈D_{i_1,i_2}〉\equiv \{\begin{array}{cc}\frac{1}{{\Phi }_{i_2}}{\displaystyle \sum _{g=1}^G\left(D_{i_1,g}+{\widehat{D}}_{i_1,g}\right)}{\varphi }_{i_2,g}& \text{if}{i}_1\text{>}{i}_{\text{2}},\\ \frac{1}{{\Phi }_{i_2}}{\displaystyle \sum _{g=1}^G\left(D_{i_1,g}-{\widehat{D}}_{i_1,g}\right)}{\varphi }_{i_2,g}& \text{if}{i}_1\text{<}{i}_{\text{2}},\end{array}$(13d) is the corrected version of that equation.

^e The eigenvalue tolerance for the outer transport sweeps is still ${10}^{-6}$.