# Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines

## Abstract

In this paper, three complementary methods have been implemented in the code Denovo that accelerate neutral particle transport calculations with methods that use leadership-class computers fully and effectively: a multigroup block (MG) Krylov solver, a Rayleigh quotient iteration (RQI) eigenvalue solver, and a multigrid in energy (MGE) preconditioner. The MG Krylov solver converges more quickly than Gauss Seidel and enables energy decomposition such that Denovo can scale to hundreds of thousands of cores. RQI should converge in fewer iterations than power iteration (PI) for large and challenging problems. RQI creates shifted systems that would not be tractable without the MG Krylov solver. It also creates ill-conditioned matrices. The MGE preconditioner reduces iteration count significantly when used with RQI and takes advantage of the new energy decomposition such that it can scale efficiently. Each individual method has been described before, but this is the first time they have been demonstrated to work together effectively. The combination of solvers enables the RQI eigenvalue solver to work better than the other available solvers for large reactors problems on leadership-class machines. Using these methods together, RQI converged in fewer iterations and in less time than PI for a full pressurized water reactor core. Thesemore »

- Authors:

- Univ. of California, Berkeley, CA (United States). Nuclear Engineering Dept.
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Radiation Transport and Criticality Group

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States); Univ. of California, Berkeley, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC); USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1427595

- Grant/Contract Number:
- AC05-00OR22725

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Nuclear Science and Engineering

- Additional Journal Information:
- Journal Volume: 190; Journal Issue: 1; Journal ID: ISSN 0029-5639

- Publisher:
- American Nuclear Society - Taylor & Francis

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 22 GENERAL STUDIES OF NUCLEAR REACTORS; 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; 42 ENGINEERING; eigenvalue; Rayleigh quotient; preconditioning

### Citation Formats

```
Slaybaugh, R. N., Ramirez-Zweiger, M., Pandya, Tara, Hamilton, Steven, and Evans, T. M.
```*Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines*. United States: N. p., 2018.
Web. doi:10.1080/00295639.2017.1413875.

```
Slaybaugh, R. N., Ramirez-Zweiger, M., Pandya, Tara, Hamilton, Steven, & Evans, T. M.
```*Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines*. United States. doi:10.1080/00295639.2017.1413875.

```
Slaybaugh, R. N., Ramirez-Zweiger, M., Pandya, Tara, Hamilton, Steven, and Evans, T. M. Tue .
"Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines". United States. doi:10.1080/00295639.2017.1413875. https://www.osti.gov/servlets/purl/1427595.
```

```
@article{osti_1427595,
```

title = {Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines},

author = {Slaybaugh, R. N. and Ramirez-Zweiger, M. and Pandya, Tara and Hamilton, Steven and Evans, T. M.},

abstractNote = {In this paper, three complementary methods have been implemented in the code Denovo that accelerate neutral particle transport calculations with methods that use leadership-class computers fully and effectively: a multigroup block (MG) Krylov solver, a Rayleigh quotient iteration (RQI) eigenvalue solver, and a multigrid in energy (MGE) preconditioner. The MG Krylov solver converges more quickly than Gauss Seidel and enables energy decomposition such that Denovo can scale to hundreds of thousands of cores. RQI should converge in fewer iterations than power iteration (PI) for large and challenging problems. RQI creates shifted systems that would not be tractable without the MG Krylov solver. It also creates ill-conditioned matrices. The MGE preconditioner reduces iteration count significantly when used with RQI and takes advantage of the new energy decomposition such that it can scale efficiently. Each individual method has been described before, but this is the first time they have been demonstrated to work together effectively. The combination of solvers enables the RQI eigenvalue solver to work better than the other available solvers for large reactors problems on leadership-class machines. Using these methods together, RQI converged in fewer iterations and in less time than PI for a full pressurized water reactor core. These solvers also performed better than an Arnoldi eigenvalue solver for a reactor benchmark problem when energy decomposition is needed. The MG Krylov, MGE preconditioner, and RQI solver combination also scales well in energy. Finally, this solver set is a strong choice for very large and challenging problems.},

doi = {10.1080/00295639.2017.1413875},

journal = {Nuclear Science and Engineering},

issn = {0029-5639},

number = 1,

volume = 190,

place = {United States},

year = {2018},

month = {2}

}