# A fully coupled two-level Schwarz preconditioner based on smoothed aggregation for the transient multigroup neutron diffusion equations

## Abstract

The neutron multigroup diffusion equations is an important approximation of the neutron transport equation, and has been widely used for studying the motion of neutrons and their interactions with the stationary background materials. Solving the multigroup diffusion equations faces challenging because many variables are coupled together and a lot of computing resources are required. In this paper, we focus on the development of a scalable parallel algorithm framework for the system of equations arising from the discretization of the multigroup diffusion equations using a finite element method in space and using an implicit scheme in time. The parallel agorithm framweork consists of an inexact Jacobian-free Newton for the nonlinear equations, a Krylov subspace method for the Jacobian system and a parallel Schwarz preconditioner for the acceleration of the algorithm convergence. A coarse space based on smoothed aggregation is introduced to construct a two-level method to improve the parallel performance of the Schwarz preconditioner. We numerically shown that compared with the one-level method, the proposed two-level method works significantly better in terms of the number of linear iterations and the compute time for a system of equations with millions of unknowns and a supercomputer with thousands of processors.

- Authors:

- Idaho National Laboratory

- Publication Date:

- Research Org.:
- Idaho National Lab. (INL), Idaho Falls, ID (United States)

- Sponsoring Org.:
- USDOE Office of Nuclear Energy (NE)

- OSTI Identifier:
- 1478629

- Report Number(s):
- INL/CON-17-40867-Rev002

Journal ID: ISSN 1070-5325

- DOE Contract Number:
- AC07-05ID14517

- Resource Type:
- Conference

- Resource Relation:
- Journal Volume: 25; Journal Issue: 3; Conference: 18th Copper Mountain Conference on Multigrid Methods, 2017, Copper Mountain, Colorado, 03/26/2017 - 03/30/2017

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 - MATHEMATICS AND COMPUTING; parallel processing; two-level Schwarz preconditioner; multigroup neutron diffusion equations; Newton-Krylov-Schwarz; smoothed aggregation

### Citation Formats

```
Kong, Fande, Wang, Yaqi, Permann, Cody J, Schunert, Sebastian, Peterson, John W, Andrs, David, and Martineau, Richard C.
```*A fully coupled two-level Schwarz preconditioner based on smoothed aggregation for the transient multigroup neutron diffusion equations*. United States: N. p., 2018.
Web. doi:10.1002/nla.2162.

```
Kong, Fande, Wang, Yaqi, Permann, Cody J, Schunert, Sebastian, Peterson, John W, Andrs, David, & Martineau, Richard C.
```*A fully coupled two-level Schwarz preconditioner based on smoothed aggregation for the transient multigroup neutron diffusion equations*. United States. doi:10.1002/nla.2162.

```
Kong, Fande, Wang, Yaqi, Permann, Cody J, Schunert, Sebastian, Peterson, John W, Andrs, David, and Martineau, Richard C. Tue .
"A fully coupled two-level Schwarz preconditioner based on smoothed aggregation for the transient multigroup neutron diffusion equations". United States. doi:10.1002/nla.2162. https://www.osti.gov/servlets/purl/1478629.
```

```
@article{osti_1478629,
```

title = {A fully coupled two-level Schwarz preconditioner based on smoothed aggregation for the transient multigroup neutron diffusion equations},

author = {Kong, Fande and Wang, Yaqi and Permann, Cody J and Schunert, Sebastian and Peterson, John W and Andrs, David and Martineau, Richard C},

abstractNote = {The neutron multigroup diffusion equations is an important approximation of the neutron transport equation, and has been widely used for studying the motion of neutrons and their interactions with the stationary background materials. Solving the multigroup diffusion equations faces challenging because many variables are coupled together and a lot of computing resources are required. In this paper, we focus on the development of a scalable parallel algorithm framework for the system of equations arising from the discretization of the multigroup diffusion equations using a finite element method in space and using an implicit scheme in time. The parallel agorithm framweork consists of an inexact Jacobian-free Newton for the nonlinear equations, a Krylov subspace method for the Jacobian system and a parallel Schwarz preconditioner for the acceleration of the algorithm convergence. A coarse space based on smoothed aggregation is introduced to construct a two-level method to improve the parallel performance of the Schwarz preconditioner. We numerically shown that compared with the one-level method, the proposed two-level method works significantly better in terms of the number of linear iterations and the compute time for a system of equations with millions of unknowns and a supercomputer with thousands of processors.},

doi = {10.1002/nla.2162},

journal = {},

issn = {1070-5325},

number = 3,

volume = 25,

place = {United States},

year = {2018},

month = {5}

}