### Abstract

Computer codes involving neutron transport theory for nuclear engineering applications always require verification to assess improvement. Generally, analytical and semi-analytical benchmarks are desirable, since they are capable of high precision solutions to provide accurate standards of comparison. However, these benchmarks often involve relatively simple problems, usually assuming a certain degree of abstract modeling. In the present work, we show how semi-analytical equivalent benchmarks can be numerically generated using convergence acceleration. Specifically, we investigate the error behavior of a 1D spatial finite difference scheme for the multigroup (MG) steady-state neutron diffusion equation in plane geometry. Since solutions depending on subsequent discretization can be envisioned as terms of an infinite sequence converging to the true solution, extrapolation methods can accelerate an iterative process to obtain the limit before numerical instability sets in. The obtained results have been compared to the analytical solution to the 1D multigroup diffusion equation when available, using FORTRAN as the computational language. Finally, a slowing down problem has been solved using a cascading source update, showing how a finite difference scheme performs for ultra-fine groups (10^{4} groups) in a reasonable computational time using convergence acceleration.

Original language | English (US) |
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Title of host publication | International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013 |

Pages | 2088-2102 |

Number of pages | 15 |

Volume | 3 |

State | Published - 2013 |

Event | International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013 - Sun Valley, ID, United States Duration: May 5 2013 → May 9 2013 |

### Other

Other | International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013 |
---|---|

Country | United States |

City | Sun Valley, ID |

Period | 5/5/13 → 5/9/13 |

### Fingerprint

### Keywords

- Convergence acceleration
- Extrapolation methods
- Multigroup
- Neutron diffusion equation
- Ultra-fine group

### ASJC Scopus subject areas

- Nuclear Energy and Engineering
- Applied Mathematics

### Cite this

*International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013*(Vol. 3, pp. 2088-2102)

**Converged accelerated finite difference scheme for the multigroup neutron diffusion equation.** / Terranova, Nicholas; Mostacci, Domiziano; Ganapol, Barry D.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013.*vol. 3, pp. 2088-2102, International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013, Sun Valley, ID, United States, 5/5/13.

}

TY - GEN

T1 - Converged accelerated finite difference scheme for the multigroup neutron diffusion equation

AU - Terranova, Nicholas

AU - Mostacci, Domiziano

AU - Ganapol, Barry D

PY - 2013

Y1 - 2013

N2 - Computer codes involving neutron transport theory for nuclear engineering applications always require verification to assess improvement. Generally, analytical and semi-analytical benchmarks are desirable, since they are capable of high precision solutions to provide accurate standards of comparison. However, these benchmarks often involve relatively simple problems, usually assuming a certain degree of abstract modeling. In the present work, we show how semi-analytical equivalent benchmarks can be numerically generated using convergence acceleration. Specifically, we investigate the error behavior of a 1D spatial finite difference scheme for the multigroup (MG) steady-state neutron diffusion equation in plane geometry. Since solutions depending on subsequent discretization can be envisioned as terms of an infinite sequence converging to the true solution, extrapolation methods can accelerate an iterative process to obtain the limit before numerical instability sets in. The obtained results have been compared to the analytical solution to the 1D multigroup diffusion equation when available, using FORTRAN as the computational language. Finally, a slowing down problem has been solved using a cascading source update, showing how a finite difference scheme performs for ultra-fine groups (104 groups) in a reasonable computational time using convergence acceleration.

AB - Computer codes involving neutron transport theory for nuclear engineering applications always require verification to assess improvement. Generally, analytical and semi-analytical benchmarks are desirable, since they are capable of high precision solutions to provide accurate standards of comparison. However, these benchmarks often involve relatively simple problems, usually assuming a certain degree of abstract modeling. In the present work, we show how semi-analytical equivalent benchmarks can be numerically generated using convergence acceleration. Specifically, we investigate the error behavior of a 1D spatial finite difference scheme for the multigroup (MG) steady-state neutron diffusion equation in plane geometry. Since solutions depending on subsequent discretization can be envisioned as terms of an infinite sequence converging to the true solution, extrapolation methods can accelerate an iterative process to obtain the limit before numerical instability sets in. The obtained results have been compared to the analytical solution to the 1D multigroup diffusion equation when available, using FORTRAN as the computational language. Finally, a slowing down problem has been solved using a cascading source update, showing how a finite difference scheme performs for ultra-fine groups (104 groups) in a reasonable computational time using convergence acceleration.

KW - Convergence acceleration

KW - Extrapolation methods

KW - Multigroup

KW - Neutron diffusion equation

KW - Ultra-fine group

UR - http://www.scopus.com/inward/record.url?scp=84883340979&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84883340979&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84883340979

SN - 9781627486439

VL - 3

SP - 2088

EP - 2102

BT - International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, M and C 2013

ER -