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**Resolution of the generalized eigenvalue problem in the neutron diffusion equation discretized by the finite volume method.**
*(English)*
Zbl 1474.65422

Summary: Numerical methods are usually required to solve the neutron diffusion equation applied to nuclear reactors due to its heterogeneous nature. The most popular numerical techniques are the Finite Difference Method (FDM), the Coarse Mesh Finite Difference Method (CFMD), the Nodal Expansion Method (NEM), and the Nodal Collocation Method (NCM), used virtually in all neutronic diffusion codes, which give accurate results in structured meshes. However, the application of these methods in unstructured meshes to deal with complex geometries is not straightforward and it may cause problems of stability and convergence of the solution. By contrast, the Finite Element Method (FEM) and the Finite Volume Method (FVM) are easily applied to unstructured meshes. On the one hand, the FEM can be accurate for smoothly varying functions. On the other hand, the FVM is typically used in the transport equations due to the conservation of the transported quantity within the volume. In this paper, the FVM algorithm
implemented in the ARB Partial Differential Equations solver has been used to discretize the neutron diffusion equation to obtain the matrices of the generalized eigenvalue problem, which has been solved by means of the SLEPc library.

### MSC:

65N08 | Finite volume methods for boundary value problems involving PDEs |

65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |

82D75 | Nuclear reactor theory; neutron transport |

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\textit{Á. Bernal} et al., Abstr. Appl. Anal. 2014, Article ID 913043, 15 p. (2014; Zbl 1474.65422)

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### References:

[1] | Stacey, W. M., Nuclear Reactor Physics (2001), New York, NY, USA: John Wiley & Sons, New York, NY, USA |

[2] | Verdú, G.; Ginestar, D.; Vidal, V.; Muñoz-Cobo, J. L., 3D \(λ\)-modes of the neutron-diffusion equation, Annals of Nuclear Energy, 21, 7, 405-421 (1994) |

[3] | Miró, R.; Ginestar, D.; Verdú, G.; Hennig, D., A nodal modal method for the neutron diffusion equation. Application to BWR instabilities analysis, Annals of Nuclear Energy, 29, 10, 1171-1194 (2002) |

[4] | Hernandez, V.; Roman, J. E.; Vidal, V., SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems, ACM Transactions on Mathematical Software, 31, 3, 351-362 (2005) · Zbl 1136.65315 |

[5] | Hoffmann, K. A.; Chiang, S. T., Computational Fluid Dynamics, 2 (2000), Wichita, Kan, USA: Engineering Education System, Wichita, Kan, USA |

[6] | Harvie, D. J. E., An implicit finite volume method for arbitrary transport equations, ANZIAM Journal, 52, C1126-C1145 (2012) · Zbl 1390.65135 |

[7] | Cueto-Felgueroso, L.; Colominas, I.; Nogueira, X.; Navarrina, F.; Casteleiro, M., Finite volume solvers and moving least-squares approximations for the compressible Navier-Stokes equations on unstructured grids, Computer Methods in Applied Mechanics and Engineering, 196, 45-48, 4712-4736 (2007) · Zbl 1173.76358 |

[8] | Geuzaine, C.; Remacle, J.-F., Gmsh: a 3-D finite element mesh generator with built-in pre- and post-processing facilities, International Journal for Numerical Methods in Engineering, 79, 11, 1309-1331 (2009) · Zbl 1176.74181 |

[9] | Hernández, V.; Román, J. E.; Vidal, V., SLEPc: scalable library for eigenvalue problem computations, High Performance Computing for Computational Science—VECPAR 2002. High Performance Computing for Computational Science—VECPAR 2002, Lecture Notes in Computer Science, 2565, 377-391 (2003), Berlin, Germany: Springer, Berlin, Germany · Zbl 1027.65504 |

[10] | Müller, E. Z.; Weiss, Z. J., Benchmarking with the multigroup diffusion high-order response matrix method, Annals of Nuclear Energy, 18, 9, 535-544 (1991) |

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