First-order corrector for the homogenization of the criticality eigenvalue problem in the even parity formulation of the neutron transport.

*(English)*Zbl 0937.35007We consider the homogenization of the critical eigenvalue problem for the even parity flux of neutron transport in a domain with isotropic and periodically oscillating coefficients. We prove that the neutron density is factored in the product of two terms. The first one describes local behavior of the density at the cell level. It is a solution of a heterogeneous transport problem with periodic boundary conditions. The second term gives global behavior on the whole domain. It satisfies a homogeneous diffusion equation posed on the whole domain with Dirichlet boundary conditions. We also give the asymptotic analysis of the corresponding eigenvalues. This expansion gives rise to errors of the order of the cell size. It does not account for neutron leakage at the boundary of the core and yields unacceptable errors in practice. We derive a more accurate expansion of the eigenelements in the case of a symmetric and cubic domain. The analysis of a boundary layer allows us to derive modified boundary conditions for the diffusion eigenvalue problem. The resulting approximation for the leading transport eigenvalue is proven to be accurate to one order higher than previously. Numerical experiments confirm the accuracy of the reconstructed eigenvectors in realistic settings.

Reviewer: Marco Biroli (Monza)

##### MSC:

35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |

82D75 | Nuclear reactor theory; neutron transport |

35P05 | General topics in linear spectral theory for PDEs |

78A35 | Motion of charged particles |