Reduced basis method for non-symmetric eigenvalue problems: application to the multigroup neutron diffusion equations

TL;DR

This paper develops a reduced basis method with error estimates for non-symmetric eigenvalue problems, specifically applied to multigroup neutron diffusion equations in nuclear core optimization.

Contribution

It introduces a posteriori error estimates and a practical method to compute the prefactor for non-symmetric eigenproblems, extending reduced basis techniques to neutronics applications.

Findings

Effective error estimation for eigenvalues and eigenvectors.

Successful application to two-dimensional neutron diffusion problems.

Numerical results demonstrate the method's practicality and accuracy.

Abstract

In this article, we propose a reduced basis method for parametrized non-symmetric eigenvalue problems arising in the loading pattern optimization of a nuclear core in neutronics. To this end, we derive a posteriori error estimates for the eigenvalue and left and right eigenvectors. The practical computation of these estimators requires the estimation of a constant called prefactor, which we can express as the spectral norm of some operator. We provide some elements of theoretical analysis which illustrate the link between the expression of the prefactor we obtain here and its well-known expression in the case of symmetric eigenvalue problems, either using the notion of numerical range of the operator, or via a perturbative analysis. Lastly, we propose a practical method in order to estimate this prefactor which yields interesting numerical results on actual test cases. We provide detailed numerical simulations on two-dimensional examples including a multigroup neutron diffusion equation.