A computational technique for evaluating eigenfunctions of symmetrical nuclear systems

Abstract

Presented here is a description of a computational technique for calculating the higher order eigenfunctions of a linear Boltzmann operator of a nuclear system symmetrical in space. In this case, the usual conjugation of filtering technique and power method fails due to eigenvalue degeneracy. To overcome this drawback, we propose here a method for computing the eigenfunctions of such a system, starting from a set of asymmetrical ones, obtained introducing an ad hoc perturbation in the original system. This perturbation allows the degenerated eigenvalues to split so that the filter can be successful. The computation of the symmetrical set is performed via an extension of the Standard Method. The coefficients of the expansion depend here on the difference between the cross sections of the original system and those of the asymmetrical one. This methodology, quoted as the Generalized Standard Method (GSM), allows us to obtain several symmetrical eigenfunctions and degenerated eigenvalues which, otherwise, could not be computed using the filter technique. Using a quite small number of non-symmetrical eigenfunctions, we have been able to reconstruct an equivalent number of eigenvalues and eigenfunctions of the original system. The use of the proposed computational procedure becomes unpractical in the general multigroup problems owing to the lack of completeness of the set of eigenfunctions when they are computed via the power method. Restoring the completeness of the set would be sufficient to make an application of the method in the general case possible.