The method of choice for describing attractive quantum systems is Hartree−Fock−Bogoliubov
(HFB) theory. This is a nonlinear model which allows for the description of
pairing effects, the main explanation for the superconductivity of
certain materials at very low temperature. This paper is the first study of
Hartree−Fock−Bogoliubov theory from the point of view of numerical analysis. We start by
discussing its proper discretization and then analyze the convergence of the simple fixed
point (Roothaan) algorithm. Following works by Cancès, Le Bris and Levitt for electrons in
atoms and molecules, we show that this algorithm either converges to a solution of the
equation, or oscillates between two states, none of them being solution to the HFB
equations. We also adapt the Optimal Damping Algorithm of Cancès and Le Bris to the HFB
setting and we analyze it. The last part of the paper is devoted to numerical experiments.
We consider a purely gravitational system and numerically discover that pairing always
occurs. We then examine a simplified model for nucleons, with an effective interaction
similar to what is often used in nuclear physics. In both cases we discuss the importance
of using a damping algorithm.