  In this work we present a numerical method to solve the set of Dyson-like
equations arising the context of non-equilibrium Green's functions. The
technique is based on the self-consistent solution of the Dyson equations for
the interacting single-particle Green's function once a choice for the
self-energy, functional of the single-particle Green's function itself, is
done. We briefly review the theory of the non-equilibrium Green's functions in
order to highlight the main point useful in discussing the proposed technique.
We also discuss the relation between our approach and the textbook approach to
solve the Kadanoff-Baym equations. We then present and discuss the numerical
implementation which is based on the distribution of the elements of the
Green's function and self-energies on a grid of processes. We discuss how the
structure of the considered self-energy approximations influences the
distribution of the matrices in order to minimize the communication time among
processes and which should be considered in the case of other approximations.
We give an example of the application of our technique to the case of quenches
in ultracold gases, also discussing to the convergence features of our scheme.
