  The Keldysh formalism for nonequilibrium Green's functions is a powerful
theoretical framework for the description of the electronic structure,
spectroscopy, and dynamics of strongly correlated systems. However, the
underlying Kadanoff-Baym equations (KBE) for the two-time Keldysh Green's
functions involve a memory kernel which results in a high computational cost
for long simulation times $t_\text{max}$, with a cubic scaling of the
computation time with $t_\text{max}$. Truncation of the memory kernel can
reduce the computational cost to linear scaling with $t_\text{max}$, but the
required memory times will depend on the model and the diagrammatic
approximation to the self-energy. We explain how a truncation of the memory
kernel can be incorporated into the time-propagation algorithm to solve the
KBE, and investigate the systematic truncation of the memory kernel for the
Hubbard model in different parameter regimes, and for different diagrammatic
approximations. The truncation is easier to control within dynamical mean-field
solutions, where it is applied to a momentum-independent self-energy. Here,
simulation times up to two orders of magnitude longer are accessible both in
the weak and strong coupling regime, allowing for a study of long-time
phenomena such as the crossover between pre-thermalization and thermalization
dynamics.
