  We combine matrix-product state (MPS) and Mean-Field (MF) methods to model
the real-time evolution of a three-dimensional (3D) extended Hubbard system
formed from one-dimensional (1D) chains arrayed in parallel with weak coupling
in-between them. This approach allows us to treat much larger 3D systems of
correlated fermions out-of-equilibrium over a much more extended real-time
domain than previous numerical approaches. We deploy this technique to study
the evolution of the system as its parameters are tuned from a charge-density
wave (CDW) phase into the superconducting (SC) regime, which allows us to
investigate the formation of transient non-equilibrium SC. In our ansatz, we
use MPS solutions for chains as input for a self-consistent time-dependent MF
scheme. In this way, the 3D problem is mapped onto an effective 1D Hamiltonian
that allows us to use the MPS efficiently to perform the time evolution, and to
measure the BCS order parameter as a function of time. Our results confirm
previous findings for purely 1D systems that for such a scenario
superconductivity forms in a transient state.
