  Real-time nonequilibrium Green functions (NEGF) have been very successful to
simulate the dynamics of correlated many-particle systems far from equilibrium.
However, NEGF simulations are computationally expensive since the effort scales
cubically with the simulation duration. Recently we have introduced the G1--G2
scheme that allows for a dramatic reduction to time-linear scaling
[Schl\"unzen, Phys. Rev. Lett. 124, 076601 (2020); Joost et al., Phys. Rev. B
101, 245101 (2020)]. Here we tackle another problem: the rapid growth of the
computational effort with the system size. In many situations where the system
of interest is coupled to a bath, to electric contacts or similar macroscopic
systems for which a microscopic resolution of the electronic properties is not
necessary, efficient simplifications are possible. This is achieved by the
introduction of an embedding selfenergy -- a concept that has been successful
in standard NEGF simulations. Here, we demonstrate how the embedding concept
can be introduced into the G1--G2 scheme, allowing us to drastically accelerate
NEGF embedding simulations. The approach is compatible with all advanced
selfenergies that can be represented by the G1--G2 scheme [as described in
Joost et al., Phys. Rev. B 105, 165155 (2022)] and retains the memory-less
structure of the equations and their time linear scaling. As a numerical
illustration we investigate the charge transfer between a Hubbard nanocluster
and an additional site which is of relevance for the neutralization of ions in
matter.
