The nonequilibrium Dyson (or Kadanoff-Baym) equation, which is an equation of motion with long-range memory kernel for real-time Green functions, underlies many numerical approaches based on the Keldysh formalism. In this paper we map the problem of solving the Dyson equation in real-time onto a noninteracting auxiliary Hamiltonian with additional bath degrees of freedom. The solution of the auxiliary model does not require the evaluation of a memory kernel and can thus be implemented in a very memory efficient way. The mapping is derived for a self-energy which is local in space and is thus directly applicable within nonequilibrium dynamical mean-field theory (DMFT). We apply the method to study the interaction quench in the Hubbard model for an optical lattice with a narrow confinement, using inhomogeneous DMFT in combination with second-order weak-coupling perturbation theory. We find that, although the quench excites pronounced density oscillations, signatures of the two-stage relaxation similar to the homogeneous system can be observed by looking at the time-dependent occupations of natural orbitals.
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