  The non-equilibrium Green's function gives access to one-body observables for
quantum systems. Of particular interest are quantities such as density,
currents, and absorption spectra which are important for interpreting
experimental results in quantum transport and spectroscopy. We present an
integration scheme for the Green's function's equations of motion, the
Kadanoff-Baym equations (KBE), which is both adaptive in the time integrator
step size and method order as well as the history integration order. We analyze
the importance of solving the KBE self-consistently and show that adapting the
order of history integral evaluation is important for obtaining accurate
results. To examine the efficiency of our method, we compare runtimes to a
state of the art fixed time step integrator for several test systems and show
an order of magnitude speedup at similar levels of accuracy.
