  We demonstrate how the dynamical coarse-graining approach can be
systematically extended to higher orders in the coupling between system and
reservoir. Up to second order in the coupling constant we explicitly show that
dynamical coarse-graining unconditionally preserves positivity of the density
matrix -- even for bath density matrices that are not in equilibrium and also
for time-dependent system Hamiltonians. By construction, the approach correctly
captures the short-time dynamics, i.e., it is suitable to analyze non-Markovian
effects. We compare the dynamics with the exact solution for highly
non-Markovian systems and find a remarkable quality of the coarse-graining
approach. The extension to higher orders is straightforward but rather tedious.
The approach is especially useful for bath correlation functions of simple
structure and for small system dimensions.
