  The simulation of open many-body quantum systems is challenging, requiring
methods to both handle exponentially large Hilbert spaces and represent the
influence of (infinite) particle and energy reservoirs. These two requirements
are at odds with each other: Larger collections of modes can increase the
fidelity of the reservoir representation but come at a substantial
computational cost when included in numerical many-body techniques. An
increasingly utilized and natural approach to control the growth of the
reservoir is to cast a finite set of reservoir modes themselves as an open
quantum system. There are, though, many routes to do so. Here, we introduce an
accumulative reservoir construction -- an ARC -- that employs a series of
partial refreshes of the extended reservoirs. Through this series, the
representation accumulates the character of an infinite reservoir. This
provides a unified framework for both continuous (Lindblad) relaxation and a
recently introduced periodically refresh approach (i.e., discrete resets of the
reservoir modes to equilibrium). In the context of quantum transport, we show
that the phase space for physical behavior separates into discrete and
continuous relaxation regimes with the boundary between them set by natural,
physical timescales. Both of these regimes "turnover" into regions of over- and
under-damped coherence in a way reminiscent of Kramers' crossover. We examine
how the range of behavior impacts errors and the computational cost, including
within tensor networks. These results provide the first comparison of distinct
extended reservoir approaches, showing that they have different scaling of
error versus cost (with a bridging ARC regime decaying fastest). Exploiting the
enhanced scaling, though, will be challenging, as it comes with a substantial
increase in (operator space) entanglement entropy.
