  A tremendous amount of recent attention has focused on characterizing the
dynamical properties of periodically driven many-body systems. Here, we use a
novel numerical tool termed `density matrix truncation' (DMT) to investigate
the late-time dynamics of large-scale Floquet systems. We find that DMT
accurately captures two essential pieces of Floquet physics, namely,
prethermalization and late-time heating to infinite temperature. Moreover, by
implementing a spatially inhomogeneous drive, we demonstrate that an interplay
between Floquet heating and diffusive transport is crucial to understanding the
system's dynamics. Finally, we show that DMT also provides a powerful method
for quantitatively capturing the emergence of hydrodynamics in static
(un-driven) Hamiltonians; in particular, by simulating the dynamics of generic,
large-scale quantum spin chains (up to L = 100), we are able to directly
extract the energy diffusion coefficient.
