  High-performance graphical processing units (GPU) are used for the repeated
parallelised propagation of non-linear partial differential equations on large
spatio-temporal grids. The main challenge results as a combination of the
requirement of large grids for exploring scaling over several orders of
magnitude, both in space and time, and the need for high statistics in
averaging over many runs, in computing correlation functions for highly
fluctuating quantum many-body states. With our simulations, we explore the
dynamics of complex quantum systems far from equilibrium, with the aim of
classifying their universal characteristics such as scaling exponents near
non-thermal fixed points. Our results are strongly relevant for the development
of synthetic quantum systems when exploring the respective physics in the
laboratory.
