  Accretion occurs across a large range of scales and physical regimes. Despite
this diversity in the physics, the observed properties show remarkably
similarity. The theory of propagating fluctuations, in which broad-band
variability within an accretion disc travel inwards and combine, has long been
used to explain these phenomena. Recent numerical work has expanded on the
extensive analytical literature but has been restricted to using the 1D
diffusion equation for modelling the disc behaviour. In this work we present a
novel numerical approach for 2D (vertically integrated), stochastically driven
{\alpha}-disc simulations, generalising existing 1D models. We find that the
theory of propagating fluctuations translates well to 2D. However, the presence
of epicyclic motion in 2D (which cannot be captured within the diffusion
equation) is shown to have an important impact on local disc dynamics.
Additionally, there are suggestions that for sufficiently thin discs the
log-normality of the light-curves changes. As in previous work, we find that
the break frequency in the luminosity power spectrum is strongly dependent on
the driving timescale of the stochastic perturbations within the disc,
providing a possible observational signature for probing the magnetorotational
instability (MRI) dynamo. We also find that thinner discs are significantly
less variable than thicker ones, providing a compelling explanation for the
greater variability seen in the hard state vs the soft state of X-ray binaries.
Finally, we consider the wide-ranging applications of our numerical model for
use in other simulations.
