  We present the results of large scale simulations of 4th order nonlinear
partial differential equations of dif- fusion type that are typically
encountered when modeling dynamics of thin fluid films on substrates. The
simulations are based on the alternate direction implicit (ADI) method, with
the main part of the compu- tational work carried out in the GPU computing
environment. Efficient and accurate computations allow for simulations on large
computational domains in three spatial dimensions (3D) and for long
computational times. We apply the methods developed to the particular problem
of instabilities of thin fluid films of nanoscale thickness. The large scale of
the simulations minimizes the effects of boundaries, and also allows for
simulating domains of the size encountered in published experiments. As an
outcome, we can analyze the development of instabilities with an unprecedented
level of detail. A particular focus is on analyzing the manner in which
instability develops, in particular regarding differences between spinodal and
nucleation types of dewetting for linearly unstable films, as well as
instabilities of metastable films. Simulations in 3D allow for consideration of
some recent results that were previously obtained in the 2D geometry (J. Fluid
Mech. 841, 925 (2018)). Some of the new results include using Fourier
transforms as well as topological invariants (Betti numbers) to distinguish the
outcomes of spinodal and nucleation types of instabilities, describing in
precise terms the complex processes that lead to the formation of satellite
drops, as well as distinguishing the shape of the evolving film front in
linearly unstable and metastable regimes. We also discuss direct comparison
between simulations and available experimental results for nematic liquid
crystal and polymer films.
