  The effects of thermal fluctuations on nanoscale flows are captured by a
numerical scheme that is underpinned by fluctuating hydrodynamics. A stochastic
lubrication equation (SLE) is solved on non-uniform adaptive grids to study a
series of nanoscale thin-film flows. The Fornberg scheme is used for
high-resolution spatial discretisation and a fully-implicit time-marching
scheme is designed for numerical stability. The accuracy of the numerical
method is verified against theoretical results for thermal capillary waves
during the linear stage of their development. The framework is then used to
study the nonlinear behaviour of three bounded thin-film flows: (i) droplet
spreading, where new power laws are derived; (ii) droplet coalescence, where
molecular dynamics results are reproduced by the SLE at a fraction of the
computational cost and it is discovered that thermal fluctuations decelerate
the process, in contrast to previously investigated phenomena; and (iii)
thin-film rupture, where, in the regime considered, disjoining pressure
dominates the final stages of rupture.
