  Microscopic processes on surfaces such as adsorption, desorption, diffusion
and reaction of interacting particles can be simulated using kinetic Monte
Carlo (kMC) algorithms. Even though kMC methods are accurate, they are
computationally expensive for large-scale systems. Hence approximation
algorithms are necessary for simulating experimentally observed properties and
morphologies. One such approximation method stems from the coarse graining of
the lattice which leads to coarse-grained Monte Carlo (GCMC) methods while
Langevin approximations can further accelerate the simulations. Moreover,
sacrificing fine scale (i.e. microscopic) accuracy, mesoscopic deterministic or
stochastic partial differential equations (SPDEs) are efficiently applied for
simulating surface processes. In this paper, we are interested in simulating
surface diffusion for pattern formation applications which is achieved by
suitably discretizing the mesoscopic SPDE in space. The proposed discretization
schemes which are actually Langevin-type approximation models are strongly
connected with the properties of the underlying interacting particle system. In
this direction, the key feature of our schemes is that controlled-error
estimates are provided at three distinct time-scales. Indeed, (a) weak error
analysis of mesoscopic observables, (b) asymptotic equivalence of action
functionals and (c) satisfaction of detailed balance condition, control the
error at finite times, long times and infinite times, respectively. In this
sense, the proposed algorithms provide a "bridge" between continuum (S)PDE
models and molecular simulations Numerical simulations, which also take
advantage of acceleration ideas from (S)PDE numerical solutions, validate the
theoretical findings and provide insights to the experimentally observed
pattern formation through self-assembly.
