Spreading fronts: numerical simulations on discrete models and continuous equations

Out-of-equilibrium systems, inherently complex and challenging to understand, are prevalent across various disciplines, including physics where they arise in contexts such as fluid dynamics. In particular, critical out-of-equilibrium systems combine this complexity with the scaling laws and universality classes observed in critical phenomena, with kinetic surface roughening, the study of how a flat surface becomes progressively rougher over time, serving as a prime example. This behavior manifests in a wide variety of contexts, including metal corrosion, cell proliferation, and, notably, the growth of thin films, which can emerge as a result of wetting processes. In this thesis, we conduct extensive numerical simulations to study critical fluctuations and identify universal features of several rough interfaces, generated by simulating discrete models of thin film growth and by performing direct numerical integration of continuum equations. To explore the universal behavior of these interfaces, we identify the critical exponents that characterize the spatio-temporal fluctuations of the front. Additionally, we analyze the dynamics of thin films across different physical scenarios to deepen our understanding of their behavior in out-of-equilibrium conditions, especially in the case where these films are formed by the action of an external force such as Surface Acoustic Waves.