Thermodynamically consistent and positivity-preserving discretization of the thin-film equation with thermal noise

Abstract:

In micro-fluidics, both capillarity and thermal fluctuations play an important role. On the level of the lubrication approximation, this leads to a quasi-linear fourth-order parabolic equation for the film height $h$ driven by space-time white noise. The (formal) gradient flow structure of its deterministic counterpart, the so-called thin-film equation, which encodes the balance between driving capillary and limiting viscous forces, provides the guidance for the thermodynamically consistent introduction of fluctuations. We follow this route on the level of a spatial discretization of the gradient flow structure, i.e., on the level of a discretization of energy functional and dissipative metric tensor.

Starting from an energetically conformal finite-element (FE) discretization, we point out that the numerical mobility function introduced by Grün and Rumpf can be interpreted as a discretization of the metric tensor in the sense of a mixed FE method with lumping. While this discretization was devised in order to preserve the so-called entropy estimate, we use this to show that the resulting high-dimensional stochastic differential equation (SDE) preserves pathwise and pointwise strict positivity, at least in case of the physically relevant mobility function arising from the no-slip boundary condition.

As a consequence, and as opposed to previous discretizations of the thin-film equation with thermal noise, the above discretization is not in need of an artificial condition at the boundary of the configuration space orthant $\{h>0\}$ (which, admittedly, could also be avoided by modelling a disjoining pressure). Thus, this discretization gives rise to a consistent invariant measure, namely a discretization of the Brownian excursion (up to the volume constraint), and thus features an entropic repulsion. The price to pay over more direct discretizations is that when writing the SDE in Itô’s form, which is the basis for the Euler-Mayurama time discretization, a correction term appears.

We perform various numerical experiments to compare the behavior and performance of our discretization to that of a particular finite difference discretization of the equation. Among other things, we study numerically the invariance and entropic repulsion of the invariant measure and provide evidence for the fact that the finite difference discretization touches down almost surely while our discretization stays away from the $\partial \{h > 0\}$.