Time integration and steady-state continuation for 2d lubrication equations

TL;DR

This paper introduces numerical algorithms for analyzing 2D lubrication equations, enabling detailed study of thin film dynamics and bifurcations through time integration and steady-state continuation techniques.

Contribution

The paper develops and applies novel exponential propagation and continuation algorithms with Cayley transforms for 2D lubrication equations, improving stability and efficiency.

Findings

Algorithms reliably analyze steady states and bifurcations.

Effective in 2D and 3D lubrication problems.

Applicable to dewetting and drop depinning scenarios.

Abstract

Lubrication equations allow to describe many structurin processes of thin liquid films. We develop and apply numerical tools suitable for their analysis employing a dynamical systems approach. In particular, we present a time integration algorithm based on exponential propagation and an algorithm for steady-state continuation. In both algorithms a Cayley transform is employed to overcome numerical problems resulting from scale separation in space and time. An adaptive time-step allows to study the dynamics close to hetero- or homoclinic connections. The developed framework is employed on the one hand to analyse different phases of the dewetting of a liquid film on a horizontal homogeneous substrate. On the other hand, we consider the depinning of drops pinned by a wettability defect. Time-stepping and path-following are used in both cases to analyse steady-state solutions and their bifurcations as well as dynamic processes on short and long time-scales. Both examples are treated for two- and three-dimensional physical settings and prove that the developed algorithms are reliable and efficient for 1d and 2d lubrication equations, respectively.