Time Integration and Steady-state Continuation Method for Lubrication Equations

The partial di erential equations describing the evolution of thin liquid lms on solid substrates in lubrication approximation are highly non-linear. Thus, the classical semi-implicit time integrator with a constant linear part fails due to a strong variation of the Jacobian between two consecutive time-steps. We propose to integrate the equations using an exponential propagation. This method requires to evaluate the rightmost eigenvalues of the Jacobian Matrix. Because the discretized system is large and sparse, we adapt di erent classical iterative methods (Chebyschev acceleration, shift-invert Cayley transform). We show that the Cayley transform is the most powerful method. Furthermore, the rightmost spectrum determined in such a way can also be employed in a continuation technique that allows to follow steady state solutions in parameter space. In this way time stepping and bifurcation analysis can be coupled. The resulting common numerical framework is exempli ed using (i) dewetting on a horizontal homogeneous substrate, and (ii) the depinning of pinned drops. Both examples are treated in twoand three-dimensional settings.