Gradient flow formulation and second order numerical method for motion by mean curvature and contact line dynamics on rough surface

We study the dynamics of a droplet moving on an inclined rough surface in absence inertial and viscous stress effects. In this case, is purely geometric motion terms wetting domain capillary surface. Using single graph representation, we interpret as gradient flow Hilbert manifold. propose unconditionally stable first/second order numerical schemes to simulate droplet, which described using by mean curvature coupled with contact lines. The are based (i) explicit boundaries, decouple dynamic updates lines surface, (ii) semi-Lagrangian method grids (iii) predictor-corrector nonlinear elliptic solver upto second accuracy. For case quasi-static continuous spatial variable schemes, prove stability convergence schemes. To demonstrate accuracy long-time validation proposed several challenging computational examples - including breathing droplets, droplets inhomogeneous surfaces Kelvin pendant constructed compared exact solutions obtained desingularized differential-algebraic system equations (DAEs).