High-order ALE schemes for incompressible capillary flows

The spatial discretization of problems with moving boundaries is considered, incorporating the temporal evolution of not just the mechanical variables, but also of the nodal positions of the moving mesh. The outcome is a system of Differential-Algebraic Equations (DAE) of index 2 or, in the case of inertialess flow, just 1. From the DAE formulation it its possible to define strategies to build schemes of arbitrary accuracy. We introduce here several schemes of order 2 and 3 that avoid the solution of a nonlinear system involving simultaneously the mechanical variables and the geometrical ones. This class of schemes has been the one adopted by the majority of practitioners of Computational Fluid Dynamics up to now. The proposed schemes indeed achieve the design accuracy, and further show stability restrictions that are not significantly more severe than those of popular first order schemes. The numerical experimentation is performed on capillary problems, discretized by both div-stable (P2/P1, P1+/P1) and equal-order (P1/P1, stabilized) finite elements, and incorporating surface tension and triple (contact) line effects.