Project Eberhard Bänsch and Stephan Weller

The fundamental problems in the numerical approximation of multiphase systems, or more generally speaking, in the treatment of flows with capillary free boundaries are the representation of the free surface, evaluating the curvature, handling of discontinuities, most importantly the pressure jump, and time discretization strategies for the decoupling of flow computation and geometry. While for the first three items there exists a vast literature and many techniques developed over the last two decades, the last problem of how to efficiently treat the time discretization has been widely ignored. The majority of the existing approaches just decouple the flow field from the geometry by a simple segregated approach, i.e. evaluating the geometric quantities from the previous time step. These strategy leads to a) a severe capillary CFL condition and b) is of first order in time at most. Existing semi-implicit discretizations exist that overcome problem a), but are still first order only and rather dissipative in certain situations. Thus this project aims at developing, implementing, and analysing time discretizations of higher order that are unconditionally stable and minimally dissipative, thus allowing for rather large time steps. To solve the arising systems, which will necessarily comprise a coupling between flow field and geometry, efficient solution techniques will be developed.