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In this work several time discretization techniques for capillary flows, i.e. either one-phase free
surface flow or two-phase flow, are compared. The focus is the development of methods that
are a) of higher order (i.e. at least second order), b) unconditionally stable and c) do not suffer
from much numerical dissipativity.
The mathematical model of the problem is initially described and a variational, dimensionless
formulation is given that can be used for both one- and two-phase capillary flows. The formula-
tion is in Arbitrary Lagrangian-Eulerian (ALE) coordinates so a problem-adapted moving mesh
can be used. Utilizing the Finite Element Method (FEM) the equations are discretized in space
and a differential algebraic matrix-vector formulation is derived.
Several fully implicit and linearly implicit time discretization techniques are introduced, their
properties with respect to stability, convergence, and dissipativity are discussed. These methods
are then applied to discretize the equations in time. Thus fully discrete equations are derived
that can be used for computer simulations.
Furthermore a space-time Galerkin approach is presented, its stability, convergence and
dissipativity properties are discussed as well. This method is then applied to the variational
formulation of the capillary flow problems and a fully discrete system is derived. For the
space-time approach an energy estimate is proved that establishes the unconditional stability
of the method by analytical means.
In an application chapter several one- and two-phase exemplary problems are introduced
and solved with the described time discretization techniques. Using Finite Element simulations
the convergence and dissipativity properties are numerically investigated.
A concluding comparison of the methods presents the advantages and drawbacks of all
methods that were compared.