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| Phase field or diffuse interface models have became a popular
modelling tool in recent years to describe the complex dynamics of two-phase flow. We are in particular
interested in Navier-Stokes-Korteweg (NSK) models for compressible liquid-vapour phase transitions. The phase-field approach
usually introduces higher-order derivatives into the governing system of equations. The classical NSK model
contains e.g.~third-order derivatives which complicates a fast numerical simulation. Moreover, the first-order part of
the NSK model is of mixed hyperbolic-elliptic type which excludes modern Riemann-solver type methods. As a consequence
computations for convection-dominated regimes are practically impossible.
To avoid these difficulties we introduce a class of relaxed approaximations with purely hyperbolic first-order part.
Moreover the relaxed systems do not contain higher-order space derivatives.\\
In the limit of infinite Korteweg parameter the relaxation approximation coincides with the original NSK model.
We will present rigorous convergence results on the level of model problems. The talk concludes with multidimensional numerical
experiments. The experiments show the numerical efficiency of the relaxation approximation. We will show furthermore
that the relaxation approximation allows stable computations for convection-dominated regimes and in the sharp-interface limit.
Finally we will give an outlook how to generalize the relaxation technique to other instances of phase field models. This is joint
work with A.~Corli, J.~Neusser, and V.~Schleper. |
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