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The isothermal Navier-Stokes-Korteweg system is used to model dynamics of a compressible fluid exhibiting phase transitions between a liquid and a vapor phase in the presence of capillarity effects close to phase boundaries. Standard numerical discretizations are known to violate discrete versions of inherent energy inequalities, thus leading to spurious dynamics of computed solutions close to static equilibria (e.g., parasitic currents). In this work, we propose a time-implicit discretization of the problem, and use piecewise linear (or bilinear), globally continuous finite element spaces for both, velocity and density fields, and two regularizing terms where corresponding parameters tend to zero as the mesh-size h > 0 tends to zero. Solvability, non-negativity of computed densities, as well as conservation of mass, and a discrete energy law to control dynamics are shown. Computational experiments are provided to study interesting regimes of coefficients for viscosity and capillarity.
@article{M2AN_2013__47_2_401_0,
author = {Braack, Malte and Prohl, Andreas},
title = {Stable discretization of a diffuse interface model for liquid-vapor flows with surface tension},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {401--420},
publisher = {EDP-Sciences},
volume = {47},
number = {2},
year = {2013},
doi = {10.1051/m2an/2012032},
mrnumber = {3021692},
zbl = {1267.76019},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2012032/}
}
TY - JOUR AU - Braack, Malte AU - Prohl, Andreas TI - Stable discretization of a diffuse interface model for liquid-vapor flows with surface tension JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 401 EP - 420 VL - 47 IS - 2 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2012032/ DO - 10.1051/m2an/2012032 LA - en ID - M2AN_2013__47_2_401_0 ER -
%0 Journal Article %A Braack, Malte %A Prohl, Andreas %T Stable discretization of a diffuse interface model for liquid-vapor flows with surface tension %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 401-420 %V 47 %N 2 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2012032/ %R 10.1051/m2an/2012032 %G en %F M2AN_2013__47_2_401_0
Braack, Malte; Prohl, Andreas. Stable discretization of a diffuse interface model for liquid-vapor flows with surface tension. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 2, pp. 401-420. doi: 10.1051/m2an/2012032
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