Stable discretization of a diffuse interface model for liquid-vapor flows with surface tension
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 2, p. 401-420

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.

DOI : https://doi.org/10.1051/m2an/2012032
Classification:  65M60,  65M12,  76W05
Keywords: diffuse interface model, surface tension, structure preserving discretization, space-time discretization
@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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {2},
     year = {2013},
     pages = {401-420},
     doi = {10.1051/m2an/2012032},
     zbl = {1267.76019},
     mrnumber = {3021692},
     language = {en},
     url = {http://www.numdam.org/item/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 - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 2, pp. 401-420. doi : 10.1051/m2an/2012032. http://www.numdam.org/item/M2AN_2013__47_2_401_0/

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