Energy consistent discontinuous Galerkin methods for a quasi-incompressible diffuse two phase flow model∗
1 University of Stuttgart, Institute for Applied Analysis and Numerical Simulation, Pfaffenwaldring 57, 70569 Stuttgart, Germany
2 Tristan Pryer, Department of Mathematics and Statistics, Whiteknights, PO Box 220, Reading RG6 6AX, UK
Received: 31 July 2013
Revised: 24 April 2014
We design consistent discontinuous Galerkin finite element schemes for the approximation of a quasi-incompressible two phase flow model of Allen–Cahn/Cahn–Hilliard/Navier–Stokes–Korteweg type which allows for phase transitions. We show that the scheme is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to discrete equivalents of those effects already causing dissipation on the continuous level, that is, there is no artificial numerical dissipation added into the scheme. In this sense the methods are consistent with the energy dissipation of the continuous PDE system.
Mathematics Subject Classification: 65M12 / 65M60 / 76T99 / 76D45
Key words: Quasi-incompressibility / Allen–Cahn / Cahn–Hilliard / Navier–Stokes–Korteweg / phase transition / energy consistent/mimetic / discontinuous Galerkin finite element method
T.P. was supported by the EPSRC grant EP/H024018/1. J.G. was supported by the German Research Foundation (DFG) project “Modeling and sharp interface limits of local and non-local generalized Navier–Stokes–Korteweg Systems” and by the EU FP7-REGPOT project “Archimedes Center for Modeling, Analysis and Computation”.
© EDP Sciences, SMAI, 2015