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Energy consistent discontinuous Galerkin methods for the Navier–Stokes–Korteweg system

Authors: Jan Giesselmann, Charalambos Makridakis and Tristan Pryer
Journal: Math. Comp. 83 (2014), 2071-2099
MSC (2010): Primary 65M60; Secondary 76T10
Published electronically: January 8, 2014
MathSciNet review: 3223325
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Abstract: We design consistent discontinuous Galerkin finite element schemes for the approximation of the Euler–Korteweg and the Navier–Stokes–Korteweg systems. We show that the scheme for the Euler–Korteweg system is energy and mass conservative and that the scheme for the Navier–Stokes–Korteweg system is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to viscous effects, that is, there is no numerical dissipation. In this sense the methods are consistent with the energy dissipation of the continuous PDE systems.

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Additional Information

Jan Giesselmann
Affiliation: Weierstrass Institute, Mohrenstrasse 39, D-10117 Berlin, Germany
Address at time of publication: University of Stuttgart, Institute of Applied Analysis and Numerical Simulation, Pfaffenwaldring 57, 70569 Stuttgart, Germany

Charalambos Makridakis
Affiliation: Department of Applied Mathematics, University of Crete, GR-71409 Heraklion, Greece – and – Institute for Applied and Computational Mathematics, Foundation for Research and Technology-Hellas, Vasilika Vouton P.O. Box 1527, GR-71110 Heraklion, Greece
Address at time of publication: Department of Mathematics, University of Sussex, Falmer Campus, Brighton BN1 9QH, United Kingdom
MR Author ID: 289627

Tristan Pryer
Affiliation: School of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury, GB-CT2 7NF, England United Kingdom
Address at time of publication: Department of Mathematics and Statistics, University of Reading, Whiteknights, P.O. Box 220, Reading RG6 6AX, United Kingdom

Received by editor(s): July 20, 2012
Received by editor(s) in revised form: December 22, 2012, and January 17, 2013
Published electronically: January 8, 2014
Additional Notes: The authors were supported by the the FP7-REGPOT project “ACMAC–Archimedes Center for Modeling, Analysis and Computations” of the University of Crete (FP7-REGPOT-2009-1-245749).
The third author was also partially supported by the EPSRC grant EP/H024018/1
The authors would like to express their gratitude to the two anomymous referees for their constructive suggestions to improve this work.
Article copyright: © Copyright 2014 American Mathematical Society