@Article{CiCP-14-1252,
author = {Praveen Chandrashekar},
title = {Kinetic Energy Preserving and Entropy Stable Finite Volume Schemes for Compressible Euler and Navier-Stokes Equations},
journal = {Communications in Computational Physics},
year = {2013},
volume = {14},
number = {5},
pages = {1252--1286},
abstract = {<p style="text-align: justify;">Centered numerical fluxes can be constructed for compressible Euler equations
which preserve kinetic energy in the semi-discrete finite volume scheme. The essential
feature is that the momentum flux should be of the form <img src="/uploads/ueditor/image/20200730/1596076603345723.png" title="1596076603345723.png" alt="image.png" width="181" height="37"/>&nbsp;where <img src="/uploads/ueditor/image/20200730/1596076562307740.png" title="1596076562307740.png" alt="image.png" width="300" height="33"/>&nbsp;are any consistent approximations to the
pressure and the mass flux. This scheme thus leaves most terms in the numerical
flux unspecified and various authors have used simple averaging. Here we enforce
approximate or exact entropy consistency which leads to a unique choice of all the
terms in the numerical fluxes. As a consequence, a novel entropy conservative flux that
also preserves kinetic energy for the semi-discrete finite volume scheme has been proposed.
These fluxes are centered and some dissipation has to be added if shocks are
present or if the mesh is coarse. We construct scalar artificial dissipation terms which
are kinetic energy stable and satisfy approximate/exact entropy condition. Secondly,
we use entropy-variable based matrix dissipation flux which leads to kinetic energy
and entropy stable schemes. These schemes are shown to be free of entropy violating
solutions unlike the original Roe scheme. For hypersonic flows a blended scheme is
proposed which gives carbuncle free solutions for blunt body flows. Numerical results
for Euler and Navier-Stokes equations are presented to demonstrate the performance
of the different schemes.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.170712.010313a},
url = {http://global-sci.org/intro/article_detail/cicp/7201.html}
}