review guermond popov paper on general euler regularization

references.bib

@@ -693,3 +693,21 @@
   journal={arXiv preprint arXiv:1706.06673},
   year={2017}
 }
+
+@article{guermond2014viscous,
+  title={Viscous regularization of the Euler equations and entropy principles},
+  author={Guermond, Jean-Luc and Popov, Bojan},
+  journal={SIAM Journal on Applied Mathematics},
+  volume={74},
+  number={2},
+  pages={284--305},
+  year={2014},
+  publisher={SIAM}
+}
+
+@article{serre153viscous,
+  title={Viscous system of conservation laws: singular limits. Nonlinear conservation laws and applications, 433-445},
+  author={Serre, D},
+  journal={IMA Vol. Math. Appl},
+  volume={153}
+}

tex/grhd.tex

@@ -3827,6 +3827,54 @@ with gravity}
 
 
 
+\subsection{Literature review artificial viscosity methods}
+
+
+\paragraph{\cite{guermond2014viscous}: Viscouse regularization of the Euler equations and entropy principles}
+
+\begin{itemize}
+  \item Intro: Positivity of density and internal energy and proving a minium principle for the specific
+    entropy for numerical solutions for the compressible Euler equations is difficult,
+    in particular for arbitrary meshes and dimensions greater than one. Note that
+    they do not assume positivity of pressure.
+    The Godunov scheme and variants of the Lax scheme can do this, but these cannot be readily
+    generalized to high-order Galerkin approximations.
+    A way around this parabolic regularization. However, this has the problem that such
+    a regularization acts on conserved variables, some of which are not Galilean invariant
+    and, thus, a change of reference frame changes the regularization.
+    A way around that is using the NS regularization, but this comes with the problem that
+    the NS equations do not contain a regularization in the continuity equation.
+    Furthermore, NS equations violate the minimum entropy principle when the thermal
+    diffusivity is non-zero.
+    The motivation of this paper is the question of whether one can find a better regularization
+    than that of the NS equations.
+  \item Monolithic parabolic regularization: This is the scheme I impelemented first.
+    All equations are augmented with a spatial second derivative of the corresponding
+    conserved variable, together with a small viscosity parameter.
+    It can be shown that the Lax-Friedrich scheme (I think this is talking about FV methods)
+    and the parabolic equivalent are approximations of this kind of regularization,
+    which provides a particular value for the viscosity basec on the CFL number, local cell
+    width and the speed the solutiona and that of sound.
+    Physicists critize this regularization as it violates Galilean and rotational invariance.
+  \item Navier-Stokes regularization: This regularization has two major shortcomings.
+    1) it does not admit a minimum principle of entropy. 2) It does not reguarlize
+    the continuity equation. The later has the problem that contact waves are difficult to
+    resolve with this regularization.
+    \todo{Look up again how contact waves, rarefaction waves etc are defined.}
+  \item General regularization: They write down the most general parabolic regularization
+    and the construct the corresponding fluxes such that the equations provide
+    positive density, a minimum principle for the specific entropy and that the
+    regularization is compatible with a large class of entropies.
+    They make reference to a paper that develops a theory of viscous systems of
+    conservation laws which establishes short-time existence results, however,
+    that paper appears to contain lots of technical details which.
+  \item The conclusion lists the final regularization.
+  \item They show a simple experiment with a discontinuous initial data where the new
+    regularization does not admit overshoots, whereas the NS regularization does.
+    Note that the result is still a viscous wave, hence, no discontinuity.
+\end{itemize}
+
+
 \printbibliography
 
 \end{document}