up

mhd/doc/review/answers.lyx

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+\begin_body
+
+\begin_layout Standard
+Answers to reviewer 1 (most of the changes are highlighted in green)
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Quotes eld
+\end_inset
+
+In my opinion the title is too long and not really focused.
+ I suggest to shorten it and rephrase it to be something like: A robust
+ and efficient solver based on kinetic schemes for MagnetoHydroDynamic (MHD)
+ equations”: 
+\begin_inset Newline newline
+\end_inset
+
+we have changed the title.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Quotes eld
+\end_inset
+
+In the abstract the author say that they obtain “precise” results.
+ I suggest to remove this word, and replace it with “accurate” (and specify
+ which is the accuracy)
+\begin_inset Quotes erd
+\end_inset
+
+: 
+\begin_inset Newline newline
+\end_inset
+
+we have changed the word and made the sentence more precise.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Quotes eld
+\end_inset
+
+Section 2 consists of only subsection 2.1.
+ Please write some text to introduce section 2.
+ Moreover, my suggestion is to include the section 3 as subsection 2.2 and
+ also to reduce the description of the kinetic representation.
+\begin_inset Quotes erd
+\end_inset
+
+:
+\begin_inset Newline newline
+\end_inset
+
+We have renumbered the sections.
+ We have made some additional comments at the beginning of section 2.
+ We have not reduced the presentation of the kinetic model, because it is
+ not so much usual.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Quotes eld
+\end_inset
+
+At the beginning of section 4 the description of the numerical discretization
+ can be put in the text, instead on displaying it in many lines.
+\begin_inset Quotes erd
+\end_inset
+
+:
+\begin_inset Newline newline
+\end_inset
+
+we have reformatted this part.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Quotes eld
+\end_inset
+
+I suggest to include in section 4 the discussion on the numerical viscosity
+ presented in section 5.
+\begin_inset Quotes erd
+\end_inset
+
+:
+\begin_inset Newline newline
+\end_inset
+
+We have done this.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Quotes eld
+\end_inset
+
+More generally, please try to shorten the presentation of the non-original
+ part (from section 2 to section 5 included), briefly introducing the schemes
+ referring to the literature.
+\begin_inset Quotes erd
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+We have compressed the presentation in less lines, but not too much so that
+ the presentation stays more or less self-contained and pedagogical.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Quotes eld
+\end_inset
+
+One of the authors deeply studied Riemann problems for MHD with Godunov
+ like schemes.
+ Could you include a comparison between the two approaches (advantages/drawbacks
+)?
+\begin_inset Quotes erd
+\end_inset
+
+:
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Other comments:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Quotes eld
+\end_inset
+
+Please write MagnetoHydroDynamic extensively at the beginning of the paper
+ and then replace it by the acronym.
+ The same applies to GPU (in the abstract is used the acronym, but never
+ introduced).
+\begin_inset Quotes erd
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+We have corrected the writing at several places.
+\end_layout
+
+\end_body
+\end_document

mhd/doc/spring_mhd_lbm.tex

@@ -25,8 +25,9 @@
 \usepackage{xspace}
 %\usepackage{amssymb}
 %\usepackage{wasysym}
+\usepackage{xcolor}
 
-
+\newcommand{\revA}[1]{{\color{olive}#1}}
 \newcommand{\bu}{\mathbf{u}}
 \newcommand{\bZero}{\mathbf{0}}
 \newcommand{\bB}{\mathbf{B}}
@@ -53,6 +54,8 @@
 %\newcommand{\Pmat}{\ensuremath{\text{P}}\xspace}
 %\newcommand{\uvec}{\ensuremath{\vec u}\xspace}
 \newcommand{\wvec}{\ensuremath{\text{w}}\xspace}
+%\newcommand{\rev1}{zob}
+%
 %\newcommand{\vvec}{\ensuremath{\text{v}}\xspace}
 %\newcommand{\n}{\ensuremath{\vec n}\xspace}
 %\newcommand{\x}{\ensuremath{\vec x}\xspace}
@@ -68,12 +71,14 @@
 %\providecommand{\remarkname}{Remark}
 %\providecommand{\theoremname}{Theorem}
 
+
+
 \begin{document}
 	
-	\title*{A kinetic method for solving the MHD equations. Application to the
-		computation of tilt instability on uniform fine meshes.}
+	\title*{\revA{A robust and efficient solver based on kinetic
+			schemes for Magnetohydrodynamics (MHD) equations.}}
 	
-	\titlerunning{Kinetic method for MHD} 
+	\titlerunning{Kinetic scheme for MHD} 
 	\authorrunning{Baty \textit{et al.}}
 	
 	\author{Name of Author\inst{1}\and
@@ -99,18 +104,18 @@
 	
 	\chapauthor{Philippe Helluy}
 	
-	\abstract{This paper is devoted to the simulation of MHD flows with complex
+	\abstract{This paper is devoted to the simulation of \revA{Magnetohydrohynamics} flows with complex
 		structures. This kind flows present instabilities that generate shock
 		waves. We propose a robust and precise numerical method based on the
 		Lattice Boltzmann methodology. We explain how to adjust the numerical
-		viscosity in order to obtain stable, precise results and reduced divergence
+		viscosity in order to obtain stable, \revA{accurate results in smooth or discontinuous parts of the flow} and reduced divergence
 		errors. This method can handle shock wave and is almost second order.
-		It is also very well adapted to GPU computing. We also give results
+		It is also very well adapted to GPU \revA{(Graphics Processing Unit)} computing. We also give results
 		for a tilt instability test case on very fine meshes. }
 	
 	\section{Introduction}
 	
-	The MagnetoHydroDynamic (MHD) system is a fundamental model used in
+	The MagnetoHydroDynamics (MHD) system is a fundamental model used in
 	many fields of physics: astrophysics, plasma physics, geophysics...
 	Indeed, the MHD model is commonly adopted as an excellent framework
 	for collisional plasma environments. The numerical approximation of
@@ -152,7 +157,7 @@
 	
 	In order to capture fine structures, it is necessary to consider very
 	fine meshes. We have programmed the algorithm in a very efficient
-	way in order to address recent GPUs (Graphic Purpose Units) or multicore
+	way in order to address recent GPUs (Graphic Processing Units) or multicore
 	CPUs. We describe the implementation, which relies on OpenCL and PyOpenCL,
 	and the memory optimizations used for reaching high performance. The
 	program allows performing full MHD simulations on grids as fine as
@@ -164,6 +169,8 @@
 	current sheets. 
 	
 	\section{Mathematical model}
+	\revA{The MagnetoHydroDynamic (MHD) system is a model used in
+	many fields of physics. It consists of an extension of the compressible Euler equations for taking into account magnetic effects. A difficulty is that the magnetic field has to satisfies a divergence free condition. This condition expresses that magnetic charges are not observed in Nature. Standard finite volume methods do not guarantee that the numerical magnetic field is divergence free. More annoying: the divergence errors generally grows with the simulation time, which leads to physically wrong results. For limiting the divergence errors, we adopt the divergence cleaning method described in \cite{dedner2002hyperbolic}.}
 	
 	\subsection{MHD equations with divergence cleaning\label{subsec:MHD}}
 	
@@ -300,7 +307,7 @@
 	an analysis of the numerical viscosity of the kinetic method in this
 	simplified framework.
 	
-	\section{Kinetic representation}
+	\subsection{Kinetic representation}
 	
 	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -398,12 +405,12 @@
 	This comes from the definition of the time derivative in the weak
 	sense. The computation of the jumps at times $t=i\Delta t$ are called
 	the collision steps.
-	
 	The rest of the time, $\v f_{k}$ is solution of the free transport
 	equation
-	\[
+	
+	\begin{equation}\label{eq:trans}
 	\partial_{t}\v f_{k}+\v v_{k}\cdot\nabla\v f_{k}=0.
-	\]
+	\end{equation}
 	The coupling indeed occurs only at times $t=i\Delta t$.
 	
 	Let us remark that when $\omega=1$ the collision step reads
@@ -417,32 +424,33 @@
 	of the original equations. See for instance \cite{dellar2013interpretation}).
 	See also below for a mathematical analysis in the one-dimensional
 	case.
+	\section{vide}
 	
 	\section{Numerical method}
 	
 	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 	
-	For solving (\ref{eq:kin}) numerically, we first construct a regular
+	\revA{For solving (\ref{eq:kin}) numerically, we first construct a structured
 	grid of the square 
-	\[
+	$
 	\mathcal{D}=]0,L[\times]0,L[.
-	\]
-	The space step 
-	\[
+	$
+	The space step is given by 
+$
 	\Delta x=\frac{L}{N}.
-	\]
-	Grid points 
+	$
+	and the  grid-points are then as follows
 	\[
 	\v x_{i,j}=\left(\begin{array}{c}
 	(i+\frac{1}{2})\Delta x\\
 	(j+\frac{1}{2})\Delta x
 	\end{array}\right),\quad i,j\in\frac{\mathbb{Z}}{N\mathbb{Z}}.
 	\]
-	We assume periodic conditions, therefore 
+	For a simpler presentation we can assume periodic boundary condition, which amounts to the following equivalences
 	\[
-	i+N=i,\quad j+N=j.
-	\]
+	i+N\equiv i,\quad j+N \equiv j.
+	\]}
 	We denote by $\v w_{i,j}^{n}$ and $\v f_{k,i,j}^{n}$ the approximation
 	of $\v w$ and $\v f_{k}$ at the grid points $\v x_{i,j}$ and time
 	$t_{n}=n\Delta t$ just after the collision step. The values of the
@@ -452,18 +460,18 @@
 	
 	\subsection{Transport solver}
 	
-	The characteristic method gives 
+	The transport equation (\ref{eq:trans}) admits an exact solution
 	\[
 	\v f_{k}(\v x,t+\Delta t)=\v f_{k}(\v x-\Delta t\v v_{k},t).
 	\]
-	We use the following notation for the transport operator
-	\[
-	\v f(\v x,t+\Delta t)=\v T(\Delta t)\v f(\v x,t).
-	\]
+%	We use the following notation for the transport operator
+%	\[
+%	\v f(\v x,t+\Delta t)=\v T(\Delta t)\v f(\v x,t).
+%	\]
 	If we assume that the time step satisfies 
-	\[
+	$
 	\Delta t=\frac{\Delta x}{\lambda},
-	\]
+	$
 	the transport operator then reduces to a simple shift. Before the
 	collision step, we have thus 
 	\begin{equation}
@@ -478,18 +486,14 @@
 	\begin{equation}
 	\v w_{i,j}^{n+1}=\sum_{k=1}^{4}\v f_{k,i,j}^{n+1,-}.\label{eq:conservative}
 	\end{equation}
-	
 	The usual projection method would be to write 
 	\begin{equation}
 	\v f_{k,i,j}^{n+1}=\v f_{k}^{eq}(\v w_{i,j}^{n+1}).\label{eq:equilibrium}
 	\end{equation}
-	
 	As stated above, it is more precise to consider an over-relaxation
-	
 	\[
 	\v f_{k,i,j}^{n+1}=2\v f_{k}^{eq}(\v w_{i,j}^{n+1})-\v f_{k,i,j}^{n+1,-}.
 	\]
-	
 	It is possible to add a small dissipation $\tau>0.$ See \cite{graille2014approximation}
 	or \cite{coulette2016palindromic} for the relaxation scheme in the
 	case $\tau>0.$ This dissipation provides a better stability in numerical
@@ -513,7 +517,7 @@
 	refer to \cite{drui2019analysis}, for a specific analysis of the
 	boundary conditions in the over-relaxation scheme.
 	
-	\section{Analysis of the numerical viscosity in the one-dimensional case}
+	\revA{\subsection{Analysis of the numerical viscosity in the one-dimensional case}}
 	
 	In this section, we state some results about the numerical viscosity
 	of the kinetic relaxation scheme in the one-dimensional case. This