Commit bb5dac00 authored by Philippe Helluy's avatar Philippe Helluy

up

parent f3781389
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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
......@@ -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
......
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