Commit 278a84ec authored by Philippe Helluy's avatar Philippe Helluy

start review 2

parent 054eb4c4
......@@ -225,8 +225,217 @@ Answers to reviewer 2 (most of the corresponding changes are highlighted
in red)
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\begin_inset Quotes eld
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This connection, and the existence of other previous work on lattice Boltzmann
schemes for MHD (e.g.
Martı́nez et al.
1994) could be made clearer in the manuscript
\begin_inset Quotes erd
\end_inset
:
\begin_inset Newline newline
\end_inset
We have commented additional bibliography.
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\begin_inset Quotes eld
\end_inset
I am, however, rather surprised that this scheme is stable.
\begin_inset Quotes erd
\end_inset
\begin_inset Newline newline
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\begin_inset Quotes eld
\end_inset
There are many examples in the LBM literature for finite Mach number flows
of schemes that become unstable at wavenumbers comparable to the grid spacing.
\begin_inset Quotes erd
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\begin_inset Newline newline
\end_inset
The vectorial kinetic scheme is indeed stable.
And there is no restriction related to the Mach number.
We give a reference and a comment on this point after formula (10).
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\begin_inset Quotes eld
\end_inset
However, ref.
[13] contains several differ- ent formulations, most of which include extra
source terms proportional to ∇·B and ∇ψ that cannot be written as divergences
of fluxes.
I think it would be worth clarifying why the authors chose the particular
variant from [13] that they did.
I suspect the authors chose the variant that can be written in conservation
form because it is the easiest to simulate in their kinetic framework.
\begin_inset Quotes erd
\end_inset
\begin_inset Newline newline
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This is indeed the case.
The vectorial kinetic theory applies only to hyperbolic conservative systems.
Among the divergence correction we therefore choose the one that is hyperbolic
and conservative.
We have commented this point.
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\begin_inset Quotes eld
\end_inset
It is not clear to me whether the authors have used this same approach for
the full MHD system, or whether they have used the collision terms in equations
(3) and (8) and accepted whatever form of dissipation these collisions
produce.
\begin_inset Quotes erd
\end_inset
\begin_inset Newline newline
\end_inset
In several space dimensions, we were not able to extend the analysis of
the dissipation.
Therefore, we accepted whatever form of dissipation the relaxation produces.
This was shortly said at the end of Section 3.4:
\begin_inset Quotes eld
\end_inset
In the following, for simplicity reasons, we only consider scalar relaxation.
\begin_inset Quotes erd
\end_inset
.
We hope however that the 1D analysis helps the reader to understand the
effect of the over-relaxation parameter.
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\begin_inset Quotes eld
\end_inset
the long un-numbered equation on page 8
\begin_inset Quotes erd
\end_inset
:
\begin_inset Newline newline
\end_inset
We have given a number to equation (10), which indeed deserves one.
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\begin_layout Enumerate
\begin_inset Quotes eld
\end_inset
I think the diffusive term on the right hand side of the long un-numbered
equation on page 8 is essentially a result from Chen et al.
(1994)
\begin_inset Quotes erd
\end_inset
\begin_inset Newline newline
\end_inset
Formula (10) expresses a more general result that Chen-Levermore.
But it is related and we have made a comment on this point in Remark 2,
page 9.
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\begin_layout Enumerate
\begin_inset Quotes eld
\end_inset
It would be useful to give the reader some measure of absolute performance
as well as, or instead, of the relative performance figures in table 2.
It is common to report absolute performance using numbers of millions of
lattice updates per second (MLUPS) in the lattice Boltzmann community.
A good way to evaluate computational efficiency is then to convert the
MLUPS into a useful memory bandwidth, assuming each degree of freedom per
lattice point is loaded and stored once, then compare this useful bandwidth
with the theoretical maximum bandwidth provided by the hardware
\begin_inset Quotes erd
\end_inset
\end_layout
\begin_layout Enumerate
\begin_inset Quotes eld
\end_inset
I do not think the output of nvtop supports the interpretation that the
authors ascribe to it on page 19.
\begin_inset Quotes erd
\end_inset
\begin_inset Newline newline
\end_inset
We agree and we removed this naive remark.
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\begin_inset Quotes eld
\end_inset
I would like the authors to give a more detailed explanation of how they
implemented the streaming step in equation (5).
They describe their memory layout in detail, but there are two further
issues that need addressing.
Firstly, the shift from i to i ± 1 leads to coalesced but misaligned loads
or stores (...)
\begin_inset Quotes erd
\end_inset
\end_layout
\begin_layout Enumerate
If my arithmetic is correct, the largest 7000 2 simulation uses more than
10 GB of memory for one set of fields in float64 format (...)
\begin_inset Quotes erd
\end_inset
\begin_inset Newline newline
\end_inset
The largest computations were done on an NVIDIA Quadro P6000 with 24 GB
of memory.
The two sets of fields thus entered the GPU memory.
\end_layout
\begin_layout Enumerate
The authors should say precisely how they computed the discrete divergence
and discrete curl of the magnetic field to create these plots.
\end_layout
\begin_layout Standard
Shorter comments:
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\end_body
......
......@@ -1691,6 +1691,39 @@ MRREVIEWER = {Philip D. Loewen},
Volume = {179},
Year = {2002}}
@article{croisille1995numerical,
title={Numerical simulation of the MHD equations by a kinetic-type method},
author={Croisille, Jean-Pierre and Khanfir, Rabia and Chanteur, G{\'e}rard},
journal={Journal of scientific computing},
volume={10},
number={1},
pages={81--92},
year={1995},
publisher={Springer}
}
@article{martinez1994lattice,
title={Lattice boltzmann magnetohydrodynamics},
author={Mart{\'\i}nez, Daniel O and Chen, Shiyi and Matthaeus, William H},
journal={Physics of plasmas},
volume={1},
number={6},
pages={1850--1867},
year={1994},
publisher={American Institute of Physics}
}
@article{toth2000b,
title={The∇{\textperiodcentered} B= 0 constraint in shock-capturing magnetohydrodynamics codes},
author={T{\'o}th, G{\'a}bor},
journal={Journal of Computational Physics},
volume={161},
number={2},
pages={605--652},
year={2000},
publisher={Elsevier}
}
@inproceedings{deshpande1986kinetic,
Author = {Deshpande, SM},
Booktitle = {24th AIAA Aerospace Sciences Meeting},
......
......@@ -28,6 +28,7 @@
\usepackage{xcolor}
\newcommand{\revA}[1]{{\color{olive}#1}}
\newcommand{\revB}[1]{{\color{red}#1}}
\newcommand{\bu}{\mathbf{u}}
\newcommand{\bZero}{\mathbf{0}}
\newcommand{\bB}{\mathbf{B}}
......@@ -256,12 +257,13 @@
can have a non-vanishing divergence. The interest of this formulation
is for numerical approximations. Indeed, standard approximations of
the usual MHD equations suffer from drifting errors along time of
the divergence constraint \cite{powell94,dedner2002hyperbolic,barth2006role}.
the divergence constraint.\revB{ This has been observed by several authors \cite{powell94,dedner2002hyperbolic,barth2006role}. A review on this topic is
developed for instance in \cite{toth2000b}. }
Approximations of the MHD-DC generally have a much better behavior.
In this generalized model, the divergence errors propagate at the
wave speed $c_{h}$. The errors are then damped at the boundaries
of the computational domain.
\revB{In \cite{dedner2002hyperbolic} other divergence corrections are proposed. We choose a correction that leads to a conservative and hyperbolic model, in order to be able to apply the general theory of vectorial kinetic representations given in \cite{bouchut1999construction, aregba2000discrete}.}
Theoretically, the parameter $c_{h}$ can take any value. But in practice
it is generally chosen larger than all the wave speeds of the MHD
system (see \cite{dedner2002hyperbolic}).
......@@ -319,13 +321,14 @@
kinetic interpretations of the Navier-Stokes equations in which the
particles velocities can take a few number of given values. This makes
it possible to solve the kinetic model directly in an efficient way.
We refer to \cite{succi2001lattice} for a history of the LBM. Initially
devised for solving Navier-Stokes equations, the LBM has more recently
been extended to any systems of conservation laws. For more generality,
it is necessary to accept vectorial kinetic distribution functions
instead of only scalar ones \cite{bouchut1999construction,aregba2000discrete,dellar2002lattice,graille2014approximation}.
It is now possible to approximate any systems of conservation laws
with minimal lattice vectorial kinetic model. This approach is very
We refer to \cite{succi2001lattice} for a history of the LBM. \revB{ Initially
devised for solving incompressible Navier-Stokes equations, the LBM is based on a single scalar particle distribution function. More recently, it has
been extended to other systems of conservation laws. Extensions to MHD can be found in \cite{dellar2002lattice, croisille1995numerical, martinez1994lattice}. Dellar in \cite{dellar2002lattice} showed that it is not
possible to rely on a single scalar kinetic function for approximating the MHD equations. He proposes to represent the magnetic part of the equations with a vectorial kinetic formulation. The resulting hybrid kinetic model (scalar for the fluid part, vectorial for the magnetic part) is however limited to low-Mach number flows.
For addressing transsonic and supersonic flows the whole kinetic model has to be fully vectorial.
The general theory of lattice vectorial kinetic model is discussed in
\cite{bouchut1999construction, aregba2000discrete, graille2014approximation}. It is valid for any hyperbolic system of conservation laws and is no more limited to low-Mach number flows.}
This approach is very
fruitful and can be used on arbitrary unstructured meshes at any order
of approximation \cite{badwaik:hal-01451393,coulette2016palindromic}.
In addition, when the lattice velocities are aligned with the mesh,
......@@ -544,16 +547,23 @@ $
If the relaxation matrix satisfies $\v I<\vom<2\v I$ and if $\v f=\vf^{eq}$
at the initial time, then, up to second-order terms in $O(\Delta t^{2})$,
$\vw$ is a solution of the following system of conservation laws
\[
\begin{equation}\label{eq:equiveq}
\partial_{t}\vw+\partial_{x}\vF(\vw)=\lambda^{2}\Delta t\partial_{x}\left((\vom^{-1}-\frac{1}{2}\v I)(\v I-\frac{1}{\lambda^{2}}\vF'(\vw)^{2})\partial_{x}\vw\right)+O(\Delta t^{2}).
\]
\end{equation}
\end{theorem}
\begin{remark}
The proof is based on standard Taylor expansions. For the scalar case
$\vom=\omega\v I$, it can be found in \cite{courtes2020vectorial}.
$\vom=\omega\v I$, it can be found (for instance) in \cite{courtes2020vectorial}.
The approach is classical in the analysis of the Lattice Boltzmann
Method (LBM). See for instance \cite{dubois2008equivalent,otomo2017two}.
\end{remark}
\revB{
\begin{remark}
For
$\vom=\v I$ the proof, based on a Chapman-Enskog expansion, is also given in \cite{chen1994hyperbolic}.
\end{remark}}
$\quad$
The above analysis allows recovering formally the so-called sub-characteristic
......@@ -564,7 +574,9 @@ $
\mathbf{V}(\lambda,\vw)=\v I-\frac{1}{\lambda^{2}}\vF'(\vw)^{2}>0.\label{eq:num_visc}
\end{equation}
We also recover the fact that the scheme is second-order in time in
\revB{A fully rigorous mathematical proof of stability of the kinetic model is given by Bouchut in \cite{bouchut1999construction}, Section 3.2, pp. 140--142. Bouchut's proof is not based on asymptotic expansions but on fully non-linear entropy estimates. Let us emphasize that the vectorial kinetic construction ensures stability even when shock waves occur and is not limited by a low Mach assumption.}
From (\ref{eq:equiveq}) we formally observe that the scheme is second-order in time in
the over-relaxation case, when
\[
\vom=2\v I.
......
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