Commit 976dc75a authored by Philippe Helluy's avatar Philippe Helluy

last modifs

parent 307e3d4d
......@@ -457,11 +457,18 @@ Shorter comments:
\end_layout
\begin_layout Enumerate
\begin_inset Quotes eld
\end_inset
Following the spirit of the first few lines of the introduction, I think
it would be worth adding an observation that the MHD system is not strictly
hyperbolic.
This is another reason why MHD supports phenomena not seen in pure hydrodynamic
s.
\begin_inset Quotes erd
\end_inset
\begin_inset Newline newline
\end_inset
......@@ -469,6 +476,9 @@ We add a short comment about the hyperbolicity of the MHD-DC system.
\end_layout
\begin_layout Enumerate
\begin_inset Quotes eld
\end_inset
The comment on page 19 about using dimensionless MHD equations with the
magnetic permeability set to unity belongs with the first appearance of
the dimensionless MHD equations on page 2.
......@@ -476,6 +486,10 @@ The comment on page 19 about using dimensionless MHD equations with the
, so I wonder why the times are quoted as “t = 0.3s” in the captions for
figures 3, 4, 5, 13, 14 and 15.
Is “s” intended to be “seconds”, and if so, why?
\begin_inset Quotes erd
\end_inset
\begin_inset Newline newline
\end_inset
......@@ -483,7 +497,217 @@ We removed the time dimension.
\end_layout
\begin_layout Enumerate
\begin_inset Quotes eld
\end_inset
It is not clear to me whether the authors’ implementation allows nonzero
values of the u 3 and B 3 vector components normal to the plane.
If I understand correctly, all the numerical experiments use u and B vectors
that are confined to the xy plane.
\begin_inset Quotes erd
\end_inset
\begin_inset Newline newline
\end_inset
Our implementation accepts non-vanishing
\begin_inset Formula $B_{3}$
\end_inset
component.
However it is true that we do not use this feature in the tests.
\end_layout
\begin_layout Enumerate
When describing properties of GPUs (middle of page 11, top of page 12) I
think it would be better to use one of the contemporary GPUs whose properties
are listed in Table 1 as an example, instead of the Nvidia GTX 470 from
2010.
It also looks odd to quote the memories as 47GB and 63GB for the AMD and
Intel systems in Table 1.
I imagine these are the memories available for user processes while running
an operating system, but the figures for the GPUs are the total memories
installed.
\begin_inset Newline newline
\end_inset
We have fixed all these glitches.
\end_layout
\begin_layout Enumerate
Something has gone wrong with the subsection numbering in section 7.
The “Tilt instability” should be subsection 7.3.
There is nothing in the text for subsection 7.2 on the Orszag–Tang vortex
about the data plotted in figures 3, 4, 5.
\begin_inset Newline newline
\end_inset
Fixed.
We add a short comment about figures 3,4,5.
\end_layout
\begin_layout Enumerate
Page 1 onwards, I would use “divergence-free” throughout, or “free of divergence
”.
GPU is conventionally “Graphics Processing Unit” not “Graphic Purpose Unit”
(pages 2 and 11).
\begin_inset Newline newline
\end_inset
fixed
\end_layout
\begin_layout Enumerate
“note” should be “denote” when introducing pieces of notation, as in the
last line of page 2 and elsewhere.
\begin_inset Newline newline
\end_inset
fixed.
\end_layout
\begin_layout Enumerate
“resolution” would be much clearer as “implementation” for “resolution of
the transport step” on page 2 and “numerical resolu- tion” would be better
as “numerical implementation” on page 23.
\begin_inset Newline newline
\end_inset
fixed
\end_layout
\begin_layout Enumerate
Page 3, “errors along time of the divegence constraint” would be better
as “the accumulation of errors over time in the divergence- free constraint.”
These errors will not be damped at the boundaries when the boundary conditions
are periodic, as they are for most of the numerical experiments.
\begin_inset Newline newline
\end_inset
That's true, the damping comes from the numerical viscosity of the scheme.
\end_layout
\begin_layout Enumerate
Page 6, elsewhere in the text the interval is written [0, L] rather than
]0, L[ with the brackets the other way round.
I think Z/N Z is rather cryptic (especially as the group quotient is typeset
as an upright fraction) so I would write i, j ∈ {0, .
.
.
, N − 1}.
\begin_inset Newline newline
\end_inset
fixed
\end_layout
\begin_layout Enumerate
Page 9, a factor of ρ 4 cancels in all the displayed fractions for the matrix
elements a i,j .
\begin_inset Newline newline
\end_inset
Yes, fixed.
\end_layout
\begin_layout Enumerate
Page 10, second line, “sheme” should be “scheme”.
\begin_inset Newline newline
\end_inset
fixed
\end_layout
\begin_layout Enumerate
Page 14, second line of §7.1 should be “allows us to .
.
.
”.
\begin_inset Newline newline
\end_inset
fixed.
\end_layout
\begin_layout Enumerate
Page 19, paragraph 1, I would use “conserved variable” instead of “macro-variabl
e” for ψ.
This matches the terminology on page 3, though there “conservative variables”
should be “conserved variables”.
\begin_inset Newline newline
\end_inset
We replace
\begin_inset Quotes eld
\end_inset
macro-variable
\begin_inset Quotes erd
\end_inset
by
\begin_inset Quotes eld
\end_inset
conservative variable
\begin_inset Quotes erd
\end_inset
.
We do not replace
\begin_inset Quotes eld
\end_inset
conservative
\begin_inset Quotes erd
\end_inset
by
\begin_inset Quotes eld
\end_inset
conserved
\begin_inset Quotes erd
\end_inset
, because this the traditional denomination in the hyperbolic system community.
\end_layout
\begin_layout Enumerate
Page 22, the first paragraph ends with “In”.
Is some text missing?
\begin_inset Newline newline
\end_inset
fixed
\end_layout
\begin_layout Enumerate
Page 23, the last word before the displayed equation should be “setting”.
\begin_inset Newline newline
\end_inset
fixed
\end_layout
\begin_layout Enumerate
Are references [2] and [3] the same document, the latter being a pre-print
of the former? If so, is it necessary to cite both versions?
\begin_inset Newline newline
\end_inset
Yes, a shameful attempt to count a single article twice.
Fixed.
\end_layout
\begin_layout Enumerate
Many of the article titles in the bibliography are incorrectly capitalised:
“galerkin”, “maxwell”, “boltzmann”, “strang” etc.
\begin_inset Newline newline
\end_inset
Fixed.
\end_layout
\end_body
......
% Encoding: UTF-8
@article{bramas2020optimization,
title={Optimization of a discontinuous Galerkin solver with OpenCL and StarPU},
title={Optimization of a discontinuous {G}alerkin solver with {OpenCL} and {StarPU}},
author={Bramas, B{\'e}renger and Helluy, Philippe and Mendoza, Laura and Weber, Bruno},
journal={International Journal on Finite Volumes},
volume={15},
......@@ -11,7 +11,7 @@
}
@article{bouchut2010multiwave,
title={A multiwave approximate Riemann solver for ideal MHD based on relaxation II: numerical implementation with 3 and 5 waves},
title={A multiwave approximate Riemann solver for ideal {MHD} based on relaxation II: numerical implementation with 3 and 5 waves},
author={Bouchut, Fran{\c{c}}ois and Klingenberg, Christian and Waagan, Knut},
journal={Numerische Mathematik},
volume={115},
......@@ -22,7 +22,7 @@
}
@article{klockner2012pycuda,
title={PyCUDA and PyOpenCL: A scripting-based approach to GPU run-time code generation},
title={{PyCUDA} and {PyOpenCL}: A scripting-based approach to GPU run-time code generation},
author={Kl{\"o}ckner, Andreas and Pinto, Nicolas and Lee, Yunsup and Catanzaro, Bryan and Ivanov, Paul and Fasih, Ahmed},
journal={Parallel Computing},
volume={38},
......@@ -33,7 +33,7 @@
}
@book{gaster2012heterogeneous,
title={Heterogeneous computing with openCL: revised openCL 1.},
title={Heterogeneous computing with {OpenCL}: revised {OpenCL} 1.},
author={Gaster, Benedict and Howes, Lee and Kaeli, David R and Mistry, Perhaad and Schaa, Dana},
year={2012},
publisher={Newnes}
......@@ -50,7 +50,7 @@
}
@article{baty2019finmhd,
title={FINMHD: An Adaptive Finite-element Code for Magnetic Reconnection and Formation of Plasmoid Chains in Magnetohydrodynamics},
title={{FINMHD}: An Adaptive Finite-element Code for Magnetic Reconnection and Formation of Plasmoid Chains in Magnetohydrodynamics},
author={Baty, Hubert},
journal={The Astrophysical Journal Supplement Series},
volume={243},
......@@ -73,7 +73,7 @@
@article{otomo2017two,
title={Two complementary lattice-Boltzmann-based analyses for nonlinear systems},
title={Two complementary lattice-{B}oltzmann-based analyses for nonlinear systems},
author={Otomo, Hiroshi and Boghosian, Bruce M and Dubois, Fran{\c{c}}ois},
journal={Physica A: Statistical Mechanics and its Applications},
volume={486},
......@@ -88,7 +88,7 @@
Number = {7},
Pages = {1441--1449},
Publisher = {Elsevier},
Title = {Equivalent partial differential equations of a lattice Boltzmann scheme},
Title = {Equivalent partial differential equations of a lattice {B}oltzmann scheme},
Volume = {55},
Year = {2008}}
......@@ -1334,7 +1334,7 @@ MRREVIEWER = {Alastair M. Rucklidge},
}
@article {gallice97,
AUTHOR = {Cargo, P. and Gallice, G.},
TITLE = {Roe matrices for ideal {MHD} and systematic construction of
TITLE = {{Roe} matrices for ideal {MHD} and systematic construction of
{R}oe matrices for systems of conservation laws},
JOURNAL = {J. Comput. Phys.},
FJOURNAL = {Journal of Computational Physics},
......@@ -1587,7 +1587,7 @@ MRREVIEWER = {Philip D. Loewen},
Date-Modified = {2019-02-01 08:48:20 +0000},
Journal = {Physics of Fluids B: Plasma Physics},
Number = {11},
Title = {Evolution of the Orszag-Tang vortex system in a compressible medium. I. Initial average subsonic flow},
Title = {Evolution of the {O}rszag-{T}ang vortex system in a compressible medium. I. Initial average subsonic flow},
Volume = {1},
Year = {1989}}
......@@ -1597,7 +1597,7 @@ MRREVIEWER = {Philip D. Loewen},
Date-Modified = {2019-02-01 08:46:20 +0000},
Journal = {Physics of Fluids B: Plasma Physics},
Number = {29},
Title = {Evolution of the Orszag-Tang vortex system in a compressible medium. II. Supersonic flow},
Title = {Evolution of the {O}rszag-{T}ang vortex system in a compressible medium. II. Supersonic flow},
Volume = {3},
Year = {1991}}
......@@ -1611,7 +1611,7 @@ MRREVIEWER = {Philip D. Loewen},
Journal = {Journal of Computational Physics},
Keywords = {Arbitrary high-order discontinuous Galerkin schemes, sub-cell finite volume limiter, Unstructured triangular and tetrahedral meshes, Conservation laws and hyperbolic PDE with non-conservative products, Element-wise checkpointing and restarting, MOOD paradigm},
Pages = {163 - 199},
Title = {A simple robust and accurate a posteriori sub-cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes},
Title = {A simple robust and accurate a posteriori sub-cell finite volume limiter for the {D}iscontinuous {G}alerkin method on unstructured meshes},
Url = {http://www.sciencedirect.com/science/article/pii/S0021999116301292},
Volume = {319},
Year = {2016},
......@@ -1692,7 +1692,7 @@ MRREVIEWER = {Philip D. Loewen},
Year = {2002}}
@article{croisille1995numerical,
title={Numerical simulation of the MHD equations by a kinetic-type method},
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},
......@@ -1703,7 +1703,7 @@ MRREVIEWER = {Philip D. Loewen},
}
@article{martinez1994lattice,
title={Lattice boltzmann magnetohydrodynamics},
title={Lattice {B}oltzmann magnetohydrodynamics},
author={Mart{\'\i}nez, Daniel O and Chen, Shiyi and Matthaeus, William H},
journal={Physics of plasmas},
volume={1},
......@@ -1768,7 +1768,7 @@ MRREVIEWER = {Philip D. Loewen},
Number = {24},
Pages = {10108--10124},
Publisher = {Elsevier},
Title = {Fast computation of multiphase flow in porous media by implicit discontinuous Galerkin schemes with optimal ordering of elements},
Title = {Fast computation of multiphase flow in porous media by implicit {D}iscontinuous {G}alerkin schemes with optimal ordering of elements},
Volume = {227},
Year = {2008}}
......@@ -1829,7 +1829,7 @@ MRREVIEWER = {Philip D. Loewen},
Booktitle = {Compatible spatial discretizations},
Pages = {69--88},
Publisher = {Springer},
Title = {On the role of involutions in the discontinuous Galerkin discretization of Maxwell and magnetohydrodynamic systems},
Title = {On the role of involutions in the {D}iscontinuous {G}alerkin discretization of {M}axwell and magnetohydrodynamic systems},
Year = {2006}}
@article{zingan2013implementation,
......@@ -1837,7 +1837,7 @@ MRREVIEWER = {Philip D. Loewen},
Journal = {Computer Methods in Applied Mechanics and Engineering},
Pages = {479--490},
Publisher = {Elsevier},
Title = {Implementation of the entropy viscosity method with the discontinuous Galerkin method},
Title = {Implementation of the entropy viscosity method with the {D}iscontinuous {G}alerkin method},
Volume = {253},
Year = {2013}}
......@@ -1892,7 +1892,7 @@ MRREVIEWER = {Philip D. Loewen},
Isbn = {978-3-319-57394-6},
Pages = {171--178},
Publisher = {Springer International Publishing},
Title = {Palindromic Discontinuous Galerkin Method},
Title = {Palindromic {D}iscontinuous {G}alerkin Method},
Url = {https://doi.org/10.1007/978-3-319-57394-6_19},
Year = {2017},
Bdsk-Url-1 = {https://doi.org/10.1007/978-3-319-57394-6_19},
......@@ -1904,7 +1904,7 @@ MRREVIEWER = {Philip D. Loewen},
Number = {1},
Pages = {201--205},
Publisher = {Elsevier},
Title = {Lattice Boltzmann formulation for Braginskii magnetohydrodynamics},
Title = {Lattice {B}oltzmann formulation for Braginskii magnetohydrodynamics},
Volume = {46},
Year = {2011}}
......@@ -1913,7 +1913,7 @@ MRREVIEWER = {Philip D. Loewen},
Booktitle = {Numerical Mathematics and Advanced Applications 2011},
Pages = {83--90},
Publisher = {Springer},
Title = {Moment-based boundary conditions for lattice Boltzmann magnetohydrodynamics},
Title = {Moment-based boundary conditions for lattice {B}oltzmann magnetohydrodynamics},
Year = {2013}}
@article{dellar2013lattice,
......@@ -1921,7 +1921,7 @@ MRREVIEWER = {Philip D. Loewen},
Journal = {Journal of Computational Physics},
Pages = {115--131},
Publisher = {Elsevier},
Title = {Lattice Boltzmann magnetohydrodynamics with current-dependent resistivity},
Title = {Lattice {B}oltzmann magnetohydrodynamics with current-dependent resistivity},
Volume = {237},
Year = {2013}}
......@@ -1931,7 +1931,7 @@ MRREVIEWER = {Philip D. Loewen},
Number = {10},
Pages = {619--635},
Publisher = {Elsevier},
Title = {Reduced Vlasov--Maxwell simulations},
Title = {Reduced {V}lasov--{M}axwell simulations},
Volume = {342},
Year = {2014}}
......@@ -1948,7 +1948,7 @@ MRREVIEWER = {Philip D. Loewen},
@book{hesthaven2007nodal,
Author = {Hesthaven, Jan S and Warburton, Tim},
Publisher = {Springer Science \& Business Media},
Title = {Nodal discontinuous Galerkin methods: algorithms, analysis, and applications},
Title = {Nodal {D}iscontinuous {G}alerkin methods: algorithms, analysis, and applications},
Year = {2007}}
@incollection{helluy-jung-2014,
......@@ -1973,14 +1973,14 @@ MRREVIEWER = {Philip D. Loewen},
Keywords = {parallel computation ; Maxwell's equations ; OpenCL ; Discontinuous Galerkin ; {\'E}quations de Maxwell ; Calcul parall{\'e}le ; Galerkin Discontinue ; MPI},
Month = Mar,
School = {{Universit{\'e} de strasbourg}},
Title = {{Resolution of tridimensional instationary Maxwell's equations on massively multicore architecture}},
Title = {{Resolution of tridimensional instationary {M}axwell's equations on massively multicore architecture}},
Type = {Theses},
Url = {https://tel.archives-ouvertes.fr/tel-01132856},
Year = {2015},
Bdsk-Url-1 = {https://tel.archives-ouvertes.fr/tel-01132856}}
@incollection{helluy2016asynchronous,
title={Asynchronous OpenCL/MPI numerical simulations of conservation laws},
title={Asynchronous {OpenCL/MPI} numerical simulations of conservation laws},
author={Helluy, Philippe and Strub, Thomas and Massaro, Michel and Roberts, Malcolm},
booktitle={Software for Exascale Computing-SPPEXA 2013-2015},
pages={547--565},
......@@ -2054,7 +2054,7 @@ MRREVIEWER = {Philip D. Loewen},
Number = {1-3},
Pages = {215--230},
Publisher = {Springer},
Title = {Building blocks for arbitrary high order discontinuous Galerkin schemes},
Title = {Building blocks for arbitrary high order {D}iscontinuous {G}alerkin schemes},
Volume = {27},
Year = {2006}}
......@@ -2074,7 +2074,7 @@ MRREVIEWER = {Philip D. Loewen},
Number = {2},
Pages = {129--141},
Publisher = {Elsevier},
Title = {An interpretation and derivation of the lattice Boltzmann method using Strang splitting},
Title = {An interpretation and derivation of the lattice {B}oltzmann method using {S}trang splitting},
Volume = {65},
Year = {2013}}
......@@ -2111,7 +2111,7 @@ MRREVIEWER = {Philip D. Loewen},
Number = {6},
Pages = {479},
Publisher = {IOP Publishing},
Title = {Lattice BGK models for Navier-Stokes equation},
Title = {Lattice {BGK} models for {N}avier-{S}tokes equation},
Volume = {17},
Year = {1992}}
......@@ -2120,7 +2120,7 @@ MRREVIEWER = {Philip D. Loewen},
Journal = {Computers \& Fluids},
Pages = {45--55},
Publisher = {Elsevier},
Title = {A compressible lattice Boltzmann finite volume model for high subsonic and transonic flows on regular lattices},
Title = {A compressible lattice {B}oltzmann finite volume model for high subsonic and transonic flows on regular lattices},
Volume = {131},
Year = {2016}}
......@@ -2130,7 +2130,7 @@ MRREVIEWER = {Philip D. Loewen},
Number = {2},
Pages = {142--159},
Publisher = {Elsevier},
Title = {Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves},
Title = {Stable lattice {B}oltzmann schemes with a dual entropy approach for monodimensional nonlinear waves},
Volume = {65},
Year = {2013}}
......@@ -2160,7 +2160,7 @@ MRREVIEWER = {Philip D. Loewen},
Number = {1},
Pages = {196--206},
Publisher = {Elsevier},
Title = {Stability analysis of lattice Boltzmann methods},
Title = {Stability analysis of lattice {B}oltzmann methods},
Volume = {123},
Year = {1996}}
......@@ -2170,7 +2170,7 @@ MRREVIEWER = {Philip D. Loewen},
Number = {1},
Pages = {329--364},
Publisher = {Annual Reviews 4139 El Camino Way, PO Box 10139, Palo Alto, CA 94303-0139, USA},
Title = {Lattice Boltzmann method for fluid flows},
Title = {Lattice {B}oltzmann method for fluid flows},
Volume = {30},
Year = {1998}}
......@@ -2223,7 +2223,7 @@ MRREVIEWER = {Philip D. Loewen},
Number = {11},
Pages = {1309--1331},
Publisher = {Wiley Online Library},
Title = {Gmsh: A 3-{D} finite element mesh generator with built-in pre-and post-processing facilities},
Title = {{G}msh: A 3-{D} finite element mesh generator with built-in pre-and post-processing facilities},
Volume = {79},
Year = {2009}}
......@@ -2257,7 +2257,7 @@ MRREVIEWER = {Philip D. Loewen},
Number = {2},
Pages = {645--673},
Publisher = {Elsevier},
Title = {Hyperbolic divergence cleaning for the MHD equations},
Title = {Hyperbolic divergence cleaning for the {MHD} equations},
Volume = {175},
Year = {2002}}
......@@ -2277,7 +2277,7 @@ MRREVIEWER = {Philip D. Loewen},
Number = {6},
Pages = {319--323},
Publisher = {Elsevier},
Title = {Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations},
Title = {Fractal decomposition of exponential operators with applications to many-body theories and {M}onte {C}arlo simulations},
Volume = {146},
Year = {1990}}
......@@ -2304,7 +2304,7 @@ MRREVIEWER = {Philip D. Loewen},
}
@article{coulette2019high,
title={High-order implicit palindromic discontinuous Galerkin method for kinetic-relaxation approximation},
title={High-order implicit palindromic {D}iscontinuous {G}alerkin method for kinetic-relaxation approximation},
author={Coulette, David and Franck, Emmanuel and Helluy, Philippe and Mehrenberger, Michel and Navoret, Laurent},
journal={Computers \& Fluids},
volume={190},
......@@ -2337,7 +2337,7 @@ MRREVIEWER = {Philip D. Loewen},
@TechReport{agullo:hal-01473475,
author = {Agullo, Emmanuel and Buttari, Alfredo and Byckling, Mikko and Guermouche, Abdou and Masliah, Ian},
title = {{Achieving high-performance with a sparse direct solver on Intel KNL}},
title = {{Achieving high-performance with a sparse direct solver on Intel {KNL}}},
institution = {{Inria Bordeaux Sud-Ouest ; CNRS-IRIT ; Intel corporation ; Universit{\'e} Bordeaux}},
year = {2017},
type = {Research Report},
......
......@@ -125,7 +125,7 @@
and more different wave speeds. It is also subject to complex phenomena
such as occurrence of shock waves, current sheet formation, magnetic
reconnection, instabilities and turbulent behaviors. An additional
specificity is that the magnetic field has to satisfy a free divergence
specificity is that the magnetic field has to satisfy a divergence-free
condition, which generally difficult to be verified by numerical solutions.
In order to deal with the divergence-free condition, we adopt here
a modified version of the MHD equations by a divergence cleaning term
......@@ -145,7 +145,7 @@
instance).
In this paper, we solve the kinetic representation with a Lattice-Boltzmann
Method (LBM), where the resolution of the transport step is done exactly
Method (LBM), where the the transport step is solved exactly
on a regular grid. An important aspect of the LBM is the choice of
the relaxation parameter in the collision step. The choice of the
parameter allows adjusting the numerical viscosity of the LBM scheme.
......@@ -171,7 +171,7 @@
\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}.}
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}}
......@@ -234,7 +234,7 @@
0
\end{array}\right).
\]
The velocity and magnetic field are noted
The velocity and magnetic field are denoted
\[
\v u=(u_{1},u_{2},u_{3})^{T},\quad\v B=(B_{1},B_{2},B_{3})^{T},
\]
......@@ -258,7 +258,7 @@
\revB{It is possible to show that the MHD-DC system, as the MHD system, is hyperbolic \cite{dedner2002hyperbolic}.
However this system is not strictly hyperbolic, which leads to some difficulties, such as non-uniquemness of the
Riemann problems in some situations \cite{torrilhon2003uniqueness}.}
Riemann problems in some situations \cite{torrilhon03}.}
The interest of the MHD-DC formulation
is for numerical approximations. Indeed, standard approximations of
......@@ -336,7 +336,7 @@
\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}.
of approximation \cite{badwaik2018task,coulette2016palindromic}.
In addition, when the lattice velocities are aligned with the mesh,
it is possible to adopt a very simple exact solver of the transport
step. This is the method that we present here.
......@@ -454,7 +454,7 @@ $
\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}}.
\end{array}\right),\quad i,j\in \{0,\ldots , N-1\}.
\]
For a simpler presentation we can assume periodic boundary condition, which amounts to the following equivalences
\[
......@@ -463,7 +463,7 @@ $
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
kinetic vectors just before the collision step are noted $\v f_{k,i,j}^{n,-}$.
kinetic vectors just before the collision step are denoted $\v f_{k,i,j}^{n,-}$.
For solving the kinetic system (\ref{eq:kin}) we treat separately
the transport and the relaxation terms.
......@@ -536,7 +536,7 @@ $
of the kinetic relaxation scheme in the one-dimensional case. This
one-dimensional analysis will give us simple intuitions for adjusting
the relaxation parameter in the 2D case. We consider the one-dimensional
MHD-DC system (\ref{eq:mhd-1d}). For more simplicity, we will note
MHD-DC system (\ref{eq:mhd-1d}). For more simplicity, we will denote
$\v F=\v F_{1}$ and $x=x_{1}$. The equations then read
\[
\partial_{t}\v w+\partial_{x}\v F(\vw)=0
......@@ -642,20 +642,20 @@ $
with
$$
a_{1,1} =\frac{\epsilon\left(\rho^{4}\left(\lambda^{2}-c^{2}\right)-3\rho^{4}u^{2}\right)}{\Delta t\left(u^{4}\rho^{4}+\left(c^{2}-\lambda^{2}\right)^{2}\rho^{4}-2u^{2}\left(c^{2}+\lambda^{2}\right)\rho^{4}\right)}+\frac{1}{2},