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

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@@ -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.

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@@ -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 divergencefree 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 divergencefree. 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}}

\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

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@@ -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