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}).

...

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@@ -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,

...

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@@ -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

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