where $\omega=\frac{2\Delta t}{2\tau+\Delta t}$. Typically, one will

choose $\omega=1.9$ in our simulations.

\revA{We see that the whole algorithm is extremely simple. It is a succession of "shifts" (\ref{eq:shift-algorithm}) and "collisions" (\ref{eq:collision}. For $\omega=2$ the scheme is second order but unstable in shocks. For $\omega=1$ the scheme is very robust, entropy dissipative, but quite diffusive. It would be interesting to construct a rigorous strategy for choosing locally the optimal value of $\omega$. Let us mention that many finite volume schemes have been designed for solving MHD equations, with specific Riemann solvers. We can mention for instance the solver of \cite{bouchut2010multiwave}, based on a relaxation approach, with proven stability and accuracy features. However, the solver presented in \cite{bouchut2010multiwave} is more complicated to program and less computationally efficient. We can thus compensate the slightly lower accuracy of the kinetic scheme by finer meshes, without increasing too much the computational load.}

\subsection{Boundary conditions}

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@@ -531,7 +535,7 @@ $

For the analysis, it is possible to replace the scalar relaxation

parameter $\omega$ by a matrix $\vom$, for more generality. In the

following, we understand the comparison of matrices in the usual way,

following, we \revA{establish the comparison between matrices} in the usual way,

by the comparison of the associated quadratic form (the resulting

order is thus not total).

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@@ -581,7 +585,7 @@ $

In order to check practically the accuracy of the approximation, we

apply the above analysis for a simplified system of two conservation

laws

laws.

We consider the one-dimensional isothermal Euler equations with a

diagonal diffusion of $\epsilon\partial_{xx}\left(\rho,\rho u\right)^{T}$.

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@@ -597,7 +601,7 @@ $

\end{align}

This system is a very simplified version of the MHD equation, where

the magnetic field is assumed to vanish and the gas is supposed to

the magnetic field is assumed to vanish and the gas is supposed to be

isothermal. We have chosen to use this specific non-physical diffusion

for our first test since it is exactly the type of diffusion that

is apparent in a standard finite volume code using a Lax-Friedrichs

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@@ -635,15 +639,15 @@ $

For our first test of the matrix relaxation, we solve (\ref{eq: Isothermal Navier-Stokes})

comparing the Lattice Boltzmann sheme using (\ref{eq:relax-with-diffusion})

as relaxation matrix and a standard explicit centered finite volume

scheme for approximating (\eqref{eq: Isothermal Navier-Stokes}).

scheme for approximating (\ref{eq: Isothermal Navier-Stokes}).

In this centered scheme, the time step is taken very small in such

way that the stability condition is satisfied and that the time integration

error can be neglected. In other words, the equivalent PDE of both

schemes is (\ref{eq:equiv-eq-2order}). Hence, given the parameters

in both schemes are set to represent the same diffusion $\epsilon$,

one should get the same type of diffusion for both schemes.\\

To test this we take the simple test case of a stationary viscous

shock:\\

To test this we take the simple case of a stationary viscous