\title*{A kinetic method for solving the MHD equations. Application to the

computation of tilt instability on uniform fine meshes.}

\title*{\revA{A robust and efficient solver based on kinetic

schemes for Magnetohydrodynamics (MHD) equations.}}

\titlerunning{Kinetic method for MHD}

\titlerunning{Kinetic scheme for MHD}

\authorrunning{Baty \textit{et al.}}

\author{Name of Author\inst{1}\and

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@@ -99,18 +104,18 @@

\chapauthor{Philippe Helluy}

\abstract{This paper is devoted to the simulation of MHD flows with complex

\abstract{This paper is devoted to the simulation of \revA{Magnetohydrohynamics} flows with complex

structures. This kind flows present instabilities that generate shock

waves. We propose a robust and precise numerical method based on the

Lattice Boltzmann methodology. We explain how to adjust the numerical

viscosity in order to obtain stable, precise results and reduced divergence

viscosity in order to obtain stable, \revA{accurate results in smooth or discontinuous parts of the flow} and reduced divergence

errors. This method can handle shock wave and is almost second order.

It is also very well adapted to GPU computing. We also give results

It is also very well adapted to GPU \revA{(Graphics Processing Unit)}computing. We also give results

for a tilt instability test case on very fine meshes. }

\section{Introduction}

The MagnetoHydroDynamic (MHD) system is a fundamental model used in

The MagnetoHydroDynamics (MHD) system is a fundamental model used in

many fields of physics: astrophysics, plasma physics, geophysics...

Indeed, the MHD model is commonly adopted as an excellent framework

for collisional plasma environments. The numerical approximation of

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In order to capture fine structures, it is necessary to consider very

fine meshes. We have programmed the algorithm in a very efficient

way in order to address recent GPUs (Graphic Purpose Units) or multicore

way in order to address recent GPUs (Graphic Processing Units) or multicore

CPUs. We describe the implementation, which relies on OpenCL and PyOpenCL,

and the memory optimizations used for reaching high performance. The

program allows performing full MHD simulations on grids as fine as

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current sheets.

\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}.}

\subsection{MHD equations with divergence cleaning\label{subsec:MHD}}

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an analysis of the numerical viscosity of the kinetic method in this