Numerical simulation for the MHD system in 2D using OpenCL

In this work we compute the MHD equations with divergence cleaning on GPU. The method is based on the finite volume approach and Strang dimensional splitting. The simplicity of the approach makes it a good candidate for a GPU implementation with OpenCL. With adequate memory optimization access, we achieve very high speedups, compared to a classical sequential implementation. Résumé. Dans ce travail, nous résolvons les équations de la MHD avec correction de divergence sur carte graphique. La méthode est basée sur les volumes finis et le splitting directionnel de Strang. La simplicité de l’algorithme en fait un bon candidat pour la programmation sur carte graphique sous OpenCL. Avec de bonnes optimisations des accès mémoire, nous obtenons de très bonnes accélérations, comparé à une programmation séquentielle classique. Introduction The aim of this work is to propose an efficient algorithm for solving the two-dimensional MHD equations. The Magneto-Hydro-Dynamics (MHD) system is a model to describe the behavior of conducting fluids inside a magnetic field. It is useful for instance for simulating the edge plasma in tokamak simulations or astrophysics plasmas. For one-dimensional computations, we can assume that the magnetic field in the x-direction is constant. In this case the divergence free condition on the magnetic field B, which reads ∇ · B = 0, is automatically satisfied. In higher dimensions, it is important to impose the divergence free condition in order to ensure a good precision of the scheme for long time simulations. We impose the condition through a divergence cleaning technique described in [3, 5]. Our numerical method is based on a the finite volume approach. We have implemented two differents fluxes for solving the problem: the Rusavov flux, the VFRoe flux [2]. One objective of our work is to compute plasma reconnection in astrophysics. Such computations require very fine meshes and thus lead to long computations. Therefore, we have implemented our algorithm on GPU using the OpenCL environment. We use several techniques in order to achieve very high memory bandwitdth: optimized transposition algorithm, cache prefetch, etc. We test our algorithms on one-dimensional and twodimensional well-known test cases: a Riemann problem with a strong shock and the Orzag-Tang vortex. 1 IRMA, 7 rue René Descartes, 67084 Strasbourg, [email protected] 2 IRMA, 7 rue René Descartes, 67084 Strasbourg, [email protected] 3 ICPS LSIIT, Boulevard Sébastien Brant, 67400 Illkirch-Graffenstaden, [email protected] c © EDP Sciences, SMAI 2013 2 ESAIM: PROCEEDINGS 1. Model The Magneto-Hydro-Dynamics (MHD) system a useful model for describing the behavior of astrophysical plasmas, or conducting fluids in a magnetic fields. The unknowns are the density ρ, the velocity u ∈ R, the internal energy e, the pressure p and the magnetic field B ∈ R. All of these unknowns depend on the time t and on the space variable x ∈ R. For numerically enforcing the zero divergence condition on the magnetic field, it was proposed [6] to modify the original MHD equations by considering an additional variable ψ. With this "divergence cleaning" the equations read: