A Parallel Vlasov solver using a Wavelet based Adaptive Mesh Refinement Matthieu Haefele, Guillaume Latu and Michael Gutnic INRIA CALVI project (http://math.u-strasbg.fr/calvi) LSIIT, UMR CNRS 7005 & IRMA, UMR CNRS 7501 Universit´e Louis Pasteur, 7 rue Descartes Strasbourg, France {haefele | latu}@lsiit.u-strasbg.fr, gutnic@math.u-strasbg.fr Abstract We are interested in solving the Vlasov equation used to describe collective effects in plasmas. This non- linear partial differential equation coupled with Maxwell equation describes the time evolution of the particle dis- tribution in phase space. The numerical solution of the full three-dimensional Vlasov-Maxwell system rep- resents a considerable challenge due to the huge size of the problem. A numerical method based on wavelet transform enables to compute the distribution function on an adaptive mesh from a regular discretization of the phase space. In this paper, we evaluate the costs of this recently developed adaptive scheme applied on a reduced one-dimensional model, and its parallelization. We got a fine grain parallel application that achieves a good scalability up to 64 processors on a shared memory architecture. 1 INTRODUCTION In the quest for a new source of energy, understand- ing plasma behavior is one of the most challenging problems to overcome. Plasma can be considered as the fourth state of matter and exists at huge tempera- ture conditions (10 4 K or more). These conditions are reached in different facilities, in particular in tokamak reactors. A kinetic description is used to model such phenomenon. The plasma is governed by the Vlasov equation coupled with Poisson or Maxwell equations to evaluate the self-consistent fields generated by the particles. Vlasov equation ∂f ∂t + v.∇ x f +(E + v × B) .∇ v f =0, (1) characterizes the evolution of particle distribution in time and space according to the electro-magnetic fields E(x, t) and B(x, t). The distribution function f (x, v,t) represents the particle density at a time t and a point (x,v) in phase space. In phase space, a point is charac- terized by a position vector x and a velocity vector v. To describe numerous cases, one need (x,v) ∈ R d × R d with d = 3. Finding an approximation of this non-linear partial differential equation enables the simulation of new ac- celerator and tokamak designs to validate them before building the devices. For a more fundamental purpose, finding such an approximate solution makes it possi- ble for physicists to represent the behavior of different physical parameter sets. Two major kinds of numerical methods exist to find an approximate solution of the Vlasov equation. The Particle In Cell method (PIC) [1] follows a large num- ber of particles (≃ 10 9 ) initially distributed randomly in phase space, whereas the semi-langrangian method approximates the particle distribution function on a uniform mesh of the phase space [10]. Although mathe- maticians have already studied wavelet transforms and adaptive numerical scheme, coupling wavelets and non- linear partial differential equations approximation rep- resents a great deal of interest. The wavelet decom- position gives a sparse representation and a natural criterion to perform local grid refinements. The paral- lelization of such method [9, 5] is interesting in order to deal with applications that manipulate large data with possibly many dimensions. The present work focuses on the parallelization of an adaptive semi-lagrangian code which considers a single physical dimension. This work corresponds to a first step in the design of a new parallel Vlasov solver. These current researches are performed in an in- terdisciplinary approach within the INRIA CALVI project: physics and mathematics with Nancy 1 Uni- versity, mathematics and computer science with Stras-