An exponential integrator for a highly oscillatory Vlasov equation

International audienceIn the framework of a Particle-In-Cell scheme for some 1D Vlasov-Poisson system depending on a small parameter, we propose a time-stepping method which is numerically uniformly accurate when the parameter goes to zero. Based on an exponential time differencing approach, the scheme is able to use large time steps with respect to the typical size of the fast oscillations of the solution

Long time behaviour of an exponential integrator for a Vlasov-Poisson system with strong magnetic field

International audienceWith the aim of solving in a four dimensional phase space a multi-scale Vlasov-Poisson system, we propose in a Particle-In-Cell framework a robust time-stepping method that works uniformly when the small parameter vanishes. As an exponential integrator, the scheme is able to use large time steps with respect to the typical size of the solution's fast oscillations. In addition, we show numerically that the method has accurate long time behaviour and that it is asymptotic preserving with respect to the limiting Guiding Center system

Semi-Lagrangian simulations of the diocotron instability

We consider a guiding center simulation on an annulus. We propose here to revisit this test case by using a classical semi-Lagrangian approach. First, we obtain the conservation of the electric energy and mass for some adapted boundary conditions. Then we recall the dispersion relation and discussions on diff erent boundary conditions are detailed. Finally, the semi-Lagrangian code is validated in the linear phase against analytical growth rates given by the dispersion relation. Also we have validated numerically the conservation of electric energy and mass. Numerical issues/diffi culties due to the change of geometry can be tackled in such a test case which thus may be viewed as a fi rst intermediate step between a classical guiding center simulation in a 2D cartesian mesh and a slab 4D drift kinetic simulation

Optimization of Particle-In-Cell simulations for Vlasov-Poisson system with strong magnetic field

International audienceWe study the dynamics of charged particles under the influence of a strong magnetic field by numerically solving the Vlasov-Poisson and guiding center models. By using appropriate data structures, we implement an efficient (from the memory access point of view) particle-in-cell method which enables simulations with a large number of particles. We present numerical results for classical one-dimensional Landau damping and two-dimensional Kelvin-Helmholtz test cases. The implementation also relies on a standard hybrid MPI/OpenMP parallelization. Code performance is assessed by the observed speedup and attained memory bandwidth

Efficient Data Layouts for a Three-Dimensional Electrostatic Particle-in-Cell Code

International audienceThe Particle-in-Cell (PIC) method is a widely used tool in plasma physics. To accurately solve realistic problems, the method requires to use trillions of particles and therefore, there is a strong demand for high performance code on modern architectures. The present work describes performance results of Pic-Vert, a hybrid OpenMP/MPI and vectorized three-dimensional electrostatic PIC code.The code simulates 3d3v Vlasov-Poisson systems on Cartesian grids with periodic boundary conditions. Overall, it processes 590 million particles/second on a 24-core Intel Skylake architecture, without hyper-threading (25 million particles per second per core).The paper presents extensions in 3d of our preliminary 2d results, with highlights on the difficulties andsolutions proposed for these extensions. Specifically, our main contributions consist in proposing a new space-filling curve in 3d (called L6D) to improve the cache reuse and an adapted loop transformation (strip-mining) to achieve efficient vectorization. The analysis of these optimization strategies is performed in two-stages, first on a 24-core socket and second on a super-computer, from 1 to 3,072 cores, demonstrating significant performance gains and very satisfactory weak scaling results of the code

Kinetic modeling and numerical simulation of plasma-wall interactions in magnetic fusion devices

International audiencePhysical model In this work we apply a 1D3V kinetic model to the study of plasma wall-interactions relevant to magnetic fusion devices such as Tokamaks. The base physical model describes a plasma in contact with one or two parallel planar material walls, standing for divertor targets plates in the two examples considered here. The direction e x normal to the plate(s) is the only one taken into account, while the system is considered invariant in the (e y , e z) plane. In addition to the self-consistent electric field along e x , particles are subject to the action of a uniform external magnetic field B 0 = B 0 (sin αe x + cos αe y) tilted with respect to the wall surface. For a given species of mass m s and charge q s , the evolution of the distribution function g s (t, x, v) in the 4D phase space is driven by the Vlasov equation ∂ t g s + v x ∂ x g s + q s m s (−∂ x φ e x + v × B 0) · ∇g s = C s (g s) + S s , (1) where the self-consistent electrostatic potential φ is obtained by the Poisson equation ∂ xx φ = −(1/ε 0) ∑ s q s n s and (C s , S s) stand respectively for the contribution of collisional processes and external sources. From a computational point of view, the specificity of our approach is the use of fully Eulerian schemes in our computational codes: the particle distribution function is sampled over a 4D phase-space grid. Smooth and accurate solutions can be obtained even in low density regions without the need for any additional smoothing procedure

Vlasov modelling of parallel transport in a tokamak scrape-off layer

International audienceA one-dimensional Vlasov-Poisson model is used to describe the parallel transport in a tokamak scrape-off layer. Thanks to a recently-developed `asymptotic-preserving' numerical scheme, it is possible to lift numerical constraints on the time step and grid spacing, which are no longer limited by, respectively, the electron plasma period and Debye length. The Vlasov approach provides a good velocity-space resolution even in regions of low density. The model is applied to the study of parallel transport during edge-localized modes (ELMs), with particular emphasis on the particles and energy fluxes on the divertor plates. The numerical results are compared to analytical estimates based on a free-streaming model, with good general agreement. An interesting feature is the observation of an early electron energy flux, due to suprathermal electrons escaping the ions' attraction. In contrast, the long-time evolution is essentially quasi-neutral and dominated by the ion dynamics

Efficient Data Structures for a Hybrid Parallel and Vectorized Particle-in-Cell Code

International audienceThe contribution of the present work relies on an innovative and judicious combination of several optimization techniques for achieving high performance when using automatic vectorization and hybrid MPI/OpenMP parallelism in a Particle-in-Cell (PIC) code. The domain of application is plasma physics: the code simulates 2d2v Vlasov-Poisson systems on Cartesian grids with periodic boundary conditions. Overall, our code processes 65 million particles/second per core on Intel Haswell (without hyper-threading) and achieves a good weak scaling up to 0.4 trillion particles on 8,192 cores. The optimizations mainly consist in using (i) a structure of arrays for the particles, (ii) an efficient data structure for the electric field and the charge density, and (iii) an appropriate code for automatic vectorization of the charge accumulation and of the positions' update. In particular, we use space-filling curves to enhance data locality while enabling vectorization: starting from a redundant cell-based data structure for the electric field and for the charge density, we compare several space-filling curves for an efficient ordering of these data and we obtain a gain of 36% in the number of L2 and L3 cache misses when using a Morton curve instead of the classical row-major one. In addition, by proposing a specific writing of the updating positions code we achieve a 31% time improvement in that step. The optimizations bring an overall gain in the execution time of 42% with respect to a standard code. The parallelization of the particle loops is simply performed by means of both distributed and shared memory paradigms, without domain decomposition. We explain the weak and the strong scalings of the code bounded as expected by the overhead of the MPI communications

Comparison of free-streaming ELM formulae to a Vlasov simulation

International audienceThe main drawbacks of the original free-streaming equations for edge localised mode transport in the scrape-off layer [W. Fundamenski, R.A. Pitts, Plasma Phys. Control Fusion 48 (2006) 109] are that the plasma potential is not accounted for and that only solutions for ion quantities are considered. In this work, the equations are modified and augmented in order to address these two issues. The new equations are benchmarked against (and justified by) a numerical simulation which solves the Vlasov equation in 1d1v. When the source function due to an edge localised mode is instantaneous, the modified free-streaming 'impulse response' equations agree closely with the Vlasov simulation results. When the source has a finite duration in time, the agreement worsens. However, in all cases the match is encouragingly good, thus justifying the applicability of the free-streaming approach

A new fully two-dimensional conservative semi-Lagrangian method: applications on polar grids, from diocotron instability to ITG turbulence

article 252International audienceWhile developing a new semi-Lagrangian solver, the gap between a linear Landau run in 1Dx1Dand a 5D gyrokinetic simulation in toroidal geometry is quite huge. Intermediate test cases are welcomefor testing the code. A new fully two-dimensional conservative semi-Lagrangian (CSL) method is presentedhere and is validated on 2D polar geometries. We consider here as building block, a 2D guiding-centertype equation on an annulus and apply it on two test cases. First, we revisit a 2D test case previouslydone with a PIC approach [18] and detail the boundary conditions. Second, we consider a 4D drift-kineticslab simulation (see [10]). In both cases, the new method appears to be a good alternative to deal withthis type of models since it improves the lack of mass conservation of the standard semi-Lagrangian (BSL)method