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Accueil > Agenda > Colloques et rencontres > Archives > Agenda 2016 > NUMKIN 2016 : International Workshop on Numerical Methods for Kinetic Equations

NUMKIN 2016 : International Workshop on Numerical Methods for Kinetic Equations

IRMA Strasbourg, 17-21 October 2016

The NUMKIN 2016 will take place in Strasbourg at IRMA between October 17 and October 21 2016.

The purpose of this workshop is to bring together a small number of specialists in the development of numerical methods for collisionless and collisional kinetic equations, and discuss the current evolutions in the field. It will also be the annual general meeting of the EUROFusion Enabling Research project on Verification and development of new algorithms for gyrokinetic codes. The last day of the Meeting, Friday 21 October, will be devoted to the SeLaLib library.

Scientific Organizers : Philippe Helluy, Michel Mehrenberger, Eric Sonnendrücker

Venue : Salle de conferences, IRMA building

Registration is required at this link.

See also the previous workshop (Garching) : http://www.ipp.mpg.de/3874756/NumKin2015

Program

For the working groups, seminar rooms 309 and 418 are booked (in th math building which is next to the IRMA building)

PDF - 48.9 ko
Program
PDF - 625.4 ko
infos

monday 17.10.2016 morning :

11h-11h15 : Welcome
11h15-12h05 : Natalia Tronko

monday 17.10.2016 afternoon :

14h-15h : Alberto Bottino
15h-16h : Francesco Palermo
16h-16h30 : coffee break
16h30-17h : Timo Kiviniemi
17h-17h30 : Aaron Scheinberg
17h30-18h : James Martin-Collar

tuesday 18.10.2016 morning :

8h30-9h20 : Nicolas Crouseilles
9h20-10h10 : Mohammed Lemou
10h10-10h40 : coffee break
10h40-11h30 : Martin Campos Pinto

tuesday 18.10.2016 afternoon :

13h30-15h30 : Working groups+coffee break
15h30-16h10 : Tao Xiong
16h10-16h50 : Sever Hirstoaga
16h50-17h : Short break
17h-17h50 : Mihai Bostan
17h50-18h20 : Thao Ha

wednesday 19.10.2016 morning

8h30-9h10 : Bruno Després
9h10-9h50 : Benjamin Graille
9h50-10h15 : David Coulette
10h15-10h45 : coffee break
10h45-11h30 : Paul Dellar

afternoon : free or working groups

thursday 20.10.2016 morning :

8h30-9h20 : Mehdi Badsi
9h20-9h45 : Nhung Pham
9h45-10h10 : Erwan Deriaz
10h10-10h40 : coffee break
10h40-11h30 : Francois Dubois

thursday 20.10.2016 afternoon :

13h30-14h : Charles Ehrlacher
14h-14h30 : Yuuichi Asahi
14h30-15h : Nicolas Bouzat
15h-15h50 : Guillaume Latu
15h50-16h20 : coffee break
16h20-17h10 : Yaman Güclü
17h10-18h : Katharina Kormann

friday : selalib day
8h30-10h : Yaman Güclü and Julien Bigot, "Inputs/Outputs strategies"
10h-10h30 : coffee break
10h30-12h : Pierre Navaro and Katharina Kormann, ’’Publication"

Speakers, Abstracts and Slides

  • Yuuichi Asahi, Benchmarking of global full-f gyrokinetic codes.
    PDF - 1.8 Mo

    Full-f gyrokinetic simulations tackle several important transport phenomena such as SOC like behaviors of intermittent bursty transport, stiff temperature profiles produced by avalanche like non-local transport, momentum transport processes leading to intrinsic rotation, and influences of radial electric fields on turbulent transport. These full-f physics has a significant impact on the global confinement, and so far, several works reported different transport levels and transport properties between full-f and delta-f simulations. Contrary to relatively easy benchmarks in delta-f models, benchmarking full-f codes raise several difficulties which stem from different source models and boundary conditions. In our case, we compare two different global full-f gyrokinetic Eulerian codes, GYSELA [V. Grandgirard et al., to appear in Comp. Phys. Com. (2016), DOI : 10.1016/j.cpc.2016.05.007] and GT5D [Y. Idomura et al., Comput. Phys. Commun. 179 391, 2008.], which use different source models and boundary conditions. Since both source and boundaries have significant impact on transport and profile formation, we keep source models and boundary conditions as close as possible.
    Another important aspect of full-f gyrokinetic codes is that they can treat turbulent and neoclassical transports consistently. This favorable feature is rather problematic for benchmarking, since the reasons for differences can be difficult to understand when the simulations results are not identical. Thus, we first decomposed the problem into small parts that correspond to idealized physical situations. For turbulence, we investigated linear Ion Temperature Gradient (ITG) modes and nonlinear decaying ITG turbulence in the collision-less limit. For neoclassical calculations, we chose plasma parameters such that ITG modes are stable, and carried out collisionality scan to compare results with analytical estimates.
    In the talk, we will present benchmarking results for linear ITG modes and nonlinear decaying ITG turbulence and collisionality scan for neoclassical physics. Preliminary benchmark results for flux driven simulations will also be presented.
    Joint work with X. Garbet, Y. Idomura, V. Grandgirard, G. Latu, Y. Sarazin, G. Dif-Pradalier, P. Donnel, C. Ehrlacher, Ch. Passeron

  • Mehdi Badsi, Bi-kinetic modeling of the stationary plasma wall transition with a parallel to the wall magnetic field.
    PDF - 4.5 Mo
  • Mihai Bostan, Multi-scale analysis for transport equations and applications.
    PDF - 206.7 ko

    This study concerns the asymptotic analysis of mathematical models for strongly magnetized plasmas. It relies on a two-scale approach, based on the mean ergodic theorem, which allows us to separate between the fast and slow dynamics. The method adapts to many models. In particular it is possible to incorporate collision operators and to compute the effective diffusion matrices of the limit models. The average advection field and average diffusion matrix field appear as the long time limit for some parabolic problems, which allows us to obtain good approximations of the limit models, in the case of non uniform magnetic fields (when exact formulae are not available).

  • Alberto Bottino, Monte-Carlo finite elements gyrokinetic simulations of Alfven modes in tokamaks.
    PDF - 1.3 Mo

    The global gyrokinetic code ORB5 can simultaneously include electromagnetic perturbations, general ideal MHD axisymmetric equilibria, zonal flow preserving sources, collisions, and the ability to solve the full core plasma including the magnetic axis.
    In this work, a Monte Carlo Particle In Cell Finite Element model, starting from a gyrokinetic discrete Lagrangian, is derived and implemented into the ORB5 code. The variations of the Lagrangian are used to obtain the time continuous equations of motion
    for the particles and the Finite Element approximation of the field equations.
    The Noether theorem for the semi-discretised system, implies a certain number of conservation properties for the final set of equation. Linear and nonlinear results, concerning Alfven instabilities, in the presence of an energetic particle population, and microinstabilities, such as electromagnetic ion temperature gradient (ITG) driven modes and kinetic ballooning modes (KBM), will be presented and discussed.

  • Alberto Bottino, Verification of GK codes on linear collisionless dynamics of axisymmetric modes in tokamaks : Current status of verification on GAMs.
    PDF - 257.9 ko
  • Nicolas Bouzat, Targeting Realistic Geometry in Tokamak code Gysela.
    PDF - 4.6 Mo

    Some of the key issues to address realistic gyrokinetic simulations of Tokamaks are : efficiency and robustness of numerical schemes, accuracy of geometric description, scalability of parallel algorithms. This project intended to improve the parallel application Gysela, which is based on a gyrokinetic model and uses polar coordinates system. To access realistic geometry and realistic physics, a new variant of the interpolation method (Lagrange and splines) which can handle the mesh singularity in the poloidal plane at r = 0 has been implemented. A new non-uniform polar mesh, allowing for more equally distributed point on the polar mesh, has been developed and tested together with a mapping allowing to match magnetic field lines to the mesh. Thus, several operators (gyroaverage operator based on Hermite interpolation, Vlasov-Poisson operator, advection operator...) have been revised in order to cope with non-circular geometry. Convergence studies for all operators show good results as well in degree as in space.

  • Martin Campos Pinto, From particle methods to hybrid semi-lagrangian schemes.
    PDF - 1.6 Mo

    In this talk I will present two classes of particle methods with remappings, which aim at improving the accuracy of existing particle codes.
    Both methods use as available data the particle trajectories computed by the reference particle code, and they compute local linearizations of the characteristic flow. A first approach consists of transporting smooth particle shapes exactly along the corresponding affine flows, and then describing the transported density as a sum of
    linearly-transformed particles (LTP). This method has good convergence properties that can be demonstrated both at the theoretical and the numerical level.
    For long remapping periods the LTP is affected by the fact that extended particle shapes deteriorate the locality, which leads to increase both the approximation errors and the CPU time of evaluating the density. To avoid this weakness we have designed a second method which uses a backward lagrangian representation of the density, based on the local linearizations of the flow. The resulting scheme is more local by construction and also has enhanced convergence properties, also validated by numerical experiments.
    This is a joint work with Frederique Charles, Antoine Le Hyaric and the Selalib group.

  • David Coulette, Implicit kinetic schemes.
    PDF - 1.6 Mo
  • Nicolas Crouseilles, Uniformly accurate particle-in-cell method for the long time behavior of the 2d Vlasov-Poisson equation with strong magnetic field.
    PDF - 1.1 Mo

    In this talk, we construct numerical schemes for the 2d Vlasov-Poisson equation
    with a strong magnetic field. The so-obtained scheme is uniformly accurate with
    respect to the size of the fast time oscillations of the solution. The strategy combines
    the Particle-In-Cell method with a two-scale formulation of the characteristics,
    involving an additional periodic variable. Numerical experiments illustrate the efficiency
    of the proposed approach. This work is done in collaboration with M. Lemou, F. Mehats and X. Zhao.

  • Paul Dellar, Lattice Boltzmann approach to magnetohydrodynamics and electromagnetism.
    PDF - 2.7 Mo
  • Erwan Deriaz, Adaptive simulation of Vlasov equations in arbitrary dimension using interpolatory hierarchical bases.
    PDF - 2.7 Mo

    Interpolatory hierarchical bases [Deslauriers-Dubuc 1987] have been used in two-dimensional adaptive Vlasov simulations with a semi-lagrangian method [Besse et al. 2008]. Here we propose an insight to higher dimensional simulations, in four and six dimensions of the phase space, with a finite difference method. Interpolets naturally associates to finite differences through point evaluation of the distribution function. It provides simple and efficient AMR schemes adpated to moderately smooth solutions.
    We present tests for the two-dimensional Landau damping, for the two- and four-dimensional two stream instability and for the six-dimensional collision of two Plummer spheres in astrophysics. The C implementation, parallelized with OpenMP, relies on a tree structure —usual in AMR schemes [Teyssier 2002, Popinet 2003]— coupled to a system of flags for the point activation [Kevlahan 2005]. Associating these technics lightens the memory share of the AMR encoding, speeds the computations up and allows an encoding in high (4/6) dimensions.

  • Bruno Després, New weak formulations for resonant Maxwell’s equations and first numerical results.
    PDF - 3.1 Mo

    The Maxwell equations with the cold plasm tensor are resonant at the
    hybrid singularity. It rules out standard numerical methods. In this
    case, new weak formulations are possible with carefull treatment of the
    1/x singularity. The first numerical results confirm the mathematical
    analysis. Joint work with M. Campos-Pinto and A. Nicolopoulos.

  • Francois Dubois, Recovering the full Navier Stokes equations with lattice Boltzmann schemes.
    PDF - 1.2 Mo

    We consider multi relaxation times lattice Boltzmann scheme with two particle distributions for the thermal Navier Stokes equations formulated with conservation of mass and momentum and dissipation of volumic entropy. Linear stability is taken into consideration to determine a coupling between two coefficients of dissipation. We present interesting numerical results for one-dimensional strong nonlinear acoustic waves with shocks. Joint work with Benjamin Graille and Pierre Lallemand

  • Charles Ehrlacher, Linear instabilities with kinetic electrons in global, full-f simulations.
    PDF - 1.3 Mo

    The ion temperature gradient (ITG) instability is responsible for a turbulent heat flux that is often larger than the collisional value in tokamak plasmas. It has been shown that the non-adiabatic response of kinetic electrons increase the energy transport due to these modes [Y. Chen, Nucl. Fusion 43 (2003)]. Besides, kinetic electrons add the possibility of particle transport. Moreover trapped electrons due to the magnetic mirror force are known to contribute significantly to energy transport, via the Trapped Electron Modes (TEM).
    This issue is addressed by means of the gyrokinetic flux-driven full-f GYSELA code, which has been upgraded to account for (fully) kinetic electrons. When the electron dynamics is computed, simulations are about (mi/me)3/2 more costly than simulations with adiabatic electrons. Two strategies are used to reduce the computational cost of these simulations. First, trapped electrons only can be treated kinetically. These are characterized by a low parallel velocity. This allows us to increase the numerical time step by a factor (mi/me)1/2. Second, the mass ratio mi/me can be artificially reduced, thus raising the question of convergence with respect to the mass ratio.
    Interestingly, treating all the electrons kinetically leads to the appearance of axisymmetric modes characterized by a high pulsation ωH = ωci (mi/me)1/2 (k///k⊥). Their excitation still remains an open issue, which will be discussed. They turn out to appear concomitantly with a large radial transport of particles and heat, which leads to strong relaxations of the initial profiles. In turn, this prevents any linear analysis. Filtering out axisymmetric modes (but keeping zonal modes), as initially suggested by [Y. Idomura, J. Comp. Phys. 313 (2016) 51], cures the problem.
    The linear analysis of ITG and TEM instabilities will be presented, providing both the growth rate and the pulsation of the modes for each poloidal wave vector kθ moderate mass ratio mi/me = 100, unstable ITG modes can exhibit a growth rate 4 times larger with kinetic electrons than with adiabatic electrons. In the ITG-TEM unstable regime, linear simulations show that the linear growth rate decreases when the mass ratio increases. A convergence study will be presented.
    Joint work with , Y. Sarazin, V. Grandgirard, X. Garbet, J.-M. Rax, Ph. Ghendrih, Y. Asahi, N. Bouzat, G. Dif-Pradalier, P. Donnel, G. Latu, Ch. Passeron

  • Benjamin Graille, The Lattice Boltzmann method for hyperbolic systems.
    PDF - 6.1 Mo
  • Yaman Güclü, Screw-pinch simulations in Selalib : from reduced drift-kinetics to gyro-kinetics.
    PDF - 3.2 Mo
  • Thao Ha, Towards the heterogeneous multiscale method for simulation of the trajectory of the particle submitted to non-uniform electric field and strong magnetic field in 2D-2V.
    PDF - 842.5 ko

    This study investigates the heterogeneous multiscale method (HMM) and its
    application in solving a singularly perturbed system of ordinary differential equations (ODEs), which is conducted from the Vlasov equation using method of characteristics. Two test cases will be done, in which one test is for the case of non-uniform electric field and uniform strong magnetic field, and the other is for the case of non-uniform electric field and non-uniform strong magnetic field. The algorithm of computing the pseudo-period of the particle’s trajectory is also presented. The study aims to provide a clearer perspective of the influence of the variation of electric field and magnetic field on the evolution of the particle’s pseudoperiod. This work is in collaboration with Emmanuel Frénod and Sever Hirstoaga.

  • Sever Hirstoaga, Sequential performance and parallelization of a Particle-in-Cell Code.
    PDF - 668.9 ko

    We develop a PIC code for simulation of a 2d2v Vlasov-Poisson equation on cartesian grid with periodic boundary conditions. We analyze different strategies for improving its performance on single core through memory locality and vectorization. Then a standard approach of hybrid OpenMP/MPI implementation is used for parallelizing the code. This is joint work with Yann Barsamian.

  • Timo Kiviniemi, Implicit solvers in full-f gyrokinetic particle-in-cell simulation and extension to SOL.
    PDF - 2.4 Mo

    Recent developments of full-f gyrokinetic particle-in-cell
    code ELMFIRE are presented including implementation of direct implicit solver for
    ion polarization and electron parallel motion as well as domain
    decomposition. The simulation region in the code has also been recently
    extended to include whole minor radius from axis to scrape-off layer.

  • Katharina Kormann, Massively parallel semi-Lagrangian solution of the 6d Vlasov-Poisson problem.
    PDF - 1.3 Mo

    Grid-based solvers for the Vlasov equation give accurate results but
    suffer from the curse of dimensionality. To enable the grid-based
    solution of the Vlasov equation in 6d phase-space, we need efficient
    parallelization schemes. In this talk, we consider the 6d Vlasov-Poisson
    problem discretized by a split-step semi-Lagrangian scheme. To optimize
    single node performance, we use vectorization, efficient data access and
    OpenMP parallelism. For distributed memory parallelism, we consider two
    parallelization strategies : A remapping strategy that works with two
    different layouts keeping parts of the dimensions sequential and a
    classical partitioning into hyperrectangles. The 1d interpolations can
    be performed sequentially on each processor for the remapping scheme. On
    the other hand, the remapping consists in an all-to-all communication
    pattern. The partitioning only requires localized communication but each
    1d interpolation needs to be performed on distributed data. We compare
    both parallelization schemes and discuss how to efficiently handle the
    domain boundaries in the interpolation for partitioning. This is joint
    work with Klaus Reuter and Eric Sonnendrücker.

  • Guillaume Latu, Numerical methods and optimizations that enhance semi-lagrangian gyrokinetic calculations.
    PDF - 1.5 Mo

    Three recent numerical improvements of the Gysela code (semi-Lagrangian scheme, gyrokinetic modelling) will be presented. They tend to shorten execution time, to reduce memory footprint, and to better model boundary conditions. First, a method consisting in field-aligned interpolation has been put into place. In the context of numerical simulations of a magnetic fusion device, this approach is motivated by the fact that gradients of the solution along the magnetic field lines are typically much smaller than along a perpendicular direction, and one can take benefit of this anisotropy. It is a valuable input for the semi-Lagrangian code : we show that number of points in the computational mesh along toroidal directions is reduced. Field-aligned interpolation leads to significant memory and computational benefits for the same level of accuracy. Second, the numerical interpolation scheme for the Vlasov solver has been upgraded. Cubic splines have been replaced by high-order Lagrange interpolants. In this way, we are able to get a substantial speedup of the application due to a better achievable vectorization of the interpolations, and also some savings in the computation of the feet of characteristics. Third, several operators have been enhanced in order to avoid the problem arising at $r=0$ in the poloidal plane. In representing the whole poloidal plane, thus removing the central hole, treatment of boundary conditions is upgraded. Joint work with
    Y. Asahi, N. Bouzat, G. Dif-Pradalier, P. Donnel, C. Ehrlacher, X. Garbet, V. Grandgirard, Ph. Ghendrih, Y. Guclu, M. Mehrenberger, M. Ottaviani, Ch. Passeron , Y. Sarazin, E. Sonnendrucker.

  • Mohammed Lemou, Averaging and simulating highly-oscillatory transport equations with varying frequency.
    PDF - 310.2 ko
  • James Martin-Collar, Comparison of ballooning modes in a global gyrokinetic code, ORB5, with an MHD code, MISHKA.
    PDF - 749.3 ko
  • Francesco Palermo, New damping mechanism of geodesic acoustic modes investigated by global gyrokinetic simulations.
    PDF - 2.1 Mo

    Due to the presence of geodesic curvature in various toroidal devices, the zonal flows contain particular oscillations named geodesic acoustic modes (GAMs). The importance of GAM oscillations resides in the different shearing efficiency that the zonal flows have in relation to their oscillatory behavior. Although a lot of progresses have been made in the understanding of the physics of GAMs, their role in the turbulence regulation and their dynamics demands further investigations. In this work, the global dynamics of GAMs is studied analytically and by means of gyrokinetic simulations by varying several parameters such as temperature, safety factor and density gradients. In particular, in the presence of a non-uniform temperature profile, a continuum spectrum of GAM exists across the magnetic flux surfaces. As a consequence, a GAM is affected by phase-mixing, whose effect is to modify the radial structure of the perturbation, with a energy cascade from low to high radial wave numbers. Moreover, in tokamak plasmas, GAMs are subject to Landau damping by the passing ions. Here we describe how phase-mixing and Landau damping can act at the same time, with a combined action resulting in a strong damping mechanism for GAM. This proposed new mechanism of GAM decay is consistent with the observed existence or non-existence of GAMs in the different confinement regimes such as L-, I- and H-modes. Moreover, the influence of several equilibrium profiles on the radial propagation of a GAM packet is studied. This work is in collaboration with A. Biancalani, C. Angioni, F. Zonca, A. Bottino, B. Scott, G. D. Conway and E. Poli.

  • Nhung Pham, Numerical methods for the reduced Vlasov equation.
    PDF - 499.7 ko

    We consider the Vlasov equation coupled with the Maxwell equations for the electromagnetic field or with the Poisson equation for electric field only.
    In the general three-dimensional (3D) case, the system is very complicated with 7 variables (three velocities, three positions and the time) so leads to very heavy numerical simulations. The Particle-In-Cell (PIC) method (see for instance [1, 2]) is a popular method for computing collisionless plasma, because it allows performing simulations in complex configurations with a relatively low amount of memory and CPU ressource. However the PIC method is based on an initial random choice of the
    particles and thus presents numerical noise. Also, it is difficult to ensure the energy conservation. Therefore, Eulerian methods for solving kinetic equations are becoming more and more popular. They allow a better control of the conservation and numerical errors (for a review of eulerian methods, see [4]).
    We construct an hyperbolic approximation of the Vlasov equation in which the dependency on the velocity variable is removed. The resulting model enjoys interesting conservation and entropy properties.
    We propose different methods numerical in order to solve this hyperbolic system : Finite volume, Semi-Lagrangian, Discontinuous Galerkin. Joint work with Philippe Helluy and
    Laurent Navoret.

    [1] C.K. Birdsall and A.B. Langdon, Plasma Physics via Computer Simulation, Institute of Physics (IOP), Series in Plasma Physics, 1991.
    [2] D. Tskhakaya, R. Schneider, Optimization of PIC codes by improved memory management, Journal of Computational Physics, Volume 225, Issue 1, 2007.
    [3] Eliasson, Out flow boundary conditions for the Fourier transformed one-dimensional Vlasov-Poisson system, JJ. Sci. Comput. , 2001.
    [4] F. Filbet, E. Sonnendrucker, Comparison of Eulerian Vlasov solvers, Comput. Phys. Commun., 2003.

  • Aaron Scheinberg, Benchmarking linear global simulations in ORB5 and GENE.
    PDF - 1.9 Mo
  • Natalia Tronko, Verification of Gyrokinetic codes : from theory to numerical implementations.
    PDF - 6.5 Mo

    In fusion plasmas the strong magnetic field allows the fast gyro motion to be systematically removed from the description of the dynamics, resulting in a considerable model simplification and gain of computational time. Nowadays, the gyrokinetic (GK) codes play a major role in the understanding of the development and the saturation of turbulence and in the prediction of the consequent transport.
    We present a new and generic theoretical framework and specific numerical applications to test the validity and the domain of applicability of existing GK codes. For a sound
    verification process, the underlying theoretical GK model and the numerical scheme must be considered at the same time, which makes this approach pioneering. At the analytical level, the main novelty consists in using advanced mathematical tools such as variational formulation of dynamics for systematization of basic GK code’s equations to access the limits of their applicability. The indirect verification of numerical scheme is proposed via the Benchmark process. In this work, specific examples of code verification are presented for two GK codes : the multi-species electromagnetic ORB5 (PIC), and the radially global version of GENE (Eulerian). The proposed methodology can be applied to any existing GK code. We establish a hierarchy of reduced GK Vlasov-Maxwell equations using the generic variational formulation. Then, we
    derive and include the models implemented in ORB5 [1] and GENE inside this hierarchy. At the computational level, detailed verification of global electromagnetic test cases based on the CYCLONE are considered, including a parametric $\beta$-scan covering the transition between the ITG to KBM and the spectral properties at the nominal $\beta$ value [2]. Joint work with T. Gorler, A. Bottino and E. Sonnendrucker.
    [1] N. Tronko, A. Bottino and E. Sonnendrucker, Phys. of Plasmas 23, 082505 (2016).
    [2] T. Gorler, N. Tronko, A. Bottino et al, Phys. of Plasmas 23, 072503 (2016).

  • Tao Xiong, On the asymptotic limit of the Vlasov-Maxwell system with a strong magnetic field.
    PDF - 3.9 Mo

    In the context of strongly magnetized plasma, the 6D Vlasov-Maxwell system can be reduced to a 4D drift-kinetic model, by averaging over the gyroradius of charged particles and assuming a uniform external magnetic field. In the plane perpendicular to the magnetic field, the plasma is governed by the 2D guiding-center model [Filbet and Yang, hal-01068223]. In this talk, we will incorparate in the self-consistent magnetic field and a 3D asymptotic model is formally derived by following [Degond and Filbet, hal-01326162]. Compared to the 2D guiding-center model,our new model illustrates the effect of the self-consistent magnetic filed in the velocity direction parallel to the strong external magnetic lines. A Hermite WENO scheme is used to simulate the new 3D model.
    Instabilities created by the magnetic induction can be clearly observed by comparing to the 2D guiding-center model.

Dernière mise à jour le 16-11-2016