Résumés

Nicolas Besse: "Adaptive multiresolution semi-Lagrangian discontinuous Galerkin methods for the Vlasov equation"

In this talk, we present an adaptive numerical schemes for the Vlasov equation by combining discontinuous Galerkin discretization, wavelet-multiresolution analysis and semi-Lagrangian time integration. Both multi-wavelets and discontinuous Galerkin rely on a local polynomial basis. The schemes are tested and validated using Vlasov–Poisson equations for plasma physics and astrophysics.file

Jacques Blum: "Algorithmes de contrôle optimal pour l'identification de l'équilibre du plasma et pour l'optimisation de scénarios dans un Tokamak"

 Les méthodes du contrôle optimal des systèmes régis par des e.d.p.seront utilisées pour résoudre des problèmes d’identification  et  d’optimisation de décharges plasma dans un Tokamak.On traitera successivement:

1) l’identification de la frontière du plasma à partir des mesures magnétiques: on exposera deux méthodes, l’une basée sur la minimisation de la fonctionnelle de Kohn-Vogelius, l’autre sur la décomposition du flux en harmoniques toroïdales et l’on montrera comment ces techniques ont été appliquées à divers tokamaks (WEST, TCV, AUG, ITER)

2) l’identification de la densité de courant du plasma et de la configuration d’équilibre du flux poloidal: on présentera des techniques de contrôle optimal avec régularisation de Tikhonov pour identifier le profil de courant à l’intérieur du plasma et sa frontière libre, en utilisant les mesures magnétiques, interférométriques et polarimétriques. On comparera la qualité de l’identification en fonction des diagnostics utilisés.

3) l’optimisation de scenarios pour réaliser une évolution donnée de la frontière du plasma, passant d’une configuration limiter à une configuration avec un point-X. On appliquera les techniques du contrôle optimal, celles de la régularisation ainsi que la méthode d’optimisation séquentielle quadratique SQP pour calculer les tensions optimales à appliquer aux circuits du système de champ poloidal. On montrera des applications à ITER.

4) utilisant les techniques développées ci-dessus, on montrera comment on peut contrôler l’équilibre du plasma dans des Tokamaks (WEST, TCV,..) en appliquant à la fois un contrôle en boucle ouverte et un contrôle feedback en vue de la stabilisation du déplacement vertical (VDE).file

Frédérique Charles: "From particle methods to hybrid semi-Lagrangian schemes for Vlasov-Poisson"

Particle methods for transport equations consist in pushing forward particles along the characteristic lines of the flow, and to describe then the transported density as a sum of weighted and smoothed particles. Conceptually simple, standard particle methods have the main drawback to produce noisy solutions or to require frequent remapping.
In this talk we present two classes of particle methods which aim at improving the accuracy of the numerical approximations with a minimal amount of smoothing. The idea of the Linearly Transformed Particle method is to transform the shape functions of particles in order to follow the local variation of the flow. This method has been adapted and analyzed for the Vlasov-Poisson system and for a compressible aggregation equation. In both cases the error estimate is improved compared to classical particle methods, with the gain of a strong convergence of the numerical solution. However, for long remapping periods, shapes of particles could become to much stretched out. The second method solves this problem of locality by combining a backward semi-Lagrangian approach and local linearizations of the flow. The convergence properties are improved and validated by numerical experiments.file 

Georges-Henri Cottet: "Semi-Lagrangian particles for Vlasov-Poisson, or how to compute 6D astrophysics on a laptop"

Since the early 80s particle methods for plasma physics and incompressible flows have followed parallel avenues, except for questions related to their numerical analysis. Implementation issues were seen from rather different viewpoints. 
I will try to show how to bring closer together the two approaches,  with semi-lagrangian particles which combine locality, conservation properties and accuracy. I will illustrate this on 4D and 6D exemples in plasma and astrophysics, with comparisons with pure particle methods and multi-resolution eulerian techniques.file

Giorgio Giorgiani: "A high-order non-aligned approach for plasma transport simulations in tokamaks"

The scale separation introduced by the magnetic field in tokamak plasmas produces different dynamics in the parallel and perpendicular directions.The resulting governing equation are strongly anisotropic, requiring specifically designed numerical schemes. The classic solution consists in aligning the computational mesh with the magnetic field lines: however, this technique is often non-satisfactory. We propose in this work a novel approach based on a hybrid discontinuous Galerkin scheme on non-aligned unstructured meshes. The cross field numerical diffusion is reduced employing high-order interpolations. Some tests in 2D are presented on tokamak geometries and realistic physical parameters.file

Virginie Grandgirard: "Gyrokinetic simulations of magnetic fusion plasmas: numerical challenges"

In magnetic fusion devices, the power gain strongly increases with the energy confinement time. As a matter of fact, the quality of the plasma energy confinement largely determines the size and therefore the cost of a fusion reactor. This confinement time turns out to be mainly governed by the plasma turbulence. Understanding its origin and properties in view of its possible control is one of the critical issues in fusion science. The inhomogeneities in density, temperature, and magnetic field place the plasma naturally out of thermodynamical equilibrium, and tend to excite several micro-instabilities over a wide spectral range. These plasmas exhibit a low collisionality so that conventional fluid models are questionable and kinetic descriptions are more appropriate. In such first-principle descriptions of plasmas, the six dimensional evolution equation for the distribution function "Vlasov or Fokker-Planck equations" is solved for each species, coupled to the self-consistent equations for the electromagnetic elds, namely Maxwell's equations. Fortunately, as far as turbulent fluctuations are concerned, they develop at much lower typical frequencies than the high frequency cyclotron motion. Therefore, this 6D problem can be reduced to a 5D, known as the gyrokinetic model [1]. But even with this dimensional reduction solving 5D gyrokinetic equations for each specie reveals extremely challenging.
This presentation will be a brief overview of the different assumptions, different numerical approaches existing now in first-principle gyrokinetic codes which have been developed for this stage [2]. A special focus will be done on the GYSELA code [3] developed at IRFM/CEA institut which is based on a semi-Lagrangian approach. Gyrokinetic codes require state-of-the-art high performance computing. We will discuss the challenges to prepare the GYSELA code for future exascale simulations. Finally, we will discussed the strong numerical challenges associated to core-edge simulations which is an important issue for next generation of gyrokinetic simulations.file

References

[1] A.J. Brizard and T.S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Mod. Phys. (2007) 2.
[2] V. Grandgirard and Y. Sarazin, Gyrokinetic simulations of magnetic fusion plasmas, Panoramas et syntheses (2013) 39-40, pp 91-176.
[3] V. Grandgirard et al., A 5D gyrokinetic full-f global semi-lagrangian code for flux-driven ion turbulencesimulations, Computer Phys. Comm., (2016) 207.