Section: New Results

Development of semi-Lagrangian methods

Participants : N. Crouseilles, P. Glanc, A. Hamiaz, S. Hirstoaga, M. Mehrenberger, J. Petri, E. Sonnendrücker, C. Steiner.

The development of numerical methods - here semi-Lagrangian schemes for plasma physic applications- is continued and strengthened in the context of the on-going library Selalib. We intend to improve the robustness of the numerical tools in order to be prepared for future more realistic test problems.

We have developed a 1D x 1D Vlasov-Poisson solver on GPU using optimized FFT of CUDA and applied it on KEEN waves test case, which needs a fine resolution in velocity [46] . An efficiency of 100 Gflops on 4096x4096 grid is obtained while using single precision, and about 30 GFlops on a 2048x3048 grid using double precision. The approach is valid: implementation effort is reduced, because we rely on external optimizations and the speed-up is quite impressive (only 1 or 2 GFlops were obtained using CPU). We emphasize that FFT is used for the implementation but not (necessarily) for the numerical method. Classical methods like splines or arbitrary high order odd Lagrange interpolation are used, as they can be fitted in this framework. In order to reduce mass conservation issues while using single precision, a delta-f method is validated. The limitation is here the grid size; we were not able to run the code for bigger sizes. We then developed other strategies based on non uniform grids in velocity with cubic splines and two grid strategies (M. Mehrenberger, N. Crouseilles, E. Sonnendrücker, B. Afeyan High-Order Numerical Methods for KEEN Wave Vlasov-Poisson Simulations, Poster at PPPS, 16-21 June 2013, San Francisco) and with the semi-Lagrangian discontinuous Galerkin (SLDG) scheme (C. Steiner, M. Mehrenberger, A semi-Lagrangian discontinuous Galerkin scheme for Vlasov-Poisson equation, poster at Vlasovia, 25-28 November 2013, Nancy). Integration of the code in Selalib with upgraded interface to deal with non uniform grids has been thought but remains to be done. Thanks to the MPI Parallelization of the Selalib code, we should be able to run the code for more interesting physical parameters, in particular, when the drive amplitude goes to zero, which leads to even more localized delta-f function in velocity.

Considering the SLDG scheme, we were able to prove a super convergence property in the case of constant linear advection [48] .

We continue our work on polar grids, which are intermediate, between cartesian and general curvilinear grids. We have revisited a diocotron simulation previously done with the PIC method [75] by using a (classical) semi-Lagrangian approach. A detailed study of boundary conditions, energy and mass conservation as well as linear growth rates is performed and validated with teh code [33] . We then have extended the code to a first drift kinetic simulation (N. Crouseilles, P. Glanc, S. Hirstoaga, E. Madaule, M. Mehrenberger, J. Pétri, Semi-Lagrangian simulations on polar grids: from diocotron instability to ITG turbulence, poster at Vlasovia, 25-28 November 2013, Nancy) using at first the classical cubic splines method and then a new 2D conservative method, called CSL2D (conservative semi-Lagrangian 2D), based on mesh intersections and displacement of volumes [11] . For the latter method to work, we had to take care of the Jacobian and we used a delta-f method, in order to treat more easily non zero boundary conditions. We have benefitted from previous experience on the FSL2D (forward semi-Lagrangian) method. Again, the integration in Selalib is under development. Linear growth rate is here validated, by solving numerically the dispersion relation using recent results of [21] .

In order to deal with more complex geometries or to consider field aligned coordinates, we work on generalizing existing methods for curvilinear grids. Guiding center simulations have been successfully performed with the classical cubic splines method and a finite element solver for the Poisson equation developed by A. Back [32] . Further works concern integration in Selalib, in a more modular way. This should help the comparison with other methods as for example the recent CSL2D method [11] but also the previous CSL1D method [5] .