## Section: New Results

### Development of numerical methods

Participants : Morgane Bergot, Anaïs Crestetto, Nicolas Crouseilles, Pierre Glanc, Michel Mehrenberger, Hocine Sellama, Eric Sonnendrücker, Christophe Steiner.

The work [19] is devoted to the numerical simulation of the Vlasov equation in the fluid limit using particles. To that purpose, we first perform a micro-macro decomposition as in [53] where asymptotic preserving schemes have been derived in the fluid limit. In [53] , a uniform grid was used to approximate both the micro and the macro part of the full distribution function. Here, we modify this approach by using a particle approximation for the kinetic (micro) part, the fluid (macro) part being always discretized by standard finite volume schemes. There are many advantages in doing so: $\left(i\right)$ the so-obtained scheme presents a much less level of noise compared to the standard particle method; $\left(ii\right)$ the computational cost of the micro-macro model is reduced in the fluid regime since a small number of particles is needed for the micro part; $\left(iii\right)$ the scheme is asymptotic preserving in the sense that it is consistent with the kinetic equation in the rarefied regime and it degenerates into a uniformly (with respect to the Knudsen number) consistent (and deterministic) approximation of the limiting equation in the fluid regime.

In [39] we present finite volumes schemes for the numerical approximation of the one-dimensional Vlasov-Poisson equation (FOV CEMRACS 2011 project). Stability analysis is performed for the linear advection and links with semi-Lagrangian schemes are made. Finally, numerical results enable to compare the different methods using classical plasma test cases.

In [40] , we test an innovative numerical scheme for the simulation of the guiding-center model, of interest in the domain of plasma physics, namely for fusion devices. We propose a 1D Discontinuous Galerkin (DG) discretization, whose basis are the Lagrange polynomials interpolating the Gauss points inside each cell, coupled to a conservative semi-Lagrangian (SL) strategy. Then, we pass to the 2D setting by means of a second-order Strang splitting strategy. In order to solve the 2D Poisson equation on the DG discretization, we adapt the spectral strategy used for equally-spaced meshes to our Gauss-point-based basis. The 1D solver is validated on a standard benchmark for the nonlinear advection; then, the 2D solver is tested against the swirling deformation ow test case; finally, we pass to the simulation of the guiding-center model, and compare our numerical results to those given by the Backward Semi-Lagrangian method.

In [44] we have developed the guiding-center model in polar coordinates; numerical issues/difficulties can be tackled in such a test case which thus may be viewed as a first intermediate step between a classical center guide simulation in a 2D cartesian mesh and a 4D drift kinetic simulation.

In [25] and [28] , we are interested in the numerical solution of the collisionless kinetic or gyrokinetic equations of Vlasov type needed for example for many problems in plasma physics. Different numerical methods are classically used, the most used is the Particle In Cell method, but Eulerian and Semi-Lagrangian (SL) methods that use a grid of phase space are also very interesting for some applications. Rather than using a uniform mesh of phase space which is mostly done, the structure of the solution, as a large variation of the gradients on different parts of phase space or a strong anisotropy of the solution, can sometimes be such that it is more interesting to use a more complex mesh. This is the case in particular for gyrokinetic simulations for magnetic fusion applications. We develop here a generalization of the Semi-Lagrangian method on mapped meshes. Classical Backward Semi-Lagrangian methods (BSL), Conservative Semi-Lagrangian methods based on one-dimensional splitting or Forward Semi-Lagrangian methods (FSL) have to be revisited in this case of mapped meshes. We consider here the problematic of conserving exactly some equilibrium of the distribution function, by using an adapted mapped mesh, which fits on the isolines of the Hamiltonian. This could be useful in particular for Tokamak simulations where instabilities around some equilibrium are investigated. We also consider the problem of mass conservation. In the cartesian framework, the FSL method automatically conserves the mass, as the advective and conservative form are shown to be equivalent. This does not remain true in the general curvilinear case. Numerical results are given on some gyrokinetic simulations performed with the GYSELA code and show the benefit of using a mass conservative scheme like the conservative version of the FSL scheme. Inaccurate description of the equilibrium can yield to spurious effects in gyrokinetic turbulence simulations. Also, the Vlasov solver and time integration schemes impact the conservation of physical quantities, especially in long-term simulations. Equilibrium and Vlasov solver have to be tuned in order to preserve constant states (equilibrium) and to provide good conservation property along time (mass to begin with). Several illustrative simple test cases are given to show typical spurious effects that one can observes for poor settings. We explain why Forward Semi-Lagrangian scheme bring us some benefits. Some toroidal and cylindrical GYSELA runs are shown that use FSL.

In [12] we present the Semi-Lagrangian method which is composed by essentially two ingredients : the computation of the characteristics along which the distribution function is constant and the interpolation step. We analyse high order schemes in time based on directional splitting, which are a succession of linear transport steps. We then study the Semi-Lagrangian methods in this particular case and we make the link between different formulations. We also obtain a convergence theorem for the Vlasov-Poisson system in this framework, which remains valid in the case of small displacements. We then develop this type of methods in a more general framework, by using one dimensionnal conservative splitting. We also consider a discontinuous Galerkin variant of such schemes. In a last part, we study the gyroaverage operator which appears in plasma physics by taking care of finite Larmor radius corrections. Finally, we discuss the problematic of zero discrete divergence which gives a compatibility between field computations and the numerical method of transport.