## Section: New Results

### Development of Vlasov solvers

Participants : Nicolas Besse, Martin Campos Pinto, Nicolas Crouseilles, Alain Ghizzo, Michaël Gutnic, Matthieu Haefelé, Olivier Hoenen, Sébastien Jund, Guillaume Latu, Michel Mehrenberger, Thomas Respaud, Stéphanie Salmon, Eric Sonnendrücker.

#### Two-dimensional solvers

In [28] , a Forward semi-Lagrangian method has been tested and validated. The main difference with the classical Backward semi-Lagrangian method are twofold. First, since the characteristics curves along with the unknown is constant are followed forward in time, this method then enables the use of classical high order time discretization (such as Runge-Kutta 4 algorithms). Second, the remapping (or the deposition) step which is based on cubic spline polynomials, enables to reconstruct the distribution function on a uniform mesh using the particles which have moved during one time step. Two main improvements have been performed during the present year. First, a Cauchy-Kovaleskaya procedure has been tested and validated. This procedure enables a completely explicit way to compute the end of the characteristics, as opposite to Runge-Kutta algorithms which needs intermediate and costly steps. Second, since the method has a lot of similarity with the PIC methods, a charge conservation preserving scheme has been implemented for both Runge Kutta and Cauchy-Kovaleskaya time algorithms.

Another strategy based on conservative semi-Lagrangian methods has been implemented and tested. Whereas semi-Lagrangian methods compute the distribution function on the grid points, conservative methods considers an average of the unknown on each cell. This approximation enables the solving of multi-dimensional problems by a successive solving of one-dimensional equations. This work started last year and has been achieved this year (see [27] ). New formulations of conservative semi-Lagrangian methods have been introduced; this unified framework enables in particular to recover several numerical methods available in the literature (PFC, PPM, PSM). Moreover, new filters are proposed to ensure the unknown respects the extrema principles and limit the spurious oscillations created by high order interpolation operators.

A 2D version of LOSS that uses GPGPU computing units has been designed. On a personal computer, we obtain speedups up to 80 using the GPU versus the conventional processor. The domain decomposition strategy is the same as in our previous MPI version of LOSS. Porting codes on GPGPU is cost-effective, because high speedups could be achieved on a low-cost machine. But it is also time-consuming for the developer.

The conservative scheme PFC (developed in [76] ) and a numerical scheme for a reformulation of the Poisson equation (similar to that in [72] ) were tested in a slight different framework: the Vlasov-Poisson system with a collisional (BGK) term, for a two-species plasma with realistic electron-to-ion mass ratio and dimensionless Debye length (this problem is relevant for the numerical study of plasma-wall interactions).

#### CALVI platform

The aim of the platform is to change the way numerical methods are implemented and tested. It has been initiated because most of the researchers of the CALVI project develop new numerical methods for almost the same equations. Until now, every researchers implemented their methods as stand-alone C or Fortran applications. So, each researcher, for each code, has to implement the validation process by himself without using previous implementation done by himself or another member of the project. The platform move the implementation from stand-alone application to a module oriented one. Thanks to standardized application programing interfaces (API), the different numerical methods can be swapped between them and can be validated within a common skeleton. This common skeleton plus the standard API is actually the platform. A better reuse of existing modules is expected as well as an increased efficiency in numerical methods implementation.

The whole implementation has been refactored this year according to remarks made by the team. So the python package called 'vlasy', which stands for 'Vlasov' + 'Python', is born. Lot's of things that were accessible to the user are now embedded in Python classes within the package. As a result, the user access objects at higher level of abstraction, thus making the usage easier. Some unit tests have been introduced in the skeleton part of the package and solver validation process is also implemented as unit tests. Two Vlasov solvers have been added as well as 4 test cases. The vlasy package is already used at CEA Cadarache in a physics team.

#### Four-dimensional solvers

A four-dimensional cubic splines interpolation has been validated in the framework of the backward semi-Lagrangian method on the Vlasov-Poisson equations. This study was motivated to study the validity of the time splitting of the method when non-conservative advection terms are involved. The method is currently tested on more complex problems and appears to be competitive from a CPU time point of view. This approach benefits from the Local Splines strategy which enables a decomposition domain well suited for parallel implementation. This non split scheme has been recently added in the Gysela code. It provides a reference numerical scheme that avoids some approximations that come with splitting. Furthermore, it will allow in the future for alternative and more accurate ways to track the feet of the characteristics.

#### Adaptive solvers

In [40] a new adaptive semi-Lagrangian scheme based on wavelet approximations for solving transport equations with underlying smooth flow is presented. Inspired by a recent method of Besse, Filbet, Gutnic, Paun and Sonnendrücker, this new approach differs in the fact that it is mostly driven by the notion of good adaptation of a wavelet tree to a given function. Moreover it comes with guaranteed error estimates. In a previous joint work with Mehrenberger, we had designed a first adaptive semi-Lagrangian scheme based on multilevel, hierarchical meshes. The method consisted in predicting a new adaptive mesh for every time step by using a low-cost strategy, and next readapt it once according to the smoothness of the transported numerical solutions. By a rigorous analysis we could prove that our scheme had a prescribed accuracy, achieved by applying the prediction and correction algorithms only once per time step.

The present scheme implements similar ideas, but now in the framework of interpolatory wavelets. For this purpose we translate the property of being (strongly) well-adapted to a given function in the context of wavelet trees, and show that it is (weakly) preserved by a low-cost prediction algorithm which transports wavelet grids along any smooth flow. As a consequence, error estimates can be established for the resulting “predict and readapt” scheme under the essential assumption that the flow underlying the transport equation, as well as its numerical approximation, is a stable diffeomorphism. One complexity result is stated in addition.

#### Electromagnetic Particle In Cell (PIC) solvers

This project funded by ANR proposes to develop and compare Finite Element Time Domain (FETD) solvers based on the one hand on high order H(curl) conforming elements and on the other hand on high order Discontinuous Galerkin (DG) finite elements and investigate their coupling to the particles. These self consistent relativistic PIC solvers will be the first of this kind in this context and promise to have an impact for the simulation of realistic problems in accelerator and plasma physics.

During the last year we further developed a previous work on charge conserving Finite Element PIC schemes on general grids [51] . We proposed a general mathematical formulation for charge conserving finite elements Maxwell solvers coupled with particle schemes. In particular, we identified the finite-element continuity equations that must be satisfied by the discrete current sources for several classes of time domain Vlasov-Maxwell simulations to preserve the Gauss law at each time step, and proposed a generic algorithm for computing such consistent sources. Since our results cover a wide range of schemes (namely curl-conforming finite element methods of arbitrary degree, general meshes in 2 or 3 dimensions, several classes of time discretization schemes, particles with arbitrary shape factors and piecewise polynomial trajectories of arbitrary degree), we believe that they provide a useful roadmap in the design of high order charge conserving FEM-PIC numerical schemes.

A parallel code (Spin) has been developed to solve bigger test cases using the same numerical method. Spin solves Maxwell equations on hybrid meshes in 2D space (quadrangles and triangles) using edge finite elements. A Poisson solver, that uses also edge finite elements, is called once at the beginning of each run to find the initial self-consistent electric field. The work distribution on processors is well balanced on several dozens of cores in this 2D version. The 3D version of Spin is currently under development.