## Section: New Results

### Numerical analysis

Participants : Martin Campos Pinto, Nicolas Crouseilles, Michel Mehrenberger, Eric Sonnendrücker.

#### Analysis of numerical methods for the Vlasov-Poisson system

In [47] , we derive the order conditions for fourth order time splitting schemes in the case of the $1D$ Vlasov-Poisson system. Computations to obtain such conditions are motivated by the specific Poisson structure of the Vlasov-Poisson system : this structure is similar to Runge-Kutta-Nyström systems. The obtained conditions are proved to be the same as RKN conditions derived for ODE up to the fourth order. Numerical tests are performed and show the benefit of using high order splitting schemes in that context.

In [37] , we prove enhanced error estimates for high order semi-lagrangian discretizations of the Vlasov-Poisson equation. It provides new insights into optimal numerical strategies for the numerical solution of this problem. The new error estimate $O(min(\frac{\Delta x}{\Delta t},1)\Delta {x}^{p}+\Delta {t}^{2})$ is based on advanced error estimates for semi-lagrangian schemes, also equal to shifted Strang schemes, for the discretization of the advection equation.

#### Analysis of a new particle method with deformable shapes

Particle methods are known to be simple and efficient in most practical cases, however they suffer from weak convergence properties: they only converge in a strong sense when the particles present an extended overlapping (i.e., when the number of overlapping particles tends to infinity as the mesh size $h$ of their initialization grid tends to 0), and additional constraints such as vanishing moments. In practice, extended particle overlapping can be expensive and it involves an additional parameter to be optimized, such as the overlapping exponent $q<1$ for which the particles radius behaves like ${h}^{q}$. In PIC codes for instance, extended overlapping requires increasing the number of particles per cell together with the number of cells, which determine the radius of the particles. In many practical cases such conditions are not met, which leads to strong oscillations in the solutions. To smooth out the oscillations some methods (like the Denavit redeposition scheme, recently revisited as a Forward semi-Lagrangian scheme) use periodic remappings, but frequent remappings introduce unwanted numerical diffusion which seems to contradict the benefit of using low-diffusion particle schemes. Moreover, the vanishing moment condition prevents high orders to be achieved with positive particles.

In [44] we present a new class of particle methods with deformable shapes for transport problems that converge in the supremum norm without requiring remappings, extended overlapping or vanishing moments for the particles. UIndeed, unlike the classical error analysis based on a smoothing kernel argument, our estimates hold for any particle collection with Lipschitz smoothness and compact supports that have the same scale than their initialization grid. Our results are threefold. On the theoretical side we first show that for arbitrarily smooth characteristic flow, high order convergence rates are obtained by deforming the particles with local polynomial mappings. On the practical side we provide an explicit implementation of the first order case: the resulting linearly-transformed particle (LTP) scheme consists of transporting the particle centers along the numerical flow, together with finite difference approximations of the local Jacobian matrices of the flow. For the fully discrete scheme we establish rigorous a priori error estimates and demonstrate the uniform boundedness of the particle overlapping. Finally, we describe an adaptive multilevel version of the LTP scheme that includes a local correction filter for positivity-preserving approximations.

In [45] we apply the LTP method to the 1+1d Vlasov-Poisson problem with a simple deposition scheme and show that deforming the particles helps removing the noise traditionally observed with standard PIC schemes.

#### Two-Scale Asymptotic-Preserving issues

In the submitted paper [48] , we build a Two-Scale Macro-Micro decomposition of the Vlasov equation with a strong magnetic field. This consists in writing the solution of this equation as a sum of two oscillating functions with circonscribed oscillations. The first of these functions has a shape which is close to the shape of the Two-Scale limit of the solution and the second one is a correction built to offset this imposed shape. The aim of such a decomposition is to be the starting point for the construction of Two-Scale Asymptotic-Preserving Schemes.

The aim of using Two-Scale Asymptotic-Preserving Schemes is first, to deal efficiently with long time scales with solutions having high frequency oscillations and second, to manage the transition between different regimes, in a unified framework.

The aim of a new starting project is to test on a simplified model the Two-Scale Asymptotic-Preserving Schemes. The model, a two dimensional in phase space Vlasov-Poisson equation with small parameter, is used for a long time simulation of a beam in a focusing channel. This work was already done in [71] in the case where the solution is approximated by the two scale limit. The goals are first to improve this approximation, by going further, to the first order one, and secondly, to replace this approximation by an exact decomposition, using the macro-micro framework. This last approach will permit to treat the case of a not necessary small parameter.

In order to accomplish the first task we started to write a PIC code which is to be integrated in SeLaLib.