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Section: New Results

Keywords : Vlasov-Maxwell equations, Vlasov-Poisson equations, Maxwell equations, Propagation speed, Guiding center approximation.

Existence and other theoretical results

Participants : Mihai Bostan, Nicolas Crouseilles, Sever Hirstoaga, Simon Labrunie, Jean Roche, Eric Sonnendrücker.

We have established several existence and uniqueness results for collisionless kinetic models, the Vlasov-Poisson and Vlasov-Maxwell equations of plasma physics. We also investigated different asymptotic regimes for the Vlasov-Maxwell equations. Finally, this section includes recent results concerning the convergence of the numerical solution towards the solution of the model (Maxwell or Vlasov).

Asymptotic regimes for the Vlasov-Poisson and Vlasov-Maxwell equations

The papers [20] , [19] are devoted to the asymptotic analysis of a system of PDEs describing the evolution of charged particles. The unknown are the distribution function of a particle population and the electro-magnetic field. The particles are subject to collisional mechanisms and to the action of electro-magnetic forces. The latter are defined in a self-consistent way by the Maxwell equations. We are interested in hydrodynamic limits where the relaxation effects induced by the collisional processes are strong enough and force the distribution function to tend towards an equilibrium state. Hence, in such a regime the behavior of the particles can be described by means of a finite set of macroscopic quantities, that is certain averages with respect to momentum of the distribution function. We distinguish two asymptotic regimes:

- the high-field regime corresponds to a situation where the force field has the same order as the collision term [19] ,

- the low-field regime corresponds to a situation where the convection and the force field are also singular terms within the equations, but at lower order than the leading contribution of the collisions [20] .

Roughly speaking, the latter regime leads to convection-diffusion limit equations, while the former yields a purely hyperbolic model. The question has been pointed out by Poupaud [76] motivated by the modeling of semi-conductors devices.

In [17] we investigate the homogenization of the one dimensional Vlasov-Maxwell system. We indicate the convergence rate for the electric fields and establish weak convergence for the particle densities. In the non relativistic case we compute explicitly the limit solution. The theoretical results are illustrated by some numerical simulations.

An important problem we have been interesting in for some time in the context of models for magnetic fusion is the gyrokinetic limit of the Vlasov equation in a large non uniform magnetic field. More precisely, we consider the Vlasov equation which describes the dynamics of charged particles in an external electromagnetic field, in terms of a distribution function. When the magnetic field is assumed to be large, the rotation period of the particles around the magnetic field lines becomes small. Since this time scale is very detrimental for the stability of numerical schemes, our aim is to find a model where this time scale is removed. There is already a large literature on the subject in physics and also some preliminary mathematical results in simplified cases. We got some new results in this direction during the past year.

In [18] we study the finite Larmor radius regime for the Vlasov-Poisson equations with strong external magnetic field. The derivation of the limit model follows by formal expansion in power series with respect to a small parameter. If we replace the particle distribution by the guiding center distribution of the Larmor circles the limit of these densities satisfies a transport equation, whose velocity is given by the gyro-average of the electric field. We justify rigorously the convergence towards the above model and we investigate the well-posedness of it.

On the other hand in [48] we consider a new scaling of the Vlasov equation in order to derive a drift-kinetic model. To this end, we first make dimensionless cyclotron period appear in the scaled Vlasov equation. Then we decompose the particle velocity into the mean velocity and its random part and we deduce a system of two equations giving the evolution of the new distribution function and the mean velocity. Afterward, an asymptotic analysis is made for this model and a formal derivation of the gyrokinetic model (in a five dimension phase space) is thus obtained.

Convergence studies of numerical methods

In [16] we present a particle method for solving numerically the one dimensional Vlasov-Maxwell equations. This method is based on the formulation by characteristics. We perform the error analysis and we investigate the properties of this new scheme. The main point here is that the computation of the electric field do not require neither the calculation of the charge and current densities nor the explicit resolution of the Maxwell equations. In fact such schemes rely only on the approximation of the characteristics and the electric field, which are generally more regular than the particle distribution (think that f can be a L1 function or even a measure). We do not need to ask for the smoothness of the particle distribution of the exact solution since no interpolation is performed.

In [43] , a numerical analysis of the semi-Lagrangian scheme applied to the reduced Vlasov-Maxwell model is performed. This model has been recently introduced in the literature for studying laser-plasma interaction and some theoretical results has been established so far. This work is devoted to the proof of convergence of a semi-Lagrangian numerical solution towards the unique solution of the problem. The main interest of this work consists in the fact that a two dimensional advection is studied whereas split problems were studied so far in this framework. Numerical experiments proved the good behavior of the scheme even if the use of higher order interpolation operators is strongly advised.

We (in collaboration with P. Ciarlet) performed in [22] a numerical analysis of the time-dependent Maxwell equations with elliptic correction and with divergence constraint. These slightly generalized formulations are used to avoid a numerical drift of the numerical solution to Maxwell's equations when the continuity equation $ \partial$t$ \varrho$+ div J= 0 (or a discrete version of it) is not exactly satisfied. This issue may arise when simulating plasmas, or more generally systems of charged particles, by PIC or Eulerian (Vlasovian) codes. This is (to our knowledge) the first comprehensive numerical analysis of the various nodal finite element methods for Maxwell's equations, including the treatment of both the divergence condition div E= $ \varrho$/ $ \varepsilon$0 and the geometrical singularities. We concentrated on continuous (nodal) finite elements because they do not create spurious discontinuities of the force experienced by the particles. We obtained error estimates for the elliptic correction and for the mixed (saddle-point) formulation of the constraint; the two approaches are not equivalent at discrete level when nodal elements are used, unlike what happens with edge elements. The estimates are valid for the three main versions of nodal element method for Maxwell's equations: the basic one (which can be used when the domain is regular or convex), the weighted regularization and the singular complement method (which are used to take geometrical singularities into account).

Domain decomposition for the resolution of nonlinear equations

The principal objective of this work was to give a result of existence and present a numerical analysis of weak solutions for the following quasi-linear elliptic problem in one and two dimensions:

Im21 ${~~~~~\mfenced o={ \mtable{...}}$(2)

where A is a second order derivatives operator in one dimension and the Laplace operator in two dimensions, G, F are Caratheodory non negative functions. The function f is given finite and non negative. The domain Im22 ${\#937 \#8834 \#8477 ^N,~~N=1,2}$ is open and bounded.

Such problems arise from biological, chemical and physical systems and various methods have been proposed for study the existence, uniqueness, qualitative properties and numerical simulation of solutions.

Another approach studied was the numerical approximation of the solution of the problem. The most important difficulties are in this approach the uniqueness and the blowup of the solution. The general algorithm for numerical solution of this equation is one application of the Newton method to the discretized version of the problem. However, in our case the matrix which appears in the Newton algorithm can be singular. To overcome this difficulty we introduced a domain decomposition to compute an approximation of the iterates by the resolution of a sequence of problems of the same type as the original problem in subsets of the given computational domain. This domain decomposition method coupled with a Yosida approximation of the non-linearity allows us to compute a numerical solution. In the 2-d case we consider the case where the data belong to L1( $ \upper_omega$) and the gradient dependent non-linearity is quadratic. We show the existence and present a numerical analysis of a weak solution. We apply this method to better understand the nickel-iron electrodeposition process, we have developed one-dimensional numerical model. This model addresses dissociation, diffusion, electromigration, convection and deposition of multiple ion species. To take account of the anisotropic behavior of the solution we introduce a domain decomposition numerical method. Simulations with experimental data show that our model can predict characteristic features of the nickel-iron system. New numerical analysis and simulation are published in   [13][14] .


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