CALVI was created in July 2003.

It is a project associating Institut Elie Cartan (IECN, UMR 7502, CNRS, INRIA and Université Henri Poincaré, Nancy), Institut de Recherche Mathématique Avancée (IRMA, UMR 7501, CNRS and Université de Strasbourg) and Laboratoire des Sciences de l'Image, de l'Informatique et de la Télédétection (LSIIT, UMR 7005, CNRS and Université de Strasbourg) with close collaboration to Laboratoire de Physique des Milieux Ionisés et Applications (LPMIA, UMR 7040, CNRS and Université Henri Poincaré, Nancy).

Our main working topic is modelling, numerical simulation and visualization of phenomena coming from plasma physics and beam physics. Our applications are characterized in particular by their large size, the existence of multiple time and space scales, and their complexity.

Different approaches are used to tackle these problems. On
the one hand, we try and implement modern computing
techniques like
**parallel computing**and
**grid computing**looking for appropriate methods and
algorithms adapted to large scale problems. On the other hand
we are looking for
**reduced models**to decrease the size of the problems in
some specific situations. Another major aspect of our
research is to develop numerical methods enabling us to
optimize the needed computing cost thanks to
**adaptive mesh refinement**or
**model choice**. Work in scientific visualization
complement these topics including
**visualization of multidimensional data**involving large
data sets and
**coupling visualization**and
**numerical computing**.

Launch of ADT SeLaLib

The
**plasma state**can be considered as the
**fourth state of matter**, obtained for example by
bringing a gas to a very high temperature (
10
^{4}
Kor more). The thermal energy of
the molecules and atoms constituting the gas is then
sufficient to start ionization when particles collide. A
globally neutral gas of neutral and charged particles,
called
**plasma**, is then obtained. Intense charged particle
beams, called nonneutral plasmas by some authors, obey
similar physical laws.

The hierarchy of models describing the evolution of charged particles within a plasma or a particle beam includes:

N-body models where each particle interacts
directly with all the others,

kinetic models based on a statistical description of the particles,

fluid models valid when the particles are at a thermodynamical equilibrium.

**Calvi team mainly focuses on kinetic models, but not
exclusively**. In particular, every kind of models are
mathematically analyzed, approximate models are built and
studied and model hierarchies are set out.

In a so-called
**kinetic model**, each particle species
sin a plasma or a particle beam is described by a
distribution function
corresponding to the statistical average of the
particle distribution in phase-space corresponding to many
realisations of the physical system under investigation.
The product
is the average number of particles of the considered
species, the position and velocity of which are located in
a bin of volume
centered around
. The distribution function contains a lot more
information than what can be obtained from a fluid
description, as it also includes information about the
velocity distribution of the particles.

A kinetic description is necessary in collective plasmas
where the distribution function is very different from the
Maxwell-Boltzmann (or Maxwellian) distribution which
corresponds to the thermodynamical equilibrium, otherwise a
fluid description is generally sufficient. In the limit
when collective effects are dominant with respect to binary
collisions, the corresponding kinetic equation is the
*Vlasov equation*

which expresses that the distribution
function
fis conserved along the particle trajectories which
are determined by their motion in their mean
electromagnetic field. The Vlasov equation which involves a
self-consistent electromagnetic field needs to be coupled
to the Maxwell equations in order to compute this field

which describes the evolution of the electromagnetic field generated by the charge and current densities

associated to the charged particles.

When binary particle-particle interactions are dominant
with respect to the mean-field effects then the
distribution function
fobeys the Boltzmann equation

where
Qis the nonlinear Boltzmann collision operator. In
some intermediate cases, a collision operator needs to be
added to the Vlasov equation.

The numerical solution of the three-dimensional Vlasov-Maxwell system represents a considerable challenge due to the huge size of the problem. Indeed, the Vlasov-Maxwell system is nonlinear and posed in phase space. It thus depends on seven variables: three configuration space variables, three velocity space variables and time, for each species of particles. This feature makes it essential to use every possible option to find a reduced model wherever possible, in particular when there are geometrical symmetries or small terms which can be neglected.

Beside this,
**enriching and making more rigorous the model
hierarchy**is also an important challenge which requires
a deep knowledge of different models in use in plasma
physics (e.g. Magneto-Hydro-Dynamics, Laser-Matter
Interaction, Waves In Plasma, etc.) and their connections
with Vlasov-like models.

The mathematical analysis of the Vlasov equation is essential for a thorough understanding of the model as well for physical as for numerical purposes. It has attracted many researchers since the end of the 1970s. Among the most important results which have been obtained, we can cite the existence of strong and weak solutions of the Vlasov-Poisson system by Horst and Hunze , see also Bardos and Degond . The existence of a weak solution for the Vlasov-Maxwell system has been proved by Di Perna and Lions . An overview of the theory is presented in a book by Glassey .

Many questions concerning for example uniqueness or existence of strong solutions for the three-dimensional Vlasov-Maxwell system are still open. Moreover, their is a realm of approached models that need to be investigated. In particular, the Vlasov-Darwin model for which we could recently prove the existence of global solutions for small initial data .

On the other hand, the asymptotic study of the Vlasov equation in different physical situations is important in order to find or justify reduced models. One situation of major importance in tokamaks, used for magnetic fusion as well as in atmospheric plasmas, is the case of a large external magnetic field used for confining the particles. The magnetic field tends to incurve the particle trajectories which eventually, when the magnetic field is large, are confined along the magnetic field lines. Moreover, when an electric field is present, the particles drift in a direction perpendicular to the magnetic and to the electric field. The new time scale linked to the cyclotron frequency, which is the frequency of rotation around the magnetic field lines, comes in addition to the other time scales present in the system like the plasma frequencies of the different particle species. Thus, many different time scales as well as length scales linked in particular to the different Debye length are present in the system. Depending on the effects that need to be studied, asymptotic techniques allow to find reduced models. In this spirit, in the case of large magnetic fields, recent results have been obtained by Golse and Saint-Raymond , as well as by Brenier . Our group has also contributed to this problem using homogenization techniques to justify the guiding center model and the finite Larmor radius model which are used by physicist in this setting , , .

Another important asymptotic problem yielding reduced models for the Vlasov-Maxwell system is the fluid limit of collisionless plasmas. In some specific physical situations, the infinite system of velocity moments of the Vlasov equation can be closed after a few of those, thus yielding fluid models.

The development of efficient numerical methods is essential for the simulation of plasmas and beams. Indeed, kinetic models are posed in phase space and thus the number of dimensions is doubled. Our main effort lies in developing methods using a phase-space grid as opposed to particle methods. In order to make such methods efficient, it is essential to consider means for optimizing the number of mesh points. This is done through different adaptive strategies. In order to understand the methods, it is also important to perform their mathematical analysis. For a few years, we also have been interesting in solvers that use Particle In Cell method. This new issue allows us to enrich some parts of our research activities previously centered on the Semi-Lagrangian approach. We also have been initiating to insert asymptotic analysis products and geometry products within numerical methods in oder to enforce their robustness and their ability to perform long term simulations.

The numerical integration of the Vlasov equation is one of the key challenges of computational plasma physics. Since the early days of this discipline, an intensive work on this subject has produced many different numerical schemes. One of those, namely the Particle-In-Cell (PIC) technique, has been by far the most widely used. Indeed it belongs to the class of Monte Carlo particle methods which are independent of dimension and thus become very efficient when dimension increases which is the case of the Vlasov equation posed in phase space. However these methods converge slowly when the number of particles increases, hence if the complexity of grid based methods can be decreased, they can be the better choice in some situations. This is the reason why one of the main challenges we address is the development and analysis of adaptive grid methods.

Exploring grid based methods for the Vlasov equation, it becomes obvious that they have different stability and accuracy properties. In order to fully understand what are the important features of a given scheme and how to derive schemes with the desired properties, it is essential to perform a thorough mathematical analysis of this scheme, investigating in particular its stability and convergence towards the exact solution.

The semi-Lagrangian method consists in computing a
numerical approximation of the solution of the Vlasov
equation on a phase space grid by using the property of the
equation that the distribution function
fis conserved along characteristics. More precisely,
for any times
sand
t, we have

where are the characteristics of the Vlasov equation which are solution of the system of ordinary differential equations

with initial conditions , .

From this property,
f^{n}being known one can induce a numerical method for
computing the distribution function
f^{n+ 1}at the grid points
consisting in the following two steps:

For all
i,
j, compute the origin of the
characteristic ending at
, i.e. an approximation of
,
.

As

f^{n+ 1}can be computed by interpolating
f^{n}which is known at the grid points at the points
.

This method can be simplified by performing a time-splitting separating the advection phases in physical space and velocity space, as in this case the characteristics can be solved explicitly.

Uniform meshes are most of the time not efficient to solve a problem in plasma physics or beam physics as the distribution of particles is evolving a lot as well in space as in time during the simulation. In this case a variants, called adaptive semi-Lagrangian methods, of semi-Lagrangian methods was set out better to better fit the distribution of particles.

The solutions to Maxwell's equations are
*a priori*defined in a function space such that the
curl and the divergence are square integrable and that
satisfy the electric and magnetic boundary conditions.
Those solutions are in fact smoother (all the derivatives
are square integrable) when the boundary of the domain is
smooth or convex. This is no longer true when the domain
exhibits non-convex
*geometrical singularities*(corners, vertices or
edges).

Physically, the electromagnetic field tends to infinity in the neighbourhood of the re-entrant singularities, which is a challenge to the usual finite element methods. Nodal elements cannot converge towards the physical solution. Edge elements demand considerable mesh refinement in order to represent those infinities, which is not only time- and memory-consuming, but potentially catastrophic when solving time dependent equations: the CFL condition then imposes a very small time step. Moreover, the fields computed by edge elements are discontinuous, which can create considerable numerical noise when the Maxwell solver is embedded in a plasma (e.g. PIC) code.

In order to overcome this dilemma, a method consists in
splitting the solution as the sum of a
*regular*part, computed by nodal elements, and a
*singular*part which we relate to singular solutions
of the Laplace operator, thus allowing to calculate a local
analytic representation. This makes it possible to compute
the solution precisely without having to refine the
mesh.

This
*Singular Complement Method*(SCM) had been
developed
and implemented
in plane geometry.

An especially interesting case is axisymmetric geometry. This is still a 2D geometry, but more realistic than the plane case; despite its practical interest, it had been subject to much fewer theoretical studies . The non-density result for regular fields was proven , the singularities of the electromagnetic field were related to that of modified Laplacians , and expressions of the singular fields were calculated . Thus the SCM was extended to this geometry. It was then implemented by F. Assous (now at Bar-Ilan University, Israel) and S. Labrunie in a PIC–finite element Vlasov–Maxwell code .

As a byproduct, space-time regularity results were obtained for the solution to time-dependent Maxwell's equation in presence of geometrical singularities in the plane and axisymmetric cases , .

In order to set out numerical methods that can be valid even for long term simulations, two strategies may be followed. The first one is to incorporate in them asymptotic analysis concepts allowing to take into account, precisely, only the resulting mean effect of oscillations. This yields Two-Scale-Numerical-Methods which were introduced in and and which constitutes an active research activity.

The second, which also gives rise to an active research activity, consists in incorporating Hamitonian mechanics and symplectic geometry concepts while building numerical schemes.

The applications we consider lead to very large size computational problems for which we need to apply modern computing techniques enabling to use efficiently many computers including traditional high performance parallel computers and computational grids.

The full Vlasov-Maxwell system yields a very large computational problem mostly because the Vlasov equation is posed in six-dimensional phase-space. In order to tackle the most realistic possible physical problems, it is important to use all the modern computing power and techniques, in particular parallelism and grid computing.

An important issue for the practical use of the methods we develop is their parallelization. We address the problem of tuning these methods to homogeneous or heterogeneous architectures with the aim of meeting increasing computing resources requirements.

Most of the considered numerical methods apply a series of operations identically to all elements of a geometric data structure: the mesh of phase space. Therefore these methods intrinsically can be viewed as a data-parallel algorithm. A major advantage of this data-parallel approach derives from its scalability. Because operations may be applied identically to many data items in parallel, the amount of parallelism is dictated by the problem size.

Parallelism, for such data-parallel PDE solvers, is achieved by partitioning the mesh and mapping the sub-meshes onto the processors of a parallel architecture. A good partition balances the workload while minimizing the communications overhead. Many interesting heuristics have been proposed to compute near-optimal partitions of a (regular or irregular) mesh. For instance, the heuristics based on space-filing curves give very good results for a very low cost.

Adaptive methods include a mesh refinement step and can highly reduce memory usage and computation volume. As a result, they induce a load imbalance and require to dynamically distribute the adaptive mesh. A problem is then to combine distribution and resolution components of the adaptive methods with the aim of minimizing communications. Data locality expression is of major importance for solving such problems. We use our experience of data-parallelism and the underlying concepts for expressing data locality , optimizing the considered methods and specifying new data-parallel algorithms.

As a general rule, the complexity of adaptive methods requires to define software abstractions allowing to separate/integrate the various components of the considered numerical methods (see as an example of such modular software infrastructure).

Another key point is the joint use of heterogeneous architectures and adaptive meshes. It requires to develop new algorithms which include new load balancing techniques. In that case, it may be interesting to combine several parallel programming paradigms, i.e. data-parallelism with other lower-level ones.

Our general approach for designing efficient parallel algorithms is to define code transformations at any level. These transformations can be used to incrementally tune codes to a target architecture and they warrant code reusability.

Controlled fusion is one of the major prospects for a long term source of energy. Two main research directions are studied: magnetic fusion where the plasma is confined in tokamaks using a large external magnetic field and inertial fusion where the plasma is confined thanks to intense laser or particle beams. The simulation tools we develop can be applied for both approaches.

Controlled fusion is one of the major challenges of the 21st century that can answer the need for a long term source of energy that does not accumulate wastes and is safe. The nuclear fusion reaction is based on the fusion of atoms like Deuterium and Tritium. Deuterium can be obtained from the water of the oceans that is widely available and Tritium can be produced from Lithium directly in a tokamak. Moreover, the reaction does not produce long-term radioactive wastes, unlike today's nuclear power plants which are based on nuclear fission.

Two major research approaches are followed towards the objective of fusion based nuclear plants: magnetic fusion and inertial fusion. In order to achieve a sustained fusion reaction, it is necessary to confine sufficiently the plasma for a long enough time. If the confinement density is higher, the confinement time can be shorter but the product needs to be greater than some threshold value.

The idea behind magnetic fusion is to use large toroidal
devices called tokamaks in which the plasma can be confined
thanks to large applied magnetic field. The international
project ITER

The inertial fusion concept consists in using intense laser beams or particle beams to confine a small target containing the Deuterium and Tritium atoms. The Laser Mégajoule which is being built at CEA in Bordeaux will be used for experiments using this approach.

Nonlinear wave-wave interactions are primary mechanisms by which nonlinear fields evolve in time. Understanding the detailed interactions between nonlinear waves is an area of fundamental physics research in classical field theory, hydrodynamics and statistical physics. A large amplitude coherent wave will tend to couple to the natural modes of the medium it is in and transfer energy to the internal degrees of freedom of that system. This is particularly so in the case of high power lasers which are monochromatic, coherent sources of high intensity radiation. Just as in the other states of matter, a high laser beam in a plasma can give rise to stimulated Raman and Brillouin scattering (respectively SRS and SBS). These are three wave parametric instabilities where two small amplitude daughter waves grow exponentially at the expense of the pump wave, once phase matching conditions between the waves are satisfied and threshold power levels are exceeded. The illumination of the target must be uniform enough to allow symmetric implosion. In addition, parametric instabilities in the underdense coronal plasma must not reflect away or scatter a significant fraction of the incident light (via SRS or SBS), nor should they produce significant levels of hot electrons (via SRS), which can preheat the fuel and make its isentropic compression far less efficient. Understanding how these deleterious parametric processes function, what non uniformities and imperfections can degrade their strength, how they saturate and interdepend, all can benefit the design of new laser and target configuration which would minimize their undesirable features in inertial confinement fusion. Clearly, the physics of parametric instabilities must be well understood in order to rationally avoid their perils in the varied plasma and illumination conditions which will be employed in the National Ignition Facility or LMJ lasers. Despite the thirty-year history of the field, much remains to be investigated.

Our work in modelling and numerical simulation of plasmas and can be applied to problems like laser-matter interaction, the study of parametric instabilities (Raman, Brillouin), the fast ignitor concept in the laser fusion research. Another application is devoted to the development of Vlasov gyrokinetic codes in the framework of the magnetic fusion programme in collaboration with the Department of Research on Controlled Fusion at CEA Cadarache.

Vlasov equation, Vlasov-Poisson and Vlasov-Maxwell systems are also used to model transport of particle beams in accelerators. As a consequence our work also applies to this field. In particular, we work in collaboration with the American Heavy Ion Fusion Virtual National Laboratory, regrouping teams from laboratories in Berkeley, Livermore and Princeton on the development of simulation tools for the evolution of particle beams in accelerators.

Kinetic models like the Vlasov equation can also be applied for the study of large nano-particles as approximate models when ab initio approaches are too costly.

In order to model and interpret experimental results obtained with large nano-particles, ab initio methods cannot be employed as they involve prohibitive computational times. A possible alternative resorts to the use of kinetic methods originally developed both in nuclear and plasma physics, for which the valence electrons are assimilated to an inhomogeneous electron plasma. The LPMIA (Nancy) possesses a long experience on the theoretical and computational methods currently used for the solution of kinetic equation of the Vlasov and Wigner type, particularly in the field of plasma physics.

Using a Vlasov Eulerian code, we have investigated in detail the microscopic electron dynamics in the relevant phase space. Thanks to a numerical scheme recently developed by Filbet et al. , the fermionic character of the electron distribution can be preserved at all times. This is a crucial feature that allowed us to obtain numerical results over long times, so that the electron thermalization in confined nano-structures could be studied.

The nano-particle was excited by imparting a small velocity shift to the electron distribution. In the small perturbation regime, we recover the results of linear theory, namely oscillations at the Mie frequency and Landau damping. For larger perturbations nonlinear effects were observed to modify the shape of the electron distribution.

For longer time, electron thermalization is observed: as the oscillations are damped, the center of mass energy is entirely converted into thermal energy (kinetic energy around the Fermi surface). Note that this thermalization process takes place even in the absence of electron-electron collisions, as only the electric mean-field is present.

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, all 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 moves the implementation from a 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. Lots of things that were accessible to the user are now embedded in Python classes within the package. As a result, the user accesses objects at a higher level of abstraction, thus making the usage easier. Some unit tests have been introduced in the skeleton part of the package and the 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.

Obiwan is an adaptive semi-Lagrangian code for the
resolution of the Vlasov equation. It has up to now a
cartesian 1Dx-1Dv version and a 2Dx-2Dv version. The 1D
version is coupled either to Poisson's equation or to
Maxwell's equations and solves both the relativistic and the
non relativistic Vlasov equations. The grid adaptivity is
based on a multiresolution method using Lagrange
interpolation as a predictor to go from one coarse level to
the immediately finer one. This idea amounts to using the
so-called interpolating wavelets. A parallel version of the
code exists and uses the OpenMP paradigm. Domain size of
512
^{4}has been considered and the method allows
to save effectively memory and computation time compared to a
non-adaptive code.

YODA is an acronym for Yet anOther aDaptive Algorithm. The sequential version of the code was developed by Michel Mehrenberger and Martin Campos-Pinto during CEMRACS 2003. The development of a parallel version was started by Eric Violard in collaboration with Michel Mehrenberger in 2003. It is currently continued with the contributions of Olivier Hoenen. It solves the Vlasov equation on a dyadic mesh of phase-space. The underlying method is based on hierarchical finite elements. Its originality is that the values required for interpolation at the next time step are determined in advance. In terms of efficiency, the method is less adaptive than some other adaptive methods (multi-resolution methods based on interpolating wavelets as examples), but data locality is improved.

The Brennus code is developed in the framework of a contract with the CEA Bruyères-Le-Châtel. It is based on a first version of the code that was developed at CEA. The new version is written in a modular form in Fortran 90. It solves the two and a half dimensional Vlasov-Maxwell equations in cartesian and axisymmetric geometry and also the 3D Vlasov-Maxwell equations. It can handle both structured and unstructured grids in 2D but only structured grids in 3D. Maxwell's equations are solved on an unstructured grid using either a generalized finite difference method on dual grids or a discontinuous Galerkin method in 2D. On the 2D and 3D structured meshes Yee's method is used. The Vlasov equations are solved using a particle method. The coupling is based on traditional PIC techniques.

The LOSS code is devoted to the numerical solution of the Vlasov equation in four phase-space dimensions, coupled with the two-dimensional Poisson equation in cartesian geometry. It implements a parallel version of the semi-Lagrangian method based on a localized cubic splines interpolation we developed. It has the advantage compared to older versions of the cubic splines semi-Lagrangian method to be efficient and scalable even when the number of processors becomes important (several hundreds). It is written in Fortran 90 and MPI. The computation kernel of LOSS has been adapted and put in the GYSELA5D code owned by the CEA-Cadarache.

The GYSELA code is a 5D global gyrokinetic code that simulates a ring of confined plasma in a torus with circular cross-section. The evolution of the ion distribution function and of the electric potential are computed self-consistently, assuming quasi-neutrality and that the electrons are in Maxwell-Boltzmann equilibrium with this potential.

In
, in the continuity of
,
**existence of weak solutions for the stationary
Nordström-Vlasov equations in a bounded domain**is set
out. The proof follows by a fixed point method. The
asymptotic behavior for large light speed is analyzed as
well. Convergence towards the stationary Vlasov-Poisson
model for stellar dynamics is also studied.

In
the
**Two-dimensional Finite Larmor Radius asymptotic
regime**(previously reached in
and
) is addressed using a two
scale convergence methods after rewriting the
Vlasov-Poisson system in a coordinate system so called
**canonical gyrokinetic coordinate**system.

analyses numerically the
so-called
**Fourier–Singular Complement Method for the
time-dependent Maxwell equations in an axisymmetric
domain**. This work completes a series of articles on the
numerical solution of the equations of electromagnetism in
this type of domain (see
for Maxwell's equations in the
case of axially symmetric data and
for Poisson's equation with
arbitrary data). The method relies on a continuous
approximation of the electromagnetic field, unlike, e.g.,
edge element methods. This has many advantages in the case
of model coupling, e.g. if the Maxwell solver is
embedded in a Vlasov–Maxwell code, either PIC or Eulerian.
The symmetry of rotation is exploited by using finite
elements in a meridian section of the domain only, and a
spectral method in the azimuthal dimension. The analysis
incorporates an approach which allows one to handle both
noisy or approximate data which fail to satisfy the charge
conservation equation, as may happen in a Vlasov–Maxwell
code and domains with geometrical singularities (non-convex
edges and/or vertices) which cause the electromagnetic
field to be less regular than in a smooth or convex
domain.

In
and
a new method for the solving of
the charge conservation problem is proposed and applied to
the Vlasov-Poisson equation. It is based on the
**Forward semi-Lagrangian method**(see
) which bears similarities with
Particle In Cell (PIC) method. Using strategies employed in
PIC methods, a charge conserving numerical method is then
obtained for a semi-Lagrangian method.

This a joint work with Noureddine Alaa, Professor at the Marrakech Cadi Ayyad University.

Strongly problems of parabolic equations have received considerable attentions, and various forms of this problems have been proposed in the literature, especially in the area of reaction-diffusion equations with cross-diffusion, such problems arise from biological, chemical and physical systems. Various methods have been proposed in the mathematical literature to study the existence, uniqueness and compute numerical approximation of solutions for quasi-linear partial differential equation problems. This year our work was about the periodic case. We develop a numerical method to solve periodic non linear parabolic equations based on domain decomposition and optimization interior points method.

This report describes a Hermite
formulation of the conservative
**PSM scheme**which is very generic and allows to
implement different semi-Lagrangian schemes. We also test and
propose
**numerical limiters**which should improve the robustness
of the simulations by diminishing spurious oscillations. This
work involved a trainee (J. Guterl, advised by J.-P.
Braeunig) and their results were incorporated in GYSELA.

Concerning plasma simulation, the purpose
of this report (
) is
**core turbulence in tokamak plasma**in toroidal geometry.
The natural 6-dimensional problem (3D in space and 3D in
velocity) is reduced to a 5D gyrokinetic model, taking
advantage of the particular motion of particles due to the
presence of a strong magnetic field. Using previously built
**Semi-Lagrangian Methods with Flux Limiters**applied to
the
**Vlasov-Poisson system, Plasma Turbulence
simulations**are performed and reported.

The
**validity of quasilinear theory (QL) describing the weak
warm beam–plasma instability**has been a controversial
topic for several decades. In
it is tackled anew, both
analytically and by numerical simulations which benefit
from the power of modern computers and from the development
in the last decade of Vlasov codes endowed with both
accuracy and weak numerical diffusion. Self-consistent
numerical simulations within the Vlasov–wave description
show that QL theory remains valid in the strong chaotic
diffusion regime. However there is a non-QL regime before
saturation, which confirms previous analytical work and
numerical simulation, but contradicts another analytical
work. We show analytically the absence of mode coupling in
the saturation regime of the instability where a plateau is
present in the tail of the particle distribution function.
This invalidates several analytical works trying to prove
or to contradict the validity of QL theory in the strongly
nonlinear regime of the weak warm beam–plasma
instability.

The results of numerical simulations performed with our
model were compared systematically to analytical solutions
obtained with a simplified free-streaming model [W.
Fundamenski et al., Plasma Phys. Control. Fus.
**48**, 109 (2006)]. Several comparisons with numerical
results obtained with a PIC code (developed at the
University of Innsbruck) were also performed. Despite the
differences in the numerical methods and in the model, a
fairly good agreement was observed. Comparisons with a
fluid code (developed at Culham, UK) and with experimental
results from JET are under way.

We report in a paper to be published a new modelling method to study multiple species dynamics in magnetized plasmas. Such a method is based on the gyro water bag modelling, which consists in using a multi step-like distribution function along the velocity direction parallel to the magnetic field. Such a model has been very recently ported to the context of strongly magnetized plasmas. We present its generalization to the case of multi species magnetized plasmas: each ion species being modelled via a multi water bag distribution function. We discuss in details the modelling procedure. As an illustration, we present results obtained in the linear framework.

For a few years, our team has been involved in several research projects involving the application of Vlasov-like equations to the physics of nano-sized objects, such as thin metal films, nanoparticles, quantum wells and quantum dots. It is a topic with tremendous potential for a broad spectrum of applications, ranging from materials science to biology and medicine. Our approach – based on a phase-space description of the dynamics – is not widely used in the nanophyics community, which constitutes one of the originalities of our project.

We have developed (see
and
) a dynamical model that
successfully explains the observed time evolution of the
magnetization in diluted magnetic semiconductor quantum
wells after weak laser excitation. Based on a many-particle
expansion of the exact
p-
dexchange interaction, our
approach goes beyond the usual mean-field approximation. It
includes both the sub-picosecond demagnetization dynamics
and the slower relaxation processes which restore the
initial ferromagnetic order on a nanosecond timescale. In
agreement with experimental results, our numerical
simulations show that, depending on the value of the
initial lattice temperature, a subsequent enhancement of
the total magnetization may be observed on a timescale of
few hundreds of picoseconds.

More recently, our model was augmented in order to include the role played by the quantum confinement and the band structure. It was shown that the sample thickness and the background hole density strongly influence the phenomenon of demagnetization. Quantitative results were given for III-V ferromagnetic GaMnAs quantum wells of thickness 4 and 6 nm.

This work is performed in collaboration with Jose Herskovits Norman of UFRJ, Rio de Janeiro, Antonio André Novotny from the LNCC, Petropolis, both from Brazil and Alfredo Canelas from the University of the Republic, Montevideo, Uruguay.

The industrial technique of electromagnetic casting allows for contactless heating, shaping and controlling of chemical aggressive, hot melts. The main advantage over the conventional crucible shape forming is that the liquid metal does not come into contact with the crucible wall, so there is no danger of contamination. This is very important in the preparation of very pure specimens in metallurgical experiments, as even small traces of impurities, such as carbon and sulphur, can affect the physical properties of the sample. Industrial applications are, for example, electromagnetic shaping of aluminum ingots using soft-contact confinement of the liquid metal, electromagnetic shaping of components of aeronautical engines made of superalloy materials (Ni,Ti, ...), control of the structure solidification.

The electromagnetic casting is based on the repulsive forces that an electromagnetic field produces on the surface of a mass of liquid metal. In the presence of an induced electromagnetic field, the liquid metal changes its shape until an equilibrium relation between the electromagnetic pressure and the surface tension is satisfied. The direct problem in electromagnetic casting consists in determining the equilibrium shape of the liquid metal. In general, this problem can be solved either directly studying the equilibrium equation defined on the surface of the liquid metal, or minimizing an appropriate energy functional. The main advantage of this last method is that the resulting shapes are mechanically stable.

The inverse problem consists in determining the electric currents and the induced exterior field for which the liquid metal takes on a given desired shape. This is a very important problem that one needs to solve in order to define a process of electromagnetic liquid metal forming.

In a previous work we studied the inverse electromagnetic casting problem considering the case where the inductors are made of single solid-core wires with a negligible area of the cross-section. In a second paper we considered the more realistic case where each inductor is a set of bundled insulated strands. In both cases the number of inductors was fixed in advance see . In this year we aim to overcome this constraint, and look for configurations of inductors considering different topologies with the purpose of obtaining better results. In order to manage this new situation we introduce a new formulation for the inverse problem using a shape functional based on the Kohn-Vogelius criterion. A topology optimization procedure is defined by means of topological derivatives, see .

This work is performed in collaboration with Yves Peysson (DRFC, CEA Cadarrache).

The aim of this project is to develop a finite element numerical method for the full-wave simulation of electromagnetic wave propagation in plasma. Full-wave calculations of the LH wave propagation is a challenging issue because of the short wave length with respect to the machine size. In the continuation of the works led in cylindrical geometry, a full toroidal description for an arbitrary poloidal cross-section of the plasma has been developed.

Since its wavelength
at the LH frequency is very small as compared to the
machine size
R, a conventional full wave description represents a
considerable numerical effort.Therefore, the problem is
addressed by an appropriate mathematical finite element
technique, which incorporates naturally parallel processing
capabilities. It is based on a mixed augmented variational
(weak) formulation taking account of the divergence
constraint and essential boundary conditions, which
provides an original and efficient scheme to describe in a
global manner both propagation and absorption of
electromagnetic waves in plasmas.

With such a description, usual limitations of the
conventional ray tracing related to the approximation
<<
_{B}<<
R, where
_{B}is the size of the beam transverse to the rf power
flow direction, may be overcome. Since conditions are
corresponding to
, the code under development may be considered as a
WKB full wave, dielectric properties being local.

This formulation provide a natural implementation for parallel processing, a particularly important aspect when simulations for plasmas of large size must be considered.

The domain considered is as near as possible of the cavity fill by a tokamak plasma. Toroidal coordinates are introduced. In our approach we consider Fourier decomposition in the angular coordinate to obtain stationary Maxwell equations in a cross-section of the tokamak cavity.

A finite element method is proposed for the simulation
of time-harmonic electromagnetic waves in a plasma, which
is an anisotropic medium. The approach chosen here is
sometimes referred to as
*full-wave modelling*in the literature: the original
Maxwell's equations are used to obtain a second order
equation for the time-harmonic electric field. These are
written in a weak form using an augmented variational
formulation (AVF), which takes into account the divergence
and boundary conditions. The variational formulation is
then discretized using modified Taylor-Hood (nodal)
elements.

One of the objectives in 2010 was the evolution of the MatLab finite element code " FullWaveFEM" developed to handle more real cases, in particular we introduce a new boundary condition in order to take account of the antenna and essential condition are considered.

Methods, results and more generally knowledge produced within Calvi team are used in other fields.

and resume a research programme on simulation of compressible multi-material fluid flows with sharp interface capturing called FVCF-NIP (Finite Volumes with Characteristic Flux).

This work is achieved in collaboration with J.-M. Ghidaglia (ENS Cachan), F. Dias (ENS Cachan) and B. Desjardins (ENS Ulm) in the frame of the Laboratoire de Recherche Commun MESO (ENS cachan - CEA Bruyères-le-Châtel).

In and asymptotic methods initially designed for tokamak plasmas were applied to coastal ocean waters linked phenomena.

Methods for mass transfer modelling was applied in the haulage context in .

In parallel simulation methods was applied on a biological question.

We have been involved for the last few years in the development and optimization of the full-f semi-Lagrangian gyrokinetic code GYSELA 5D originally written by V. Grangirard at CEA-Cadarache. The code is based on a 5D gyrokinetic approximation of the Vlasov equation (for the description of the ions) coupled to a quasi-neutrality equation for the computation of the self-consistent electrostatic potential. The code, which is written in toroidal geometrie, has been optimized to run efficiently on up to 4096 processors to study the effects of zonal flows on the development of turbulence in a Tokamak plasma.

The major achievements of the year have been the following:

Introduction of magnetic coordinates to separate the fast motion of particles along the magnetic field lines from the slow perpendicular motion.

Introduction of a conservative semi-Lagrangian scheme (PSM) in the GYSELA code that allows a 1D directional splitting procedure to solve the 4D Vlasov equations in the gyrokinetic model.

Modification of the parallelization strategy to take benefits of the 1D directional splitting procedure.

Development of a new very accurate quasi-neutral solver based on NURBS that can handle arbitrary geometries.

The numerical techniques to deal with the gyroaverage operator have been adapted to the GYSELA code framework and especially to deal with boundary conditions. Moreover, a new formulation of the quasi-neutral equation has been proposed in the frame of a collaboration IRMA-LATP Marseille that has to be further studied and validated.

Development of a new Forward Semi-Lagrangian scheme for the Vlasov equation.

The goal of this work is to develop a full wave method to describe the dynamics of lower hybrid current drive problem in tokamaks.

Calvi members are involved in ANR projects.

- Non thematic ANR: Study of wave-particle interaction for Vlasov plasmas (leader A. Ghizzo). In collaboration with F. Califano from the University of Pisa in Italy.

- ANR Calcul Intensif et Simulation: HOUPIC
(ANR-06-CIS6-013-01, leader E. Sonnendrücker): Development
of 3D electromagnetic PIC codes comparing conforming Finite
Elements and Discontinuous Galerkin Solvers on unstructured
grids.
http://

- Non thematic ANR: EGYPT project (leader Ph. Ghendrih). Study of gyrokinetic models and their numerical approximation. In collaboration with DRFC/CEA-Cadarache.

- Eric Sonnendrücker is heading the Large Scale
Initiative Fusion energy that started at the beginning of
2009
http://

- Every member of Calvi is involved within this Large Scale Initiative.

- ADT SeLaLib was launch in fall 2010.

- Edwin Chacon-Golcher joined Calvi team in december as Senior Engineer as leader of the future Engineer team.

- An API for SeLaLib is being defined (in collaboration with the CEA Gysela team).

(Details on Cemracs projects may be found on the web
page:
http://

- Guillaume Latu, Ahmed Ratnani and Eric Sonnendrücker were involved in Cemracs Project "Gyronurbs" whose target was to solve the Valsov equation in complex geometry using Nurbs.

- Aurore Back, Anaïs Crestetto, Ahmed Ratnani and Eric Sonnendrücker were involved in Cemracs Project "IsoPic" whose target was to develop an axisymmetric PIC code based on isogeometric analysis.

- Nicolas Crouseilles and Michel Mehrenberger were involved in Cemracs Project "VlasovDG" whose target was to develop a Discontinuous Galerkin Vlasov code.

This project involves several French and German teams both in the applied mathematics and in the fluid dynamics community. Its aim is the development of numerical methods for the computation of noise generated in turbulent flows and to understand the mechanisms of this noise generation.

The project is subdivided into seven teams each
involving a French and a German partner. Our German partner
is the group of C.-D. Munz at the University of Stuttgart.
More details can be found on the web page
http://

This project is funded by European Union under the
Seventh Framework Program (FP7) which will provide a
comprehensive framework and infrastructure for core and
edge transport and turbulence simulation, linking grid and
High Performance Computing, to the fusion modelling
community. It has started in January 2008 and ends in
December 2010. CALVI is involved in this project to provide
efficient and reliable visualization tools. Our proposal is
based on the use of two tools: Python with numPy and
Matplotlib packages and VisIt Software. Our contribution
consists in three packages: getting data from fusion
community into VisIt and Python, accessing VisIt and Python
from Kepler which is the central software of the project,
and providing 4D compression and visualization. This year
we made the first point which was quite straight forward.
More details can be found on the web page
http://

Funded by a Fulbright grant this project is developed in collaboration with A. Friedman from the Lawrence Berkeley National Laboratory and J. Verboncoeur at the University of California, Berkeley.

Nicolas Crouseilles and Eric
Sonnendrücker organized Cemracs 2010 in Marseille (120
participants over 6 weeks) (
http://

During the first week : Summer school on numerical methods for fusion.

During the five other weeks : Projects supported by INRIA, CEA, Universities, CNRS.

Giovanni Manfredi jointly edited the proceedings of the international workshop "Vlasovia 2009" (Luminy, Marseille, 2009), which are to be published in the journal " Transport theory and statistical physics".

Jean Rodolphe Roche is the research coordinator of the L.R.C. projet - Full wave modelling of lower hybrid current drive in tokamaks.

Jean-Philippe Braeunig

was invited to give a talk at the meeting of ANR Project Espoir in June.

was invited by Prof. Dias to give the following talk: "Some improvements and applications of the Finite Volumes FVCF-NIP method for the simulation of compressible multi-material fluid flows"at the "Seminary of the School of Mathematical Sciences", University College Dublin, Irland in January.

gave the following conferences:

"Some numerical aspects of conservative schemes in a 4D Vlasov drift-kinetic code"

"The Enhanced NIP method for multi-material fluid flows"

at the "Workshop on the Physical and Numerical Modeling of Turbulent and Multi-Phase Flows" in Cargèse, Corsica in September.

Nicolas Crouseilles gave talks in

Canum 2010 in
Carcans-Maubuisson; June 2 (
http://

Program "Partial Differential
Equations in Kinetic Theories" in Cambridge -
satellite workshop in Edinburgh; 8-12 november :
"PDEs in kinetic theories: kinetic description of
biological models" (
http://

Emmanuel Frénod was invited by "Doctorials" of "École Doctorale de Mathématiques et d'Informatique de Dakar" to give a conference on "Usefullness of Modelling and Asymptotic Analysis", February 8 and 9.

Paul-Antoine Hervieux gave the following talk : "Electron dynamics and ultrafast ionization of clusters and fullerenes", at the European Conference on Atoms, Molecules and Photons in Salamanca (Spain); July.

Simon Labrunie was invited to make conferences:

Sixth Singular Days on Asymptotic Methods for PDEs in Berlin; April 29 - May 1.

ECCM 10 ("Fourth European Conference on Computational Mechanics") in Paris; May 16 - 21.

Giovanni Manfredi gave the following talks:

"Electron dynamics and ultrafast ionization of clusters and fullerenes", at the European Conference on Atoms, Molecules and Photons, Salamanca (Spain), July.

"Electron Thermalization and Decoherence in Thin Metal Films", at the annual meeting of the European COST action CUSPFEL (Chemistry With Ultrashort Pulses and Free-Electron Lasers) in Heraklion (Greece); October 23-25.

"Quantum fidelity for many-particle systems", at QCHAOS 2010: 4th Workshop on Quantum Chaos, Theory and Applications, Castro Urdiales (Spain), September 13-17.

Eric Sonnendrücker was invited for talks at:

Plasma day at university Paris 6, mars 15.

Mini-symposium at the SIAM meeting on Dynamical Systems and Partial Differential Equations, Barcelona, May 31-June 4.

Symposium on new trends in numerical methods for plasma physics, Garching, Germany, July 8.

Workshop Frontiers in Computational Astrophysics, ENS Lyon, October 11-15.

Day on mathematics for energy organized by GdR Momas and CHANT, Paris, November 5.

Workshop Classical and Quantum Mechanical Models of Many-Particle Systems, Oberwolfach, Germany, December 6-10.

Nicolas Crouseilles was member of board of examiners for the position "MdC chaire INRIA" at Pau.

Jean Rodolphe Roche is the research coordinator of a CAPES-COFECUB bilateral agreement with the Federal University of Rio de Janeiro and the National Laboratory of Scientific Computing of Brazil.

Eric Sonnendrücker is a member of CNU 26, applied mathematics and applications of mathematics.

Eric Sonnendrücker was a member of the board of examiners for professor positions at the university of Mulhouse and the ENS Cachan.

Eric Sonnendrücker is a member of panel 6 (mathematics and computer science) of GENCI for the attribution of computing hours on the french supercomputing systems

Emmanuel Frénod was invited by "Département de Mathématiques et Informatique" to give a 10-hour course for Master and Ph. D. students in Mathematics and Computer Sciences on "Two-Scale Convergence and Application to Seabed Morpho-dynamics in Tide-Influenced Coastal Ocean Waters".

Vladimir Latocha gave lectures on Scientific Computation in the Master in mathematics of UHP.

Giovanni Manfredi and Paul-Antoine Hervieux taught a 28-hour course at the Master "Condensed Matter and Nanophysics". Title of the course: Photon-matter interactions.

Jean Rodolphe Roche gave lectures on Domain Decomposition in the M2 in mathematics of UHP.

Eric Sonnendrücker taught a course on numerical methods for the Vlasov-Maxwell equations in the Master 2 of mathematics at the University of Strasbourg.

Thomas Respaud,
*Numerical coupling of Maxwell and Vlasov
equations*. Advisor: Eric Sonnendrücker, defended:
2nd November.

Aurore Back,
*Hamiltonian derivation of gyrokinetic models*.
Advisors: Emmanuel Frénod and Eric Sonnendrücker.

Anaïs Crestetto,
*High order moments fluid model for plasmas and
multiphase media.*Advisors: Philippe Helluy and Marc
Massot.

Céline Caldini, since October 2010,
*Mathematical and numerical analysis of gyro-kinetic
models - Application to magnetic
confinment.*Advisor: MihaiBostan

Pierre Glanc, since October 2010,
*Numerical approximation of Vlasov's equations with
conservative remapping methods.*Advisors: Nicolas
Crouseilles, Emmanuel Frénod, Philippe Helluy and
Michel Mehrenberger

Takashi Hattori,
*Domain decomposition methods for full wave
simulation in cold plasma*. Advisors: Simon Labrunie
and Jean Rodolphe Roche.

Sandrine Marchal,
*Domain decomposition methods to solve a system of
hyperbolic equations*. Advisors: Simon Labrunie and
Jean Rodolphe Roche.

Mathieu Lutz, since September 2010,
*Mathematical and numerical study of a gyrokinetic
model including electromagnetic effects for Tokamak
plasma simulation*. Advisors: Emmanuel Frénod and
Eric Sonnendrücker.

Ahmed Ratnani,
*Study of the quasi-neutrality equation and its
coupling with Vlasov equations*. Advisors: Nicolas
Crouseilles and Eric Sonnendrücker.

Morgane Bergot, since December 2010,
*High order time schemes for conservative
semi-Lagrangian Vlasov solvers*. Advisors: Eric
Sonnendrücker.

Olivier Hoenen,
*Visualisation tools for fusion*. Advisors: Eric
Sonnendrücker