A fundamental and enduring challenge in science and technology is the quantitative prediction of time-dependent nonlinear phenomena. While dynamical simulation (for ballistic trajectories) was one of the first applications of the digital computer, the problems treated, the methods used, and their implementation have all changed a great deal over the years. Astronomers use simulation to study long term evolution of the solar system. Molecular simulations are essential for the design of new materials and for drug discovery. Simulation can replace or guide experiment, which often is difficult or even impossible to carry out as our ability to fabricate the necessary devices is limited.

During the last decades, we have seen dramatic increases in computing power, bringing to the fore an ever widening spectrum of applications for dynamical simulation. At the boundaries of different modeling regimes, it is found that computations based on the fundamental laws of physics are under-resolved in the textbook sense of numerical methods. Because of the vast range of scales involved in modeling even relatively simple biological or material functions, this limitation will not be overcome by simply requiring more computing power within any realistic time. One therefore has to develop numerical methods which capture crucial structures even if the method is far from “converging" in the mathematical sense. In this context, we are forced increasingly to think of the numerical algorithm as a part of the modeling process itself. A major step forward in this area has been the development of structure-preserving or “geometric" integrators which maintain conservation laws, dissipation rates, or other key features of the continuous dynamical model. Conservation of energy and momentum are fundamental for many physical models; more complicated invariants are maintained in applications such as molecular dynamics and play a key role in determining the long term stability of methods. In mechanical models (biodynamics, vehicle simulation, astrodynamics) the available structure may include constraint dynamics, actuator or thruster geometry, dissipation rates and properties determined by nonlinear forms of damping.

In recent years the growth of geometric integration has been very
noticeable. Features such as *symplecticity*
or *time-reversibility* are now widely recognized as essential properties to preserve,
owing to their physical significance. This has motivated a lot
of research , , and led to many
significant theoretical achievements (symplectic and symmetric methods,
volume-preserving integrators, Lie-group methods, ...).
In practice, a few simple schemes such as the Verlet method or the Störmer method
have been used for years with great success in molecular dynamics or astronomy. However, they now need to be further improved in order to fit the tremendous increase of complexity and size of the models.

To become more specific, the project *IPSO *aims at finding and implementing new
structure-preserving schemes and at understanding the behavior of existing ones for the following type of problems:

systems of differential equations posed on a manifold.

systems of differential-algebraic equations of index 2 or 3, where the constraints are part of the equations.

Hamiltonian systems and constrained Hamiltonian systems (which are special cases of the first two items though with some additional structure).

highly-oscillatory systems (with a special focus of those resulting from the Schrödinger equation).

Although the field of application of the ideas contained in geometric integration is extremely wide (e.g. robotics, astronomy, simulation of vehicle dynamics, biomechanical modeling, biomolecular dynamics, geodynamics, chemistry...), *IPSO *will mainly concentrate on applications for *molecular dynamics simulation* and *laser simulation*:

There is a large demand in biomolecular modeling for models that integrate microscopic molecular dynamics simulation into statistical macroscopic quantities. These simulations involve huge systems of ordinary differential equations over very long time intervals. This is a typical situation where the determination of accurate trajectories is out of reach and where one has to rely on the good qualitative behavior of structure-preserving integrators. Due to the complexity of the problem, more efficient numerical schemes need to be developed.

The demand for new models and/or new structure-preserving schemes is also quite large in laser simulations. The propagation of lasers induces, in most practical cases, several well-separated scales: the intrinsically highly-oscillatory *waves* travel over long distances. In this situation, filtering the oscillations in order to capture the long-term trend is what is required by physicists and engineers.

ordinary differential equation, numerical integrator, invariant, Hamiltonian system, reversible system, Lie-group system

In many physical situations, the time-evolution of certain quantities may be written as a Cauchy problem for a differential equation of the form

For a given *flow* of (). From this point of view, a numerical scheme with step size *geometric integration* is whether *intrinsic* properties of

This question can be more specifically addressed in the following situations:

The system () is said to be

It is then natural to require that *symmetric*. Symmetric methods
for reversible systems of ODEs are just as much important as *symplectic*
methods for Hamiltonian systems and offer an interesting alternative
to symplectic methods.

The system () is said to have an invariant manifold

is kept *globally* invariant by

As an example, we mention Lie-group equations, for which the manifold has an additional group
structure. This could possibly be exploited for the space-discretisation.
Numerical methods amenable to this sort of problems have been
reviewed in a recent paper and divided into two
classes, according to whether they use

Hamiltonian problems are ordinary differential equations of the form:

with some prescribed initial values

Besides the Hamiltonian function, there might exist other invariants for
such systems: when there exist *integrable*. Consider now the parallelogram *oriented* areas of the projections over the planes

where *canonical symplectic* matrix

A continuously differentiable map

A fundamental property of Hamiltonian systems is that their exact flow is symplectic.
Integrable Hamiltonian systems behave in a very remarkable way: as a matter of fact, their invariants persist under small perturbations, as shown in the celebrated theory of Kolmogorov, Arnold and Moser. This behavior motivates the introduction of *symplectic* numerical flows that share most of the properties of the exact flow. For practical simulations
of Hamiltonian systems, symplectic methods possess an important advantage: the error-growth as a function of time is indeed linear, whereas it would typically be quadratic for non-symplectic methods.

Whenever the number of differential equations is insufficient to determine the solution of the system, it may become necessary to solve the differential part and the constraint part altogether. Systems of this sort are called differential-algebraic systems. They can be classified according to their index, yet for the purpose of this expository section, it is enough to present the so-called index-2 systems

where initial values

and of the so-called hidden manifold

This manifold

There exists a whole set of schemes which provide a numerical approximation lying on

second-order ODEs, oscillatory solutions, Schrödinger and wave equations, step size restrictions.

In applications to molecular dynamics or quantum dynamics for instance, the right-hand side of () involves *fast* forces (short-range interactions) and *slow* forces (long-range interactions). Since *fast* forces are much cheaper to evaluate than *slow* forces, it seems highly desirable to design numerical methods for which the number of evaluations of slow forces is not (at least not too much) affected by the presence of fast forces.

A typical model of highly-oscillatory systems is the second-order differential equations

where the potential

where *fast* forces deriving from *slow* forces deriving from

Another prominent example of highly-oscillatory systems is encountered in quantum dynamics where the Schrödinger equation is the model to be used. Assuming that the Laplacian has been discretized in space, one indeed gets the *time*-dependent Schrödinger equation:

where

Schrödinger equation, variational splitting, energy conservation.

Given the Hamiltonian structure of the Schrödinger equation, we are led to consider the question of energy preservation for time-discretization schemes.

At a higher level, the Schrödinger equation is a partial differential equation which may exhibit Hamiltonian structures. This is the case of the time-dependent Schrödinger equation, which we may write as

where

with the kinetic and potential energy operators

where

The multiplication by

The numerical approximation of () can be obtained using projections onto submanifolds of the phase space, leading to various PDEs or ODEs: see , for reviews. However the long-time behavior of these approximated solutions is well understood only in this latter case, where the dynamics turns out to be finite dimensional. In the general case, it is very difficult to prove the preservation of qualitative properties of () such as energy conservation or growth in time of Sobolev norms. The reason for this is that backward error analysis is not directly applicable for PDEs. Overwhelming these difficulties is thus a very interesting challenge.

A particularly interesting case of study is given by symmetric splitting methods, such as the Strang splitting:

where

waves, Helmholtz equation, high oscillations.

The Helmholtz equation models the propagation of waves in a medium with variable refraction index. It is a simplified version of the Maxwell system for electro-magnetic waves.

The high-frequency regime is characterized by the fact that the typical wavelength of the signals under consideration is much smaller than the typical distance of observation of those signals. Hence, in the high-frequency regime, the Helmholtz equation at once involves highly oscillatory phenomena that are to be described in some asymptotic way. Quantitatively, the Helmholtz equation reads

Here,

One important scientific objective typically is to
describe the high-frequency regime in terms of *rays* propagating
in the medium, that are
possibly refracted at interfaces, or bounce on boundaries,
etc. Ultimately, one would like to replace the true numerical resolution
of the Helmholtz equation by that of a simpler, asymptotic model,
formulated in terms of rays.

In some sense, and in comparison with, say, the wave equation,
the specificity of the Helmholtz equation is the following.
While the wave equation typically describes the evolution of waves
between some initial time and some given observation time,
the Helmholtz equation takes into account at once
the propagation of waves over *infinitely long*
time intervals. Qualitatively, in order to have a good understanding
of the signal observed in some bounded region of space, one readily
needs to be able to describe the propagative phenomena
in the whole space, up to infinity. In other words, the “rays” we refer to
above need to be understood from the initial time up to infinity.
This is a central difficulty in the analysis of the high-frequency behaviour
of the Helmholtz equation.

Schrödinger equation, asymptotic model, Boltzmann equation.

The Schrödinger equation is the appropriate way to describe transport phenomena at the scale of electrons. However, for real devices, it is important to derive models valid at a larger scale.

In semi-conductors, the Schrödinger equation is the ultimate model that allows to obtain quantitative information about electronic transport in crystals. It reads, in convenient adimensional units,

where

Here, the unknown is

E. Faou received the SIAM Germund Dahlquist prize in september 2015.

In earlier works, it has been shown how formal series like those used nowadays to investigate the properties of numerical integrators may be used to construct high- order averaged systems or formal first integrals of Hamiltonian problems. With the new approach the averaged system (or the formal first integral) may be written down immediately in terms of (i) suitable basis functions and (ii) scalar coefficients that are computed via simple recursions. In , we show how the coefficients/basis functions approach may be used advantageously to derive exponentially small error bounds for averaged systems and approximate first integrals.

The kinetic theory of fluid turbulence modeling developed by Degond and Lemou "Turbulence models for incompressible fluids derived from kinetic theory" (J. Math. Fluid Mech. 2002) is considered for further study, analysis and simulation. Starting with the Boltzmann like equation representation for turbulence modeling, a relaxation type collision term is introduced for isotropic turbulence. In order to describe some important turbulence phenomenology, the relaxation time incorporates a dependency on the turbulent microscopic energy and this makes difficult the construction of efficient numerical methods. To investigate this problem, we focus here on a multi-dimensional prototype model and first propose an appropriate change of frame that makes the numerical study simpler. Then, a numerical strategy to tackle the stiff relaxation source term is introduced in the spirit of Asymptotic Preserving Schemes. Numerical tests are performed in a one-dimensional framework on the basis of the developed strategy to confirm its efficiency.

In , which is the continuation of , we propose numerical schemes for linear kinetic equation which are able to deal with the fractional diffusion limit. When the collision frequency degenerates for small velocities it is known that for an appropriate time scale, the small mean free path limit leads to an anomalous diffusion equation. From a numerical point of view, this degeneracy gives rise to an additional stiffness that must be treated in a suitable way to avoid a prohibitive computational cost. Our aim is therefore to construct a class of numerical schemes which are able to undertake these stiffness. This means that the numerical schemes are able to capture the effect of small velocities in the small mean free path limit with a fixed set of numerical parameters. Various numerical tests are performed to illustrate the efficiency of our methods in this context.

In and ,
we present a solution to the well-known Test&Set operation in an asynchronous system prone to process crashes. Test&Set is a synchronization operation that, when invoked by a set of processes, returns yes to a unique process and returns no to all the others. Recently many advances in implementing Test&Set objects have been achieved, however all of them target the shared memory model. In this paper we propose an implementation of a Test&Set object in the message passing model. This implementation can be invoked by any number

The project, entitled "LODIQUAS" (for: Low DImensional QUANtum Systems), received fundings for 4 postdocs (48 months) and one pre-doc (36 months). The whole project involves the following researchers : Norbert Mauser (Vienna), Erich Gornik (Vienna), Mechthild Thalhammer (Innsbruck), Christoph Naegerl (Innsbruck), Jörg Schmiedmayer (Vienna), Hans-Peter Stimming (Vienna),Francis Nier (Rennes), Raymond El Hajj (Rennes), Claudia Negulescu (Toulouse), Fanny Delebecque (Toulouse), Stéphane Descombes (Nice), Christophe Besse (Lille).

The expected scientific and technological progress brought by the present project are as follows.

Quantum technology as the application of quantum effects in macroscopic devices has an increasing importance, not only for far future goals like the quantum computer, but already now or in the near future. The present project is mainly concerned with the mathematical and numerical analysis of these objects, in conjunction with experimental physicists. On the side of fermions quantum electronic structures like resonant tunnelling diodes show well studied non classical effects like a negative differential resistance that are exploited for novel devices. On the side of bosons the creation and manipulation of Bose Einstein Condensates (the first creation of BECs by Ketterle et al merited a Nobel prize) has become a standard technique that allows to study fundamental quantum concepts like matter-wave duality with increasingly large objects and advanced quantum effects like decoherence, thermalization, quantum chaos. In state-of-the-art experiments e.g. with ultracold atoms in optical lattices the bosonic or fermionic nature of quantum objects can change and it makes a lot of sense to treat the models in parallel in the development of mathematical methods. The experimental progress in these fields is spectacular, but the mathematical modelling and analysis as well as the numerical simulation are lagging behind. Low dimensional models are mostly introduced in a heuristic way and there is also a need for systematic derivations and comparison with the 3-d models. To close the gap is a main goal of this project that aims to deliver reliable tools and programme packages for the numerical simulation of different classes of quantum systems modelled by partial differential equation of NLS type. Virtually all participants have a strong track record of international collaboration, they grew up with the concept of the European Research Area where science knows no boundaries and scientists used to work in different countries, as it was the case in a pronounced way in mathematics and in quantum physics in the thirties of the last century. The Pre- and Post-Docs to be funded by this project will be trained in this spirit of mobility between scientific fields and between places.

This project gave rise to the following scientific achievements

**PhD students**

*Boris Pawilowski*, has been hired as a PhD student,
under the supervision of F. Nier and N. Mauser. His contract started
october 2012, and the PhD thesis was defended on December 2015.
His PhD subject is
"Mean field limit for discrete models and nonlinear discrete
Schrödinger equations".

**Postdocs**

*Loïc Le Treust* has been hired as a Postdoc,
under the supervision of F. Méhats (main) and N. Mauser. His contract
started October 2013, and it did last
two years, in Rennes and Vienna.

*Yong Zhang*, under contract in Vienna, has been invited for several one month periods in Rennes.
There are works in progress with F. Méhats and P. Chartier.

*Kristelle Roidot*, had a six months contract in Vienna, and this gave rise to
works with N. Mauser, C. Klein, J.-C. Saut, S. Descombes.

**Workshops**

**July 2012**, kick-off meeting of the LODIQUAS project, WPI, Vienna (one week, approx. 40 people, amongst which most of the participants of the project).

**February 2013**, WPI, Vienna, with a similar organization as the kick-off meeting.

**July 2013**, WPI, Vienna.
At the WPI for one week.
"Quantized Vortices in Superfluidity and
Superconductivity and Related Problems", organisers W. Bao, C. Bardos, Q.
Du, N. Mauser.

**September 2013**, WPI Vienna, "Modified dispersion for dispersive equations and
systems ", organisers R. Carles, Mauser, J.C. Saut.

**September 2013**, WPI Vienna, "Modified dispersion for dispersive equations and
systems ", organisers R. Carles, Mauser, J.C. Saut.

**October 2014**, WPI Vienna, "Blow-up and Dispersion in nonlinear Schrödinger and Wave equations",
organizers G. Lebeau, A. Jüngel,
O. Ivanovici, J.-C. Saut, H.-P. Stimming.

**December 2014**, Saint-Malo, "Lodiquas Meeting",
organisers F. Castella and P. Chartier.

**December 2015**, Dinard, "Joint Lodiquas and Ipso Meeting",

The project *Moonrise* submitted by F. Méhats has been funded by the ANR for 4 years, for the period 2015-2019. This project aims at exploring modeling, mathematical and numerical issues originating from the presence of high-oscillations in nonlinear PDEs from the physics of nanotechnologies (quantum transport) and from the physics of plasmas (magnetized transport in tokamaks). The partners of the project are the IRMAR (Rennes), the IMT (Toulouse) and the CEA Cadarache. In the IPSO team, F. Castella, P. Chartier, N. Crouseilles and M. Lemou are members of the project Moonrise.

IPSO is associated to IPL FRATRES which started in june 2015. The aim of this project is to organize Inria teams activities which develop mathematical and numerical tools in magnetically confined nuclear fusion. The ambition is to prepare the next generation of numerical modeling methodologies able to use in an optimal way the processing capabilities of modern massively parallel architectures. This objective requires close collaboration between a) applied mathematicians and physicists that develop and study mathematical models of PDE; b) numerical analysts developing approximation schemes; c) specialists of algorithmics proposing solvers and libraries using the many levels of parallelism offered by the modern architecture and d) computer scientists. The project road map ambitions to contribute in close connection with National and European initiatives devoted to nuclear Fusion to the improvement and design of numerical simulation technologies applied to plasma physics and in particular to the ITER project for magnetic confinement fusion.

Title: Numerical integration of Geometric Partial Differential Equations

Programm: FP7

Duration: September 2011 - August 2016

Coordinator: E. Faou

Inria contact: E. Faou

'The goal of this project is to develop new numerical methods for the approximation of evolution equations possessing strong geometric properties such as Hamiltonian systems or stochastic differential equations. In such situations the exact solutions endow with many physical properties that are consequences of the geometric structure: Preservation of the total energy, momentum conservation or existence of ergodic invariant measures. However the preservation of such qualitative properties of the original system by numerical methods at a reasonable cost is not guaranteed at all, even for very precise (high order) methods. The principal aim of geometric numerical integration is the understanding and analysis of such problems: How (and to which extend) reproduce qualitative behavior of differential equations over long time? The extension of this theory to partial differential equations is a fundamental ongoing challenge, which require the invention of a new mathematical framework bridging the most recent techniques used in the theory of nonlinear PDEs and stochastic ordinary and partial differential equations. The development of new efficient numerical schemes for geometric PDEs has to go together with the most recent progress in analysis (stability phenomena, energy transfers, multiscale problems, etc..) The major challenges of the project are to derive new schemes by bridging the world of numerical simulation and the analysis community, and to consider deterministic and stochastic equations, with a general aim at deriving hybrid methods. We also aim to create a research platform devoted to extensive numerical simulations of difficult academic PDEs in order to highlight new nonlinear phenomena and test numerical methods.'

N. Crouseilles and M. Lemou are members of the EUROFusion project entitled "Enabling research project for the implementation of the fusion roadmap". The leader is E. Sonnendrücker (IPP Garching, Germany).

Several IPSO members have international collaborations

L. Einkemmer, University of Innsbruck. Collaboration on numerical schemes for Vlasov-Maxwell equations with N. Crouseilles and E. Faou.

M. Thalhammer, University of Innsbruck. Collaboration on multi-revolution methods for the Schrödinger equation and Dirac equation with F. Méhats and P. Chartier.

S. Jin, University of Madison. Collaboration on numerical schemes for highly-oscillatory problems with N. Crouseilles and M. Lemou.

G. Vilmart, University of Geneva. Collaboration on uniformly accurate methods for highly-oscillatory problems with F. Méhats and P. Chartier

F. Casas, University Jaume. Collaboration on splitting methods for Vlasov equations with N. Crouseilles and E. Faou.

S. Jin (University of Madison) spent 2 months at IRMAR (University of Rennes) within the framework of the Labex H. Lebesgue semester "PDEs and long time behavior", to collaborate with N. Crouseilles and M. Lemou.

P. Chartier: invitation at the University of Geneva (Switzerland), by G. Vilmart (one month in July).

F. Méhats: invitation at the University of Geneva (Switzerland), by G. Vilmart (one week).

F. Méhats: invitation at the Beijing Computational Science Research Center (China), by W. Bao (10 days).

M. Lemou: invitation at the University of Wisconsin-Madison (US), by S. Jin (two weeks, october 2015).

M. Lemou: invitation at the University of Geneva (Switzerland), by G. Vilmart (two weeks, july 2015).

A. Debussche participated to the semester "New challenges in PDE: Deterministic dynamics and randomness in high and infinite dimensional systems" at MSRI (Berckeley, US).

M. Lemou and F. Méhats organized the CHL (Labex) workshop *Mathematical problems and modelization in kinetic theory*, Rennes, May 26-29 2015.

E. Faou organized with B. Grébert, E. Paturel and L. Thomann (Univ. Nantes)
the CHL (Labex) Summer school *PDE and large time asymptotics*, Nantes, June 22 - July 3 2015.

N. Crouseilles and P. Chartier organized the CHL (Labex) workshop *Multiscale numerical methods for differential equations*, Rennes, August 25-27, 2015.

F. Castella and P. Chartier organized the IPSO-LODIQUAS workshop with the support of the ANR LODIQUAS and MOONRISE, Dinard, December 9-11 2015.

M. Lemou organized a mini-symposium at the 9th International Conference on
Computational Physics (ICCP9): *numerical methods for quantum and kinetic problems*,
Singapore, Junary 7-11 2015.

N. Crouseilles and M. Lemou organized a mini-symposium at the national Congrès SMAI 2015:
*Numerical Approaches for Stiff PDEs*, Les Karellis, June 8-12, 2015.

P. Chartier was member of the scientific committee of ENUMATH 2015, 14-18 september, Ankara, Turkey.

P. Chartier is member of the editorial board of *Mathematical Modelling and Numerical Analysis*.

A. Debussche is editor in chief of *Stochastic Partial Differential Equations: analysis and computations*.

A. Debussche is member of the editorial board of *Potential Analysis*.

A. Debussche is member of the editorial board of *Journal of Evolution Equations*.

A. Debussche is member of the editorial board of *Differential and Integral Equations*.

A. Debussche is member of the editorial board of *ESAIM: proceedings*.

A. Debussche is member of the editorial board of the collection *Mathématiques et Applications*.

The members of the team reviewed numerous papers for numerous international journals
(Comm. Math. Phys., SIAM journals, J. Comput. Phys.,

P. Chartier gave a talk at *The 9th International Conference on Computational Physics*,
Singapore, January 7-11, 2015.

E. Faou gave a talk at the seminar of analysis, at the university of Toulouse, January, 2015.

E. Faou gave a talk at the Seminar ANEDP, University of Lille 1, January, 2015.

P. Chartier gave a seminar at the University of Geneva, March 10, 2015.

E. Faou gave a talk at ENS Lyon in the physics department, May 2015.

E. Faou gave a talk at the workshop *Mathematical Methods in Quantum Molecular Dynamics*, organized by G. Hagedorn, C. Lasser and C. Le Bris, Oberwolfach, Germany, June, 2015.

N. Crouseilles gave a talk in the seminar of the analysis team at the University of Toulouse III, June, 2015.

P. Chartier gave a talk at the workshop *Modelling and Numerics for Quantum Systems*,
Toulouse, September 2-4, 2015.

E. Faou gave the Dahlquist prize lecture at the *Scicade conference*, Potsdam, Germany, September, 2015.

P. Chartier gave a colloquium at the `*School of Mathematics in Georgia Tech*, Atlanta, USA,
October 29, 2015.

E. Faou gave a talk in the seminar of the analysis team at the University of Bordeaux I, October, 2015.

F. Méhats gave seminars in Geneva (Switzerland), Beijing (China), Reims and Lyon.

M. Lemou gave a talk at the Ki-Net international conference:
*Asymptotic Preserving and Multiscale Methods for Kinetic and Hyperbolic Problems*, Madison (USA), May 4-8, 2015.

M. Lemou gave a talk at the *International Congress on
Industrial and Applied Mathematics, ICIAM 2015. Mini-symposium "Analysis and
algorithm for coupling of kinetic and fluid equations*, Beijing (China), August 10-14, 2015.

M. Lemou gave a talk at the *International workshop on kinetic problems
in the honor of W. Strauss, R. Glassey and J. Schaeffer: Recent progress in collisionless models*,
Imperial College, London, September 7-11, 2015.

M. Lemou gave two conferences at the university of Wisconsin-Madison during his visit, between october 3 and october 17, 2015.

A. Debussche gave a talk at the workshop *New challenges in PDE: Deterministic dynamics and randomness in high and infinite dimensional systems*, MSRI, Berkeley (USA), October 19-30, 2015.

F. Méhats has been the head of the IRMAR (UMR CNRS 6625), since June 2015,

P. Chartier is scientific vice-deputy of the Inria-Rennes center.

P. Chartier is member of the "Commission d'évaluation" of Inria.

N. Crouseilles is member of the scientific council of ENS Rennes.

N. Crouseilles is partly in charge of the weekly numerical analysis seminar at ENS Rennes.

A. Debussche leads the H. Lebesgue Center (Labex) with San Vu Ngoc (coordinator) and L. Guillopé.

E. Faou is member of the scientific council of the Pôle universitaire Léonard de Vinci, since september 2015.

E. Faou is member of the CNU 26, since december 2015.

M. Lemou is member of the scientific council of ENS Rennes.

M. Lemou is member of the scientific council of the H. Lebesgue Center (Labex).

M. Lemou is head of the "numerical analysis IRMAR team".

F. Castella is head of the european ANR project "Lodiquas", described above.

F. Castella is member of the "Conseil d'UFR de Mathématiques".

Licence : P. Chartier, “Ordinary differential equations", 36 ETDH, L3, ENS Rennes.

Master : P. Chartier, “Numerical geometric integration and averaging methods", 36ETDH, M2, University of Rennes I.

Master : N. Crouseilles, “Numerical methods for kinetic equations", 18ETDH, M2, University of Rennes I.

Licence : N. Crouseilles, “Numerical methods", 36 ETDH, L3, ENS Rennes.

Master : M. Lemou, “Introduction to PDEs: hyperbolic systems and conservation laws.", 36ETDH, M2, University of Rennes I.

Master : F. Castella, “Pseudo-differential calculus.", 24ETDH, M2, University of Rennes I.

Master : F. Castella, “Kinetic equations.", 60ETDH, M1, University of Rennes I.

PhD G. Leboucher, "Stroboscopic averaging methods for highly-oscillatory partial differential equations", University of Rennes I, defended on December 8, 2015. Advisors: P. Chartier and F. Méhats.

PhD H. Hivert, started in september 2012. Advisors: N. Crouseilles and M. Lemou.

PhD R. Horsin, started in september 2013. Advisors: E. Faou and F. Rousset.

PhD M. Malo, started in september 2015. Advisors: M. Lemou and F. Méhats.

PhD J. Sauzeau, started in september 2012. Advisors: P. Chartier and F. Castella.

PhD M. Tusseau, started in september 2013. Advisors: A. Debussche and F. Méhats.

F. Méhats was referee on the thesis of A. Trescases (ENS Cachan). Defended on September 11, 2015.

M. Lemou was member of the thesis committee, Phd of Xavier Valentin at Ecole Centrale de Paris. Defended on december 16, 2015.

M. Lemou was referee on the thesis of T. Leroy (Paris 6) that will be defended on Junary 5, 2016.