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

Laser physics considers the propagation over long space (or time) scales
of high frequency waves. Typically, one has to deal with the propagation
of a wave having a wavelength of the order of

Generally speaking, the demand in developing such models or schemes in the context of laser physics, or laser/matter interaction, is large. It involves both modeling and numerics (description of oscillations, structure preserving algorithms to capture the long-time behaviour, etc).

In a very similar spirit, but at a different level of modelling, one would like to understand the very coupling between a laser propagating in, say, a fiber, and the atoms that build up the fiber itself.

The standard, quantum, model in this direction is called the Bloch model: it is a Schrödinger like equation that describes the evolution of the atoms, when coupled to the laser field. Here the laser field induces a potential that acts directly on the atom, and the link between this potential and the laser itself is given by the so-called dipolar matrix, a matrix made up of physical coefficients that describe the polarization of the atom under the applied field.

The scientific objective here is twofold. First, one wishes to obtain
tractable asymptotic models that average out the high oscillations of the atomic
system and of the laser field. A typical phenomenon here is the *resonance*
between the field and the energy levels of the atomic system. Second, one
wishes to obtain good numerical schemes in order to solve
the Bloch equation, beyond the oscillatory phenomena entailed by this model.

In classical molecular dynamics, the equations describe the evolution of atoms or molecules under the action of forces deriving from several interaction potentials. These potentials may be short-range or long-range and are treated differently in most molecular simulation codes. In fact, long-range potentials are computed at only a fraction of the number of steps. By doing so, one replaces the vector field by an approximate one and alternates steps with the exact field and steps with the approximate one. Although such methods have been known and used with success for years, very little is known on how the “space" approximation (of the vector field) and the time discretization should be combined in order to optimize the convergence. Also, the fraction of steps where the exact field is used for the computation is mainly determined by heuristic reasons and a more precise analysis seems necessary. Finally, let us mention that similar questions arise when dealing with constrained differential equations, which are a by-product of many simplified models in molecular dynamics (this is the case for instance if one replaces the highly-oscillatory components by constraints).

The development of efficient numerical methods is essential for the simulation of plasmas and beams at the kinetic level of description (Vlasov type equations). It is well known that plasmas or beams give rise to small scales (Debye length, Larmor radius, gyroperiod, mean free path...) which make numerical simulations challenging. Instead of solving the limit or averaged models by considering these small scales equal to zero, our aim is to explore a different strategy, which consists in using the original kinetic equation. Specific numerical scheme called `Asymptotic Preserving" scheme is then built to discretize the original kinetic equation. Such a scheme allows to pass to the limit with no stability problems, and provide in the limit a consistent approximation of the limit or average model. A systematic and robust way to design such a scheme is the micro-macro decomposition in which the solution of the original model is decomposed into an averaged part and a remainder.

E. Faou was plenary speaker at the CANUM, Congrès d'analyse numérique, France, June 2014

E. Faou was invited to give two presentations in the Analysis and applied mathematics seminars, Cambridge, UK, February 2014.

fixed velocity grid numerical schemes. What is needed is the capability of resolving locally in velocity while maintaining a coarse grid outside the highly perturbed region of phase space. We here report on a new Semi-Lagrangian Vlasov-Poisson solver based on conservative non-uniform cubic splines in velocity that tackles this problem head on. An additional feature of our approach is the use of a new high-order time-splitting scheme which allows much longer simulations per computational effort. This is needed for low amplitude runs. There, global coherent structures take a long time to set up, such as KEEN waves, if they do so at all. The new code's performance is compared to uniform grid simulations and the advantages are quantified. The birth pains associated with weakly driven KEEN waves are captured in these simulations. Canonical KEEN waves with ample drive are also treated using these advanced techniques. They will allow the efficient simulation of KEEN waves in multiple dimensions, which will be tackled next, as well as generalizations to Vlasov-Maxwell codes. These are essential for pursuing the impact of KEEN waves in high energy density plasmas and in inertial confinement fusion applications. More generally, one needs a fully-adaptive grid-in-phase-space method which could handle all small vorticlet dynamics whether pealing off or remerging. Such fully adaptive grids would have to be computed sparsely in order to be viable. This two-velocity grid method is a concrete and fruitful step in that direction.

In , we study the fermionic King model which may provide a relevant model of dark matter halos.

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).

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.

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.

Leader: Ph. Gendrih.

The full description is available at https://

Leader: P. Beyer

Type: FP7

Defi: NC

Instrument: ERC Starting Grant

Objectif: NC

Duration: September 2011 - August 2016

Coordinator: E. Faou

Inria contact: E. Faou

Project acronym: EUROFusion CfP-WP14-ER-01/IPP-03: 2014

Project title: verification of global gyrokinetic codes and development of new algorithms for gyrokinetic and kinetic codes

Duration: 2013-2014

Participants: N. Crouseilles and M. Lemou

Coordinator:E. Sonnendrücker

Project acronym: EUROFusion

Project title: Enabling Research Project for the implementation of the fusion roadmap

Duration: 2015-2017

Participants: N. Crouseilles and M. Lemou

Coordinator:E. Sonnendrücker

L. Einkemmer, University of Innsbruck, two weeks, november 2014.

Y. Zhang, WPI, Vienna, 3 months.

N. Crouseilles visited the group of P. Coelho (Universitad tecnico de Lisboa, Portugal), one week (november 2014).

M. Lemou and N. Crouseilles visited the India Institute of Science at Bangalore (India): from december 2d to december 17th, 2013. Visited team: around Raghurama Rao.

M. Lemou visited the Wisconsin university, Madison (USA): from February 1st to February 16th, 2014. Visited team: around Shi Jin.

P. Chartier, M. Lemou and F. Méhats visited the university of San Sebastien, Pays Basque (Spain): from June 8th to June 13th 2014.

F. Castella organised, jointly with P. Chartier, a meeting held in Saint-Malo (25 participants) in the framework of the european ANR project Lodiquas.

A. Debussche was member of the scientific committee of the conference
*Stochastic Partial Differential Equations and Applications - IX*, Trento, Italy, january 7-11, 2014.

N. Crouseilles is member of the editorial board of Hindawi review "International Journal of Analysis"

M. Lemou is associate editor in the journal "Annales de la faculté des sciences de Toulouse"

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 the "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", SMAI, Springer.

P. Chartier is member of the editorial board of M2AN (Mathematical Modelling and Numerical Analysis).

P. Chartier is member of the editorial board of ISRN Mathematical Analysis.

Master 2 lectures: N. Crouseilles, Numerical methods for kinetic equations.

Master 1 lectures: M. Lemou, Theory of distributions, University of Rennes 1 and ENS Cachan (Ker Lann), 24 hours.

Master 2: M. Lemou was the manager of Master 2 courses in "Analysis and Applications", university of Rennes 1.

E. Faou gave a series of lectures on *Stochastic methods for PDEs*, Heriot-Watt University, Edinburgh, UK, october 2014.

E. Faou gave a series of lectures on *Geometric Numerical Integration for PDE*, KIT, Karlsruhe, Germany, August 2014.

E. Faou gave a series of lectures on *Stochastic computation* and on *Geometric Numerical Integration for PDE*, Chinese Academy of Sciences, Beijing, May 2014

A. Debussche gave a mini-course on *Introduction aux EDPS* in the school *EDP avec conditions aleatoires*, Toulouse, April 22-25, 2014.

Licence 3: P. Chartier gave a lecture on ODEs at ENS Rennes, september-december, 24 hours.

N. Crouseilles and M. Lemou co-advise H. Hivert's PhD (first year in Rennes university), ENS grant.

N. Crouseilles and M. Lemou co-advise (with R. Raghurama and M. Lemou) A. Ruhi's PhD (third year in IISc), Indian grant.

M. Lemou and F. Méhats co-advised P. Carcaud's PhD: University of Rennes 1. Thesis defense on june 2nd 2014.

P. Chartier and F. Méhats co-supervise the PhD thesis of G. Leboucher.

P. Chartier and F. Castella co-supervise the PhD thesis of J. Sauzeau.

A. Debussche and F. Méhats co-supervise the PhD thesis of M. Tusseau.

E. Faou co-supervises the PhD thesis of R. Horsin.

A. Debussche and E. Faou co-supervised the thesis of M. Kopec, ENS Rennes.

N. Crouseilles: member of the PHD jury of P. Glanc, 20 january 2014 (Strasbourg); co-advising (with M. Mehrenberger) of Pierre Glanc PhD (Strasbourg University), Inria-Cordi grant.

N. Crouseilles: member of the PHD jury of Ch. Steiner, 11 december 2014 (Strasbourg); co-advising (with M. Mehrenberger) of Christophe Steiner PhD (Strasbourg University), ministry grant.

N. Crouseilles: member of the PHD jury of M. Kuhn, 29 september 2014 (Strasbourg); co-advising (with S. Genaud) of Matthieu Kuhn PhD (Strasbourg University and Inria IPSO), ANR "E2T2" grant.

N. Crouseilles: member of the Master 2 jury of P. Pereira, 26 november 2014 (Lisboa, Portugal).

F. Méhats was referee of the thesis of L. Hari (Cergy, supervised by T. Duyckaerts and C. Fermanian).

F. Méhats was referee of the thesis of X. Zhao (Singapore, supervised by W. Bao).

P. Chartier was referee of the PhD thesis of Philipp Bader, University of Valencia, june.