A fundamental and enduring challenge in science and technology is the quantitative prediction of timedependent 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 underresolved 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 structurepreserving 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 symplecticityor timereversibilityare 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, volumepreserving integrators, Liegroup 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 IPSOaims at finding and implementing new structurepreserving 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 differentialalgebraic 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).
highlyoscillatory 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...), IPSOwill mainly concentrate on applications for molecular dynamics simulationand 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 structurepreserving integrators. Due to the complexity of the problem, more efficient numerical schemes need to be developed.
The demand for new models and/or new structurepreserving schemes is also quite large in laser simulations. The propagation of lasers induces, in most practical cases, several wellseparated scales: the intrinsically highlyoscillatory wavestravel over long distances. In this situation, filtering the oscillations in order to capture the longterm trend is what is required by physicists and engineers.
In many physical situations, the timeevolution of certain quantities may be written as a Cauchy problem for a differential equation of the form
For a given
y_{0}, the solution
y(
t)at time
tis denoted
. For fixed
t,
becomes a function of
y_{0}called the
flowof (
). From this point of view, a numerical scheme with step size
hfor solving (
) may be regarded as an approximation
_{h}of
. One of the main questions of
geometric integrationis whether
intrinsicproperties of
may be passed on to
_{h}.
This question can be more specifically addressed in the following situations:
The system ( ) is said to be reversible if there exists an involutive linear map such that
It is then natural to require that _{h}satisfies the same relation. If this is so, _{h}is said to be symmetric. Symmetric methods for reversible systems of ODEs are just as much important as symplecticmethods for Hamiltonian systems and offer an interesting alternative to symplectic methods.
The system (
) is said to have an invariant manifold
gwhenever
is kept
globallyinvariant by
. In terms of derivatives and for sufficiently differentiable functions
fand
g, this means that
As an example, we mention Liegroup equations, for which the manifold has an additional group structure. This could possibly be exploited for the spacediscretisation.
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
gexplicitly or through a projection step. In both cases, the numerical solution is forced to live on the manifold at the expense of some Newton's iterations.
Hamiltonian problems are ordinary differential equations of the form:
with some prescribed initial values
(
p(0),
q(0)) = (
p
_{0},
q
_{0})and for some scalar function
H, called the Hamiltonian. In this situation,
His an invariant of the problem. The evolution equation (
) can thus be regarded as a differential equation on the manifold
Besides the Hamiltonian function, there might exist other invariants for such systems: when there exist
dinvariants in involution, the system (
) is said to be
integrable. Consider now the parallelogram
Poriginating from the point
and spanned by the two vectors
and
, and let
(
,
)be the sum of the
orientedareas of the projections over the planes
(
p
_{i},
q
_{i})of
P,
where
Jis the
canonical symplecticmatrix
A continuously differentiable map
gfrom
to itself is called symplectic if it preserves
, i.e. if
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 symplecticnumerical flows that share most of the properties of the exact flow. For practical simulations of Hamiltonian systems, symplectic methods possess an important advantage: the errorgrowth as a function of time is indeed linear, whereas it would typically be quadratic for nonsymplectic 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 differentialalgebraic systems. They can be classified according to their index, yet for the purpose of this expository section, it is enough to present the socalled index2 systems
where initial values
(
y(0),
z(0)) = (
y
_{0},
z
_{0})are given and assumed to be consistent with the constraint manifold. By constraint manifold, we imply the intersection of the manifold
and of the socalled hidden manifold
This manifold
is the manifold on which the exact solution
(
y(
t),
z(
t))of (
) lives.
There exists a whole set of schemes which provide a numerical approximation lying on . Furthermore, this solution can be projected on the manifold by standard projection techniques. However, it it worth mentioning that a projection destroys the symmetry of the underlying scheme, so that the construction of a symmetric numerical scheme preserving requires a more sophisticated approach.
In applications to molecular dynamics or quantum dynamics for instance, the righthand side of ( ) involves fastforces (shortrange interactions) and slowforces (longrange interactions). Since fastforces are much cheaper to evaluate than slowforces, 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 highlyoscillatory systems is the secondorder differential equations
where the potential
V(
q)is a sum of potentials
V=
W+
Uacting on different timescales, with
^{2}Wpositive definite and
. In order to get a bounded error propagation in the linearized equations for an explicit numerical method, the step size must be restricted according to
where
Cis a constant depending on the numerical method and where
is the highest frequency of the problem, i.e. in this situation the square root of the largest eigenvalue of
^{2}W. In applications to molecular dynamics for instance,
fastforces deriving from
W(shortrange interactions) are much cheaper to evaluate than
slowforces deriving from
U(longrange interactions). In this case, it thus 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.
Another prominent example of highlyoscillatory 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 timedependent Schrödinger equation:
where
H(
t)is finitedimensional matrix and where
typically is the squareroot of a massratio (say electron/ion for instance) and is small (
or smaller). Through the coupling with classical mechanics (
H(
t)is obtained by solving some equations from classical mechanics), we are confronted once again to two different timescales, 1 and
. In this situation also, it is thus desirable to devise a numerical method able to advance the solution by a timestep
h>
.
Given the Hamiltonian structure of the Schrödinger equation, we are led to consider the question of energy preservation for timediscretization schemes.
At a higher level, the Schrödinger equation is a partial differential equation which may exhibit Hamiltonian structures. This is the case of the timedependent Schrödinger equation, which we may write as
where
=
(
x,
t)is the wave function depending on the spatial variables
with
(e.g., with
d= 1or 3 in the partition) and the time
. Here,
is a (small) positive number representing the scaled Planck constant and
iis the complex imaginary unit. The Hamiltonian operator
His written
H=
T+
V
with the kinetic and potential energy operators
where
m_{k}>0is a particle mass and
_{xk}the Laplacian in the variable
, and where the realvalued potential
Vacts as a multiplication operator on
.
The multiplication by
iin (
) plays the role of the multiplication by
Jin classical mechanics, and the energy
is conserved along the solution of (
), using the physicists' notations
where
denotes the Hermitian
L^{2}product over the phase space. In quantum mechanics, the number
Nof particles is very large making the direct approximation of (
) very difficult.
The numerical approximation of ( ) can be obtained using projections onto submanifolds of the phase space, leading to various PDEs or ODEs: see , for reviews, and Section for the case of Gaussian wave packets dynamics. However the longtime 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:
i(
t)
V/2)exp(
i(
t)
)exp(
i(
t)
V/2)
_{0}
where
tis the time increment (we have set all the parameters to 1 in the equation). As the Laplace operator is unbounded, we cannot apply the standard methods used in ODEs to derive longtime
properties of these schemes. However, its projection onto finite dimensional submanifolds (such as Gaussian wave packets space or FEM finite dimensional space of functions in
x) may exhibit Hamiltonian or Poisson structure, whose longtime properties turn out to be more tractable.
The Helmholtz equation modelizes the propagation of waves in a medium with variable refraction index. It is a simplified version of the Maxwell system for electromagnetic waves.
The highfrequency 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 highfrequency 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,
is the small adimensional parameter that measures the typical wavelength of the signal,
n(
x)is the spacedependent refraction index, and
is a given (possibly dependent on
) source term. The unknown is
. One may think of an antenna emitting waves in the whole space (this is the
), thus creating at any point
xthe signal
along the propagation. The small
term takes into account damping of the waves as they propagate.
One important scientific objective typically is to describe the highfrequency regime in terms of rayspropagating 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 longtime 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 highfrequency behaviour of the Helmholtz equation.
The Schrödinger equation is the appropriate 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 semiconductors, 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
V(
x)is the potential and
(
t,
x)is the time and spacedependent wave function. However, the size of real devices makes it important to derive simplified models that are valid at a larger scale.
Typically, one wishes to have kinetic transport equations. As is wellknown, this requirement needs one to be able to describe “collisions” between electrons in these devices, a concept that
makes sense at the macroscopic level, while it does not at the microscopic (electronic) level. Quantitatively, the question is the following: can one obtain the Boltzmann equation (an equation
that describes collisional phenomena) as an asymptotic model for the Schrödinger equation, along the physically relevant micromacro asymptotics? From the point of view of modelling, one wishes
here to understand what are the “good objects”, or, in more technical words, what are the relevant “crosssections”, that describe the elementary collisional phenomena. Quantitatively, the
Boltzmann equation reads, in a simplified, linearized, form :
Here, the unknown is
f(
x,
v,
t), the probability that a particle sits at position
x, with a velocity
v, at time
t. Also,
(
v,
v^{'})is called the crosssection, and it describes the probability that a particle “jumps” from velocity
vto velocity
v^{'}(or the converse) after a collision process.
The technique consists in solving an approximate initial value problem on an approximate invariant manifold for which an atlas consisting of easily computablecharts exists. The numerical solution obtained is this way never drifts off the exact manifold considerably even for longtime integration.
Instead of solving the initial Cauchy problem, the technique consists in solving an approximate initial value problem of the form:
on an invariant manifold
, where
and
approximate
fand
gin a sense that remains to be defined. The idea behind this approximation is to replace the differential manifold
by a suitable approximation
for which an atlas consisting of
easily computablecharts exists. If this is the case, one can reformulate the vector field
on each domain of the atlas in an
easyway. The main obstacle of
parametrizationmethods
or of
Liemethods
is then overcome.
The numerical solution obtained is this way obviously does not lie on the exact manifold: it lives on the approximate manifold . Nevertheless, it never drifts off the exact manifold considerably, if and are chosen appropriately closeto each other.
An obvious prerequisite for this idea to make sense is the existence of a neighborhood
of
containing the approximate manifold
and on which the vector field
fis welldefined. In contrast, if this assumption is fulfilled, then it is possible to construct a new admissible vector field
given
. By admissible, we mean tangent to the manifold
, i.e. such that
where, for convenience, we have denoted . For any , we can indeed define
where is the projection along .
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
10
^{6}
m, over distances of the order
10
^{2}
mto
10
^{4}
m. In these situations, the propagation produces both a shortscale oscillation and exhibits a long term trend (drift, dispersion, nonlinear interaction with the
medium, or so), which contains the physically important feature. For this reason, one needs to develop ways of filtering the irrelevant highoscillations, and to build up models and/or
numerical schemes that do give information on the longterm behavior. In other terms, one needs to develop highfrequency models and/or highfrequency schemes.
This task has been partially performed in the context of a contract with Alcatel, in that we developed a new numerical scheme to discretize directly the highfrequency model derived from physical laws.
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 longtime 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 bewteeen this potential and the laser itself is given by the socalled 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's field. A typical phenomenon here is the resonancebetween 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 shortrange or longrange and are treated differently in most molecular simulation codes. In fact, longrange 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 byproduct of many simplified models in molecular dynamics (this is the case for instance if one replaces the highlyoscillatory components by constraints).
The work described in this section has been conducted in collaboration with Chr. Lubich and Vasile Gradinaru, from the University of Tübingen (Germany).
Gaussian wavepacket dynamics is widely used in quantum molecular dynamics, see for instance
,
. In this case, an approximation to the wave function
(
x,
t)solution of (
) is sought for in the form
with
where ·and ·denote the Euclidean norm and inner product on , respectively.
The DiracFrenkelMcLachlan variational principle yields equations of motion for these parameters. It turns out that this system of ordinary differential equations has a Poisson structure inherited from the Hamiltonian structure of the Schrödinger equation, and that to the semi classical limit 0, these equations tend to the finite dimensional Hamiltonian system , .
In a previous work, C. Lubich and E. Faou show that the projection of the splitting scheme ( ) onto the submanifold made of Gaussian wave packets yields a numerical scheme that is a Poisson integrator, which can be computed explicitly. Using backward error analysis, this shows in particular the preservation of energy for exponentially long time. If the potential has a rotational symmetry so that the angular momentum is conserved in the full quantum dynamics, then the numerical integrator also preserves the angular momentum.
The natural extension of this work is to consider the product of the previous Gaussian by polynomials. As the degrees of these polynomials increase, the corresponding submanifold of
L^{2}is expected to fill in the whole
L^{2}space, making this representation more accurate than Gaussians only. This work is at present still in progress in collaboration with C. Lubich and V. Gradinaru.
This is a joint work with E. Hairer, from the University of Geneva.
The above idea has been taken up in
. In a more general context (no restriction to Bseries) the following problem is considered: for a given onestep
method (typically very simple and of low order), find a differential equation written as a formal series in powers of the step size
h, such that the numerical solution of the method applied to this modified differential equation yields the exact solution in the sense of formal power series. Truncating the series gives
raise to new integrators of arbitrarily high order. The article
, written in honour of Michel Crouzeix, summarizes the main results of
and shows possible applications of the new integrators. The implicit midpoint rule is used as an illustrating
example.
and two numerical integrators
y_{n+ 1}=
_{f,
h}(
y_{n})and
y_{n+ 1}=
_{f,
h}(
y_{n}). The problem that we address in this article is the study of a modified differential equation, written as a formal series in powers of the step size
h,
such that the numerical solution of the method applied to ( ) is (formally) equal to the numerical solution of the method applied to the modified differential equation ( ), i.e.,
This permits us to present a unified theory and extensions of topics like:
backward error analysis,which is obtained by letting
be the exact flow of the differential equation (
). Consequently, the numerical solution of
_{f,
h}(
y)becomes the exact flow of (
). This theory is fundamental for the analysis of geometric integrators and it is treated in much detail in the monographs
of SanzSerna & Calvo
, Hairer, Lubich & Wanner
, and Leimkuhler & Reich
.
exact integration methods,which are obtained by letting
_{f,
h}(
y)be the exact flow of (
) and by taking for
_{f,
h}(
y)a simple numerical integrator. Truncating the modified equation, high order numerical integrators are constructed in this way. This approach is popular for
Hamiltonian systems through the work of Feng Kang. It also permits the construction of symplectic elementary differential Runge–Kutta methods as first considered by Murua.
This paper also explains the connection with some exact integration methods – generating function methods for Hamiltonian systems, and a recent modification by McLachlan & Zanna of the discrete Moser–Veselov algorithm for the free rigid body.
This a joint work with Ander Murua, from the University of San Sebastian.
Preserving volume forms is a necessary requirement in several wellidentified applications, such as molecular dynamics or meteorology, while preserving first integrals is vastly recognized as fundamental in a very large number of physical situations. Although the requirements appear somehow disconnected, they lead to algebraic conditions which have strong similarities and this is the very reason why we address these questions together.
It is however interesting to consider specific classes of problems, for which volumepreserving integrators can be constructed. For instance, it is clear that symplectic methods are volumepreserving for Hamiltonian systems: we show that symplectic conditions are in general necessary for a method to be volumepreserving and indeed sufficient for the special class of Hamiltonian problems. In a similar spirit, we derive simplified conditions for partitioned systems with two functions and three functions. The results obtained for two functions corroborate already known ones and results for more than three functions (and their straightforward generalization to more functions) appear to be completely new.
The equation considered in this work is the linear Schrödinger equation in one space dimension
where
is the complex unknown wave function depending on the space variable
and the time
t0. The potential
Vis a real function and the function
is the initial value of the wave function at
t= 0.
For a given time step
h>0, we consider the approximation scheme
where by definition,
is the solution
(
t)at the time
t=
hof the equation
and similarly
is the solution
(
t)at the time
t=
hof the equation
If the potential is smooth enough, it can be shown that this approximation is a first order approximation of the solution of the Schrödinger equation. But the question of the long time behaviour of the numerical solution corresponding to the splitting scheme is a much more difficult issue.
In the finite dimensional case, the behavior of splitting methods for hamiltonian systems is now well understood, see for instance . In particular, the use of the BakerCampbellHausdorff formula shows that for a sufficiently small stepsize depending on the highest eigenvalue of the problem, there exists a modified hamiltonian for the propagator ( ). The numerical flow can thus be interpreted as the exact solution of a hamiltonian system, at least for exponentially long time with respect to the stepsize. This result holds true for the linear and the nonlinear case.
In our case, though the initial equation is linear, the splitting propagator can be viewed as a nonlinear function of the infinite dimensional operators

_{xx}and
V, and we use techniques similar to the one used in classical perturbation theory to put the propagator (
) under a normal form that will give information on the long time behavior of its solution.
The idea is to consider for a fixed time step
hthe family of propagators
and to assume that
Vis analytic. For
= 0, we see that
L(0)is the free linear Schrödinger propagator. The corresponding solution can be written explicitly in terms of Fourier coefficients. The dynamics is periodic in time
and there is no mixing between the different Fourier modes. The regularity of the initial value is preserved.
To show this result, we use the following nonresonance condition on the stepsize (see ): there exist >0and >1such that
It can be shown that for a given
h_{0}>0close to 0, the set of time steps
h(0,
h_{0})that do not satisfy this condition has a Lebesgue measure
for some
r>1.
Using this almost Xshaped representation, we can analyze the long time behavior of the numerical solution and show that the dynamics can be reduced to two dimensional linear symplectic
systems mixing the two modes
kand

kfor
. This implies in particular the quasiconservation of the regularity of the initial solution for these asymptotically large modes. The method is close to standard techniques used in
finite dimensional perturbation theory, but extended here to infinite dimensional operators. The main results were announced in
.
Consider a Hamiltonian system
where
, and with a separable Hamiltonian
Hof the form
where
V(
q)is the potential function. In many applications, such as for instance molecular dynamics, it is of importance that the numerical flow used to compute the solution of
preserves the volume form and the Hamiltonian. However, it is generally admitted that no standard method can satisfy both
requirements, apart from exceptional situations such as for instance a quadratic Hamiltonian. A possible approach could be to solve in sequence the
dHamiltonian systems with Hamiltonians
obtained by freezing all components (denoted with a bar) except the two conjugate coordinates
q_{i}and
p_{i}. If each subsystem can be solved exactly and the same stepsize is used for all, the resulting “numerical" method preserves the desired quantities, since each substep is symplectic and
preserves
H^{[
i]}(and thus
H). Considering that each subsystem is of dimension 2 and thus integrable, it can be hoped that an exact solution is indeed obtainable in some specific situations. Nevertheless, such
situations are rather nongeneric, though it is important to mention at this stage the special case of
multiquadraticpotentials, i.e. potentials such that for all
i= 1, ...,
dand all
,
V^{[
i]}is
quadraticin
q_{i}. In this context, the method described above has been introduced in by R. Quispel and R.I. McLachlan in
.
In order to retain the possibility of solving exactly each subsystem and at the same time to cover more general problems, we give up the requirement of exact Hamiltonian preservation and we
consider a multiquadratic piecewise approximation of
H. If instead of
we now solve
where
is a
C^{1, 1}multiquadratic approximation of
H, the aforementioned procedure applied with exact solution of the subsystems gives a firstorder method which preserves
exactly as well as the volume form. If
for a compact subset
Kof
containing the numerical solution, then
His conserved up to an error of size
over arbitrarily long intervals of integration (including infinite ones).
Note that this approach remains valid for more general Hamiltonians (nonseparable for instance), provided an exact solution can be computed, so that all theoretical results concerning the conservation of energy and volume will be stated for general Hamiltonians. In contrast, we will describe the implementation of the method with quadratic Bsplines only for the case of separable Hamiltonians.
In the two texts and , we consider atoms interacting with a highfrequency signal, typically a laser. We propose an original model, which is purely classical, based on the parallel with the modelling of the similar situation at the quantum level. Our model provides a kinetic equation with a highfrequency signal, and involves a fast relaxation operator that takes into account the observed trend of atoms to relax towards equilibrium states of the atomic Hamiltonian. We study the physically relevant limit when the highfrequency signal becomes infinite frequency, while the relaxation becomes infinitely fast. We carefully analyze the precise physical regime involved. Combining tools borrowed from the averaging theory of ordinary differential equations (and conveniently adapted to the partial differential equation under study), together with tools borrowed from the fluid limits for kinetic equations, we completely identify the asymptotic dynamics. We prove that the atoms tend to have diffusive behaviour in the energy variable, while the diffusion process involves coefficients that are appropriate timeaverages of the original laser signal. Last, we comment on the parallel between the present classical description of matter, and the corresponding quantum situation. Our analysis does not require periodic, nor multiperiodic highfrequency signals: general, oscillating signals are allowed.
In the two texts and , we consider a gas of electrons which is strongly confined along a plane, resp. a nanowire (a line). Such situations naturally occur in the context of semiconductors, when dealing with heterojunctions e.g. The gas is apriori described by a nonlinear Schrödinger equation (the nonlinearity takes the coupling between the electrons into account), yet the strong confinement creates a highly oscillatory term. The point is, we wish to compute the asymptotic behaviour of the gas in the limit, so as to read off its limiting 2D, resp. 1D dynamics. To do so, we develop an original functional framework, and our key observation lies in the fact that the oscillations occur in an almost periodic fashion. This allows us to perform a complete averaging procedure, using the tools of averaging for ordinary differential equations in the almost periodic context. We here completely justify physical models that have been used previously, and actually extend their domain of validity.
The INGEMOL project is concerned with the numerical simulation of differential equations by socalled geometric methods, i.e. methods preserving some of the qualitative features of the exact solution. Conserving the energy or the symmetry is often physically relevant and of paramount importance in some applications such as molecular simulation or propagation of laser waves in fibers (these are the main applications considered within the project, though several others are possible: robotics, celestial mechanics…). Though a lot has been achieved by numerical analysts in the domain of numerical integration during the last two decades, with most significantly the introduction of symplectic schemes and their analysis through backward error techniques, a lot remains to be done in situations where the existing theory fails to give a useful answer; the goal of the INGEMOL project is to help solving these difficulties in some wellidentified cases : 1. whenever symmetric multistep methods are used for Hamiltonian systems, 2. whenever splitting methods are used for the Schrödinger equation, 3. whenever the system under consideration has highlyoscillating solutions.
Taking into account in the theory the unboundedness of the operators or the high oscillations of the solutions allows for the construction, in a second step, of more appropriate numerical schemes with fewer or none of the present restrictions.
Eventually, it is planned to implement the new schemes with in view their application to the simulation of laser waves and to molecular simulation.
P. Chartier is coordinator of the project. INGEMOL associates the following persons and teams:
F. Castella, P. Chartier, M. Crouzeix, G. Dujardin, E. Faou, G. Vilmart: IPSO
Ch. Chipot: Structure et réactivité des systèmes moléculaire complexes, CNRS, Nancy.
S. Descombes: ENS LYON.
E. Cancès, C. Le Bris, F. Legoll, T. Lelièvre, G. Stoltz: CERMICS, ENPC, MarnelaVallée.
This is an exchange program between the ipso team and the numerical analysis group in Tübingen, headed by C. Lubich. E. Faou is the coordinator of the french part of this project. In 2007, this program financed the following oneweek visits:
L. Gauckler, V. Gradinaru and Chr. Lubich from Tübingen
E. Faou (1 time), G. Dujardin from IPSO.
This program was valid for two years (2006 and 2007).
P. Chartier was chair of the scientific committee of the international conference SciCADE'07.
P. Chartier is member of the editorial board of M2AN.
P. Chartier is member of the editorial board of ESAIM Proceedings.
P. Chartier is guest editorinchief of a special issue of M2AN devoted to numerical methods for the integration of ODEs.
E. Faou was chair of the organization committee of the international conference SciCADE'07.
E. Faou was coorganiser of a minisymposium at he international conference SciCADE'07.
F. Castella is a member of the organizing commitee of the international conference SciCADE'07.
F. Castella is the director of the GdR CNRS 'CHANT' ('equations Cinetiques et Hyperboliques : Aspects Numeriques, Theoriques, et de modelisation'). [budget=15000 Euros per year, approximately 300 persons, and about 4 events organized per year].
P. Chartier is member of the COST (Advisory Committee for Scientific and Technological Orientations) at INRIA.
P. Chartier is member of the Comité des Projets at INRIARennes.
E. Faou is member of the Commission d'Evaluation at INRIA.
E. Faou is member of commission de spécialistes, section 26, of the Ecole Normale Supérieure de Cachan.
F. Castella is member of commission de spécialistes, section 26, of INSA, University of Rennes I.
F. Castella is member of commission de spécialistes, section 26, of the Ecole Normale Supérieure de Cachan.
E. Faou is oral examiner at ENS Cachan Bruz (“agrégation”).
P. Chartier gave a lecture at the summer school CEAEDFINRIA, " Optimal Control: Algorithms and Applications", June 2007.
P. Chartier was invited to give a talk at the University of the Basque Country, October 2007.
P. Chartier was invited to give a talk at the University of Geneva, November 2007.
E. Faou was invited to give a talk in the "SŽminaire d'analyse", University of Nantes, January 2007.
E. Faou was invited to give a seminar at Inria Sophia, May 2007.
E. Faou was invited to give a talk at Basel University, November 2007.
E. Faou gave a seminar at the Isaac Newton Institute, March 2007.
E. Faou was invited to give a talk in workshop on "Applying Geometric Integrators" at the Maxwell Institute, Edimborough, April 2007.
F. Castella was invited to give a talk at the Workshop 'Inhomogeneous Random Systems', I.H.P. Paris.
G. Dujardin was invited to give a talk at SciCADE 07, International Conference on SCIentific Computation And Differential Equations, SaintMalo (France), July 2007
G. Dujardin was invited to give a talk at JournŽe de l'Žquipe d'analyse numŽrique de l'IRMAR, Rennes, March 2007.
G. Dujardin was invited to participate and to give a talk at the "Manifold And Geometric Integration Colloquium 2007 (MAGIC 07)" in Atnasj¿en, Norway, May 2007.
G. Vilmart was invited to give a talk at SciCADE 07, International Conference on SCIentific Computation And Differential Equations, SaintMalo (France), July 2007
G. Vilmart was invited to give a talk at the Second Graduate Colloquium, Swiss Doctoral Program in mathematics, Basel (Switzerland), May 2007.
P. Chartier visited the University of San Sabastian for two weeks, at the invitation of A. Murua.
P. Chartier visited the Isaac Newton Institute for one week, at the invitation of A. Iserles.
E. Faou visited the University of Tübingen in april and in december 2007 using the PAI exchange program between the numerical analysis group in Tübingen and the team IPSO.
E. Faou visited the Isaac Newton Institute for two weeks, at the invitation of A. Iserles.
F. Castella visited the Isaac Newton Institute for one week, at the invitation of A. Iserles.
G. Dujardin visited the University of Tübingen in december 2007 using the PAI exchange program between the numerical analysis group in Tübingen and the team IPSO.
G. Dujardin visited the Isaac Newton Institute for two weeks, at the invitation of A. Iserles.
G. Vilmart visited the Isaac Newton Institute for three weeks, at the invitation of A. Iserles and E. Hairer.
The team has invited the following persons :
L. Gauckler on a oneweek visit.
A. Iserles on a a twodays visit.
P. Joly on a twodays visit.
V. Gradinaru on a twoweeks visit.