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IPSO Research team

Invariant Preserving SOlvers

Team presentation

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 methods have been used for years with great success in molecular dynamics or astronomy. However, they now need to be understood more deeply and improved further in order to fit the tremendous increase of complexity and size of the models.

Research themes

The IPSO project-team 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 space-discretisation of the Schrödinger equation).

International and industrial relations

The IPSO group aims at addressing the potential of direct simulation of molecular dynamics in chemical processes, ranging from applications in nanotechnology to the identification and prediction of biomolecular functionalities. Up to now, many challenges are presently not yet available for large-scale simulations : on the one hand, the mathematically founded extraction of macroscopic properties from molecular simulations is still in its beginning; on the other hand, the incorporation of quantum effects into molecular simulation presents is a real challenge. On this topic, IPSO has interactions with the following teams, mainly through the PRESTISSIMO action (ARC):

- Team MICMAC (CERMICS-INRIA),
- C. Chipot (CNRS Nancy),
- O. Coulaud (Team ScAlApplix, INRIA Bordeaux),
- E. Darve (Stanford University, USA),
- B. Leimkuhler (Leicester University, England),
- G. Zerah (CEA).


IPSO is also concerned with laser simulation. 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 1 micro-meter, over distances of the order 0.01 m to 1000 m. In these situations, the propagation produces both a short-scale 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 high-oscillations, and to build up models and/or numerical schemes that do give information on the long-term behavior. In other terms, one needs to develop high-frequency models and/or high-frequency schemes. This task has been partially performed in the context of the Alcaltel contract, for which we developed a new numerical scheme to discretize directly the high-frequency model derived from physical laws. With respect to this domain of application, we have interactions with the following teams:

- D. Bayart, F. Leplingard, C. Martinelli (Transmission Department of Alcatel, Marcoussis),
- Th. Colin and G. Métivier (MAB, University of Bordeaux I and II).

Keywords: Ordinary differential equations Geometric integration Laser-matter interaction Molecular dynamics.