Section: Overall Objectives

Overall objectives

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.