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 be used for both ODEs and PDEs, either deterministic or stochastic.
The IPSO project-team aims at finding and implementing new structure-preserving schemes and at understanding the behavior of existing ones for Hamiltonian ODEs and PDEs, constrained systems, and highly-oscillatory systems (with a special focus of those resulting from the space-discretisation of the Schrödinger equation).