2.1 Robust geometric integration methods for long time dynamics are a rapidly growing field within numerical analysis. However little is known of the behaviour of such methods when applied to deterministic systems with stochastic solutions, especially in the presence of symmetries, first integrals and/or adiabatic invariants. Strong assumptions, such as uniform hyperbolicity and the existence of Poincaré maps, are needed for current theoretical results. This project will extend these results to the types of systems that are being studied by this network.
2.2 These theoretical investigations will be closely linked with the development of new simulation techniques for small molecular and atomic systems, possibly including quantum effects. Recent progress has been made on adaptive and multi-scale methods for systems with multiple time scales and rapid changes in the dynamics. Special techniques are needed for the Coulombic few-body problem of celestial and atomic mechanics, including multiple long sampling trajectories, e.g. for scattering cross-sections and stability diagrams. Regularizing transformations are needed to stabilize dynamics during close-approaches and there is some work on the use of geometric integrators for this, but much more is still needed, particularly for problems involving close approaches of three or more bodies.
2.3 It has been demonstrated that multi-symplectic methods are particularly well suited to the integration of Hamiltonian PDEs arising in oceanography, atmospheric dynamics and in optical fibre design. In particular they give excellent numerical preservation of the local energy and momentum conservation laws and in turn excellent preservation of densities and fluxes. These techniques will be develped further in this project and applied to the continuum systems studied in section 4 of the project.