2.1 We will apply recent results from backward error analysis, numerical shadowing and spectral properties of the associated Frobenius-Perron operator to study the long time dynamics of numerical methods. In particular these methods will be applied to the numerical study of small molecular systems. The time evolution of densities, the approximation of time averages along numerically computed solutions, and the decay towards an equilibrium density will be studied.
2.2 The development of novel integration techniques for molecular and atomic systems will be based on adaptive regularization methods and averaging techniques. Smoothly-switched geometric integrators for Coulombic N-body problems will be extended to problems involving three-body close approaches and used to handle difficult orbits in few-body atomic models (e.g. near the Wannier ridge in helium). Geometric properties of smoothly-switched integrators will also be investigated. Multiple-time stepping and averaging techniques will be applied to quantum-classical simulations.
2.3 The development of novel integration techniques for continuum mechanics will be within the multi-symplectic framework. Multi-symplectic methods based on finite volume, finite element and spectral discretizations are also to be investigated. Furthermore, the development of adaptive and problem specific multi-symplectic methods is required. Here the idea of adapting moving mesh methods to the Hamiltonian setting will be explored, as will the idea of adapting splitting techniques to PDEs. Long time simulations of wave interactions are to be carried out and used as a benchmark against existing codes. Theoretical issues include formulation of a backward error analysis for multi-symplectic integration methods, to give rigorous estimates of their properties.