NUMERICAL METHODS

Topics

2.1	Stochastic long-time behaviour of geometric integration methods
2.2	Adaptive and structure preserving methods for molecules and atoms
2.3	Multi-symplectic integration methods for Hamiltonian PDEs

Project objectives

This part of the project is concerned with the development of new geometric numerical methods especially suited to the theoretical projects and mechanical systems described in this proposal. Specific objectives include:

Long time dynamics of geometric integration methods when applied to Hamiltonian systems with stochastic dynamics and possible underlying symmetries. Development of methods that reproduce the correct stochastic behavior of such system, for example, evolution of densities, decay of correlation, and time averages.
Studies of small molecular and atomic systems with robust and efficient adaptive and structure preserving methods. Inclusion of quantum dynamics and simulation methods for combined quantum-classical systems.
Development of methods for Hamiltonian PDEs which conserve symplecticity, including methods that preserve the multi-symplectic structure of phase space and energy-momentum conservation laws. Simulation of wave phenomena in continuum dynamics.

Major breakthroughs are likely to include the development of numerical techniques which preserve the geometrical structure under adaptivity and the development of novel geometric integration methods for PDEs.

Scientific originality

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.

Research method

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.

2000-06-22