NUMERICAL METHODS

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:

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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.

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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.

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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.