Forward Symplectic Algorithms for Solving Physical Evolution Equations

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This project seeks to develop a new class of higher order decomposition, or split operator algorithms for solving both time-reversible (Hamilton, Schreodinger, Gross-Pitaevskii) and time-irreversible (finite temperature, Fokker-Planck, Navier-Stokes) evolution equations. For time-reversible systems, these algorithms are automatically symplectic or unitary. Up to now, it has not been possible to develop similar algorithms for solving time-irreversible systems beyond second order because conventional higher order decomposition algorithms all have negative time steps not implementable in time-irreversible systems. Fourth and higher order forward time step algorithms developed in this project can solve both types of equation with very large time steps, allowing much longer time simulations of classical, stochastical and quantum dynamical systems. The success of this project will provide a set of fundamentally new and very powerful numerical tools for scientists and engineers to solve a variety of problems with greater precision and longer simulation time than before. This can impact the long term prediction of satellite motions near earth, the long time dynamics of pharmacological macromolecules in solutions, the propagation of electromagnetic wave in fibra cables, the response of atoms to extremely short and intense laser pulses, the understanding of superfluid behavior in finite systems as the temperature is lowered, the theoretical design of new materials, and the use of Bose-Einstein condensate as a magnifying, detection, or quantum device.