Forward Symplectic Algorithms for Solving Physical Evolution Equations

该项目旨在开发一类新的高阶分解或分裂算子算法，用于解决时间可逆（Hamilton, Schreodinger, Gross-Pitaevskii）和时间不可逆（有限温度，Fokker-Planck, Navier-Stokes）进化方程。对于时间可逆系统，这些算法是自动辛的或酉的。到目前为止，由于传统的高阶分解算法都具有负时间步长，无法在时间不可逆系统中实现，因此不可能开发出类似的算法来求解二阶以上的时间不可逆系统。本项目开发的四阶和高阶前向时间步长算法可以用非常大的时间步长求解这两种类型的方程，从而允许对经典、随机和量子动力系统进行更长时间的模拟。该项目的成功将为科学家和工程师提供一套全新的、非常强大的数值工具，以比以前更高的精度和更长的模拟时间解决各种问题。这可能会影响近地卫星运动的长期预测，溶液中药理学大分子的长期动力学，光纤电缆中电磁波的传播，原子对极短且强烈的激光脉冲的响应，对温度降低时有限系统中超流体行为的理解，新材料的理论设计，以及将玻色-爱因斯坦凝聚物用作放大，检测或量子装置的使用。

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.