Mathematical Sciences: Numerical Methods for Hamiltonian Systems with Many Degrees of Freedom: Constraints, Nonconservative Perturbations, and Parallel Computing

这个项目涉及在自然力作用下大群粒子或物体相互作用的动力学的有效数值解。例如，在生物大分子动力学的研究、药物设计、等离子体模拟、飞行器和空间结构模拟以及宇宙学中，都会遇到多体和多粒子系统。相关问题是由含时偏微分方程组的空间半离散化引起的。通过超越各种代码和数值技术的特定应用特征，该项目将为动态模拟寻找新的、广泛适用的数学和计算工具。数学研究的重点将是保持和利用运动方程中的哈密顿结构，例如通过使用新的辛离散化格式。该项目将通过考虑受约束的哈密顿系统和几乎哈密顿系统，扩大当前数值分析人员之间的研究对话。将寻求适合在并行计算机上实现的方法。在所有阶段，将强调计算效率和实用性，而不仅仅是数学新颖性，并将采用符合物理科学传统的实验方法。

This project is concerned with efficient numerical solution of the dynamics of large groups of particles or bodies interacting under the forces of nature. Many-body and many-particle systems are encountered, for example, in studies of the dynamics of biological macromolecules, in drug design, in the simulation of plasmas, in vehicle and space-structure simulation, and in cosmology. Related problems arise through spatial semidiscretization of time-dependent partial differential equations. By looking beyond the application-specific features of the various codes and numerical techniques, this project will seek new, broadly applicable mathematical and computational tools for dynamical simulations. The focus of the mathematical study will be on preserving and exploiting Hamiltonian structure in the equations of motion, e.g. by using new symplectic discretization schemes. The project will broaden the current research dialogue among numerical analysts by considering constrained Hamiltonian systems and nearly Hamiltonian systems. Methods suited to implementation on parallel computers will be sought. At all stages, computational efficiency and practicality will be emphasized over mere mathematical novelty, and an experimental approach consistent with the tradition of the physical sciences will be adopted.