LTB: Generalized Variational Integrators for Large-Scale Scientific Computation

几何积分涉及的是保持连续动力系统几何结构的数值方法的构建。在科学和工程中出现的许多问题，如太阳系动力学和分子动力学，都是高度非线性的，对微小的扰动很敏感，并且具有影响解的定性行为的潜在几何结构。这些动力系统的混沌特性使得精确计算长时间积分的特定轨迹变得非常昂贵。因此，研究保留问题几何形状的数值方法是可取的，因为它们产生更定性准确的模拟。该项目的目标是将基于离散汉密尔顿原理的变分积分器推广到由天体动力学、分子动力学和计算力学引起的更大规模问题。这将涉及结合大规模科学计算的方法，如自适应、光谱方法、多分辨率分层技术和区域分解，同时保留哈密顿ode和偏微分方程的变分积分器的几何保存特性。复杂物理系统的计算机模拟已经成为传统实验技术的重要补充，作为验证和指导科学理论发展的工具，以及技术和工程的实际进步。这项研究将提高我们精确和有效地计算复杂系统长期行为的能力，这是合理设计药物和高性能复合材料的一个基本方面。此外，它有可能通过允许新的和创新的工业设计的快速原型直接在计算机上加速技术发展的步伐。

Geometric integration is concerned with the construction of numerical methods that preserve the geometric structure of a continuous dynamical system. Many problems arising in science and engineering, such as solar system dynamics and molecular dynamics, are highly nonlinear, sensitive to small perturbations, and have underlying geometric structure that affects the qualitative behavior of solutions. The chaotic properties of these dynamical systems render prohibitively expensive the accurate computation of particular trajectories for long-time integration. As such, it is instead desirable to study numerical methods that preserve the geometry of a problem as they yield more qualitatively accurate simulations. The goal of this project is to generalize variational integrators based on a discrete Hamilton's principle to larger-scale problems arising from astrodynamics, molecular dynamics, and computational mechanics. This will involve incorporating methods from large-scale scientific computation, such as adaptivity, spectral methods, multi-resolution hierarchical techniques, and domain decomposition, while retaining the geometric preservation properties of variational integrators for Hamiltonian ODEs and PDEs.Computer simulations of complex physical systems have become an increasingly important complement to traditional experimental techniques as a tool for validating and guiding theoretical developments in science, as well as practical advances in technology and engineering. This research will improve our ability to accurately and efficiently compute the long-time behavior of complex systems, which is a fundamental aspect of the rational design of pharmaceuticals and high-performance composite materials. In addition, it has the potential to accelerate the pace of technological development by allowing the rapid prototyping of new and innovative industrial designs directly on the computer.