Numerical Methods for Multiscale Physical Problems

许多物理问题具有多个时间和空间尺度，这带来了巨大的数学挑战和数值困难。建议发展几类数值方法来有效地计算某些具有多个尺度的重要物理问题。正在研究的问题是基本性质的，从量子力学、经典力学、运动学理论到流体力学。特别是，将有有效的数值方法用于：1.可以有效地扩展到(半)经典区域的薛定谔方程；2.经典力学中的多相计算和相关问题，以及它在速调管中调制电子束中的应用；3.在跃迁和流体区域中也有效的动力学方程。数值方法依赖于从一个物理尺度到另一个物理尺度的深刻的数学理解。对这些问题的研究是许多数学物理领域的核心，也是现代技术(如MEMS、纳米科学和技术、等离子体、流体力学、连续介质力学等)的数学基础。如果这些项目成功实施，将提供新的和最先进的数值方法来有效地模拟多尺度的重要物理问题。这些将大大提高我们对现代工业应用中出现的各种重要物理问题的理解。

Many physical problems have multiple time and space scales that impose tremendous mathematical challenges and numerical difficulties. It is proposed to develop several classes of numerical methods for efficiently computing some important physical problems that admit multiple scales. The problems under study are of a fundamental nature, ranging from quantum mechanics, classical mechanics, kinetic theory to hydrodynamics. In particular, there will be efficient numerical methods developed for: 1. the Schroedinger equations that can be extended efficiently into the (semi)classical regime; 2. multiphase computation in classical mechanics and related problems, and its application in modulated electron beams in a Klystron; 3. kinetic equations that are also efficient in the transition and fluid regimes. The numerical methods rely on the deep mathematical understanding of the transition from one physical scale to the other. It will provide several essential ingredients to many numerical methods for multiscale physical problems.The study of these problems is central to many areas of mathematical physics and to the mathematical foundation of modern technology (e.g., MEMS, nanoscale science and technology, plasmas, fluid mechanics, continuum mechanics, ...). These projects, if successfully carried out, will provide novel and state-of-the-art numerical methods to efficiently simulate important physical problems of multiple scales. These will significantly enhance our understanding of a variety of important physical problems arising in modern industrial applications.