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。提出了发展几类数值方法来有效地计算一些重要的物理问题，承认多尺度。 所研究的问题是一个基本的性质，从量子力学，经典力学，动力学理论流体力学。 特别是，将有高效的数值方法开发：1。薛定谔方程，可以有效地扩展到（半）经典政权; 2。经典力学中的多相计算及其相关问题，以及它在速调管中调制电子束中的应用; 3.动力学方程，也是有效的过渡和流体制度。数值方法依赖于对从一个物理尺度到另一个物理尺度的转变的深刻数学理解。它将为多尺度物理问题的许多数值方法提供几个基本成分。这些问题的研究是数学物理许多领域和现代技术数学基础的核心（例如，MEMS、纳米科学和技术、等离子体、流体力学、连续介质力学.）。这些项目，如果成功地进行，将提供新的和国家的最先进的数值方法，以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.