Numerical Methods for Multiscale Physical Problems

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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.