SCREMS: Multiscale Computing in Astrophysics, Geophysics, Hydrodynamics, Kinetic and Quantum Applications

该奖项支持由威斯康星大学麦迪逊分校数学系的四名教员和他们的学生研究伙伴购买一个中型计算系统，该系统将支持计算数学研究。这四个项目位于不同的科学领域。其中一个项目旨在开发和测试计算方法，以研究质量是如何吸进黑洞的。黑洞是巨大而致密的天体，产生的引力如此之大，以至于连光都无法逃脱，并以几种形式存在于我们的宇宙中，包括星系中心的超大质量黑洞和恒星规模的x射线双星。关于吸积过程的详细物理和由此产生的动力学还没有完全了解。这个项目的目标是开发准确而有效的计算方法来进一步了解这些过程。第二个项目是研究尚未明确包括在当前大气、海洋和气候计算模型中的小尺度物理效应的影响。这些影响要么因为计算机能力不足而被忽视，要么因为缺乏计算机能力而被忽略。即使在今天和明天最先进的超级计算机上，或通过粗略的半经验公式间接包含在内。与天气和气候模拟相关的“简化”数学模型(即用于旋转和分层流动的3D Boussinesq方程)的计算机计算和理论表明，这些小尺度效应可能是非常显著的，与半经验公式相反。第三个项目旨在开发和测试能够有效处理量子力学和经典力学中的多尺度问题的数值方法。对这些问题的研究是数学物理和现代技术(例如纳米科学、等离子体、微型机电系统[MEMS]等)许多领域的核心。近年来，我们在设计稳健和有效的数值方法来计算量子力学和几何光学中出现的多值解方面取得了重大进展。在这个提议中，这项工作将被扩展到计算具有耦合经典和量子体制的界面的解。我们的目标是提供一种半经典的方法，这种方法比经典方法的成本略高，但比直接求解薛定谔方程的量子模拟成本低得多。一个特别重要的应用是纳米级量子点，这是现代半导体制造程序的标准产品。第四个项目继续研究在简单流体的流动中发现的反复出现的不稳定相干态，例如在管道、通道中流动的空气或水，或者在飞机机翼、船只、汽车或其他船只周围流动的空气或水。这种流动通常处于复杂的“湍流”状态，其特征是明显的随机涡流运动，但新发现的相干态(即有结构和有组织的)捕捉到了湍流的许多关键特征和特征。这些发现表明，在“混沌”理论中对小系统成功的一些深刻的理论观点可能被应用并扩展到更复杂的流体系统。该项目的一个目标是提供一个关键的不稳定相干态的目录，这可能会被认为是一个“湍流基因组计划”。

This award supports the purchase of a mid-size computationalsystem that will support computational mathematics researchby four faculty members and their student research associatesin the department of Mathematics at the University of Wisconsin-Madison.The four projects are in distinct areas of science. One project aims to develop and test computational methods tostudy how mass is accreted onto (sucked into) black holes. Black holes are massive compact objects that induce such stronggravity that not even light can escape them andoccur in our universe in several forms includingas super-massive black holes at the center of galaxies andat the scale of stars in the form of x-ray binaries.The detailed physics of the accretion process andthe resulting dynamics are not yet fully understood.The goal of this project is the developmentof accurate and efficient computational methods for furtherunderstanding of these processes.A second project is to study the impact of `small scale' physical effects that have not been explicitly included in currentcomputational models of the atmosphere, oceans and the climate.Those effects are either ignored for lack of computer power,even on today's -- and tomorrow's-- state of the art supercomputers,or are indirectly included through crude, semi-empirical formulas.Computer calculations and theory on `simplified'mathematical models (i.e. the 3D Boussinesq equations for rotatingand stratified flow) of relevance to weather and climate modeling, haveshown that these small scales effects can be very significantand contrary to the semi-empirical formulas.A third project aims to develop and test numerical methods that canefficiently deal with multiscale problems in quantum and classicalmechanics. The study of these problems is central to many areas ofmathematical physics and modern technology (e.g. Nanoscience,plasmas, micro-electro-mechanical-systems [MEMS],...). In recentyears we have made significant progress in devising robust andefficient numerical methods to compute multivalued solutions thatarise in quantum mechanics and geometric optics. In this proposalthat work will be extended to compute solutions with interfacesthat couple classical and quantum regimes. The goal is to provide a semiclassical approach that is slightly more expensive than a classical approach but much less expensive than a quantum simulation based on solving directly the Schrodinger equation. One particularly important application is the nano-scale quantum dots that are a standard product of modern semiconductor manufacturing procedures.The fourth project continues the study of recurrent unstable coherentstates that have been discovered in the flow of simple fluids suchas air or water flowing in pipes, channels or over airplane wings,around boats, cars or other vessels. Such flows are typically in a complex`turbulent' state characterized by apparently random eddy motions,but the newly discovered coherent states (i.e. structured and well-organized)capture many key features and characteristics of turbulent flows.These discoveries suggest that some deep theoretical ideas that havebeen successful for small systems in `chaos' theory may be appliedand extended to much more complex fluid systems. One aim of the projectis to provide a catalog of key unstable coherent states that might beconsidered a `turbulence genome project'.