Analysis, Algorithms and Computations for Model Problems in Physical Sciences

拟议的项目涉及开发、分析和应用数值模拟工具来解决计算科学中的一些问题，特别是计算物理和计算材料科学中的问题。它是PI过去在这一领域研究工作的继续，对超导、玻色-爱因斯坦凝聚和二元和多元合金相变中的各种问题的建模、分析和计算做出了贡献。在拟议的工作中，PI将在过去进展的基础上，在不同时间和空间尺度上研究一些有趣的物理问题，并在设计新的算法和有效的解算器方面采取新的举措，从而可以用来理解实验现象和潜在的物理和材料性质。拟议的工作将在以下几个方面进行：为潜在的物理问题建立或完善数学模型，以扩大这些模型的有效范围；分析这些模型，以进一步了解它们的性质和解决方案；开发、分析和实现算法，特别是并行和自适应算法，用于这些模型的数值模拟；并使用我们的算法和代码，与物理学家和材料科学家一起，研究物理和材料科学中的一些有趣现象，包括进一步研究超导体和玻色-爱因斯坦凝聚体中的量子化涡旋，对运气的影响，以及对介观唯象模型的有效性和扩展的研究。虽然我们将专注于开发创新的数学理论和计算算法，但也将尽可能与物理实验进行比较。

The proposed project is concerned with the development,analysis and applications of numerical simulation toolsto a number of problems in computational sciences,in particular, in computational physics and computationalmaterial sciences. It is a continuation of PI's past research work in this area that has contributed to the modeling, analysis, and computation of various problems in superconductivity,Bose-Einstein condensation, and phase transitions in binary and multicomponent alloys. In the proposed work, while building upon the past progress,the PI will take on new initiatives in the study of some interesting physical problems at various time and spatial scalesand in the design of new algorithms and efficientsolvers which can then be used to understand experimental phenomenaand the underlying physical and material properties.The proposed works are to be carried out in the following aspects: to develop or refine mathematical models for the underlying physical problems, so to enlarge the range of validity of such models; to analyze these models in order to gain further understanding of their properties and of their solutions; to develop, analyze, andimplement algorithms, in particular, parallel and adaptive algorithms,for the numerical simulation of these models; and to use our algorithms and codes, together with physicists and material scientists, to study some interesting phenomena in physics and material sciences,including the further studies on the quantized vortices in superconductors and Bose-Einstein condensates, on the effects offluctuation and on the validity and the extension of mezoscopic phenomenological models. While we will focus on developing innovative mathematical theory and computational algorithms, comparisons with the physical experiments will also be made whenever possible.