Mathematical Sciences: Development, Analysis, and Applications for Numerical Methods for Nonlinear Partial Differential Equations

主要研究人员及其合作者研究非线性偏微分方程数值方法的发展、分析和应用。主要包括：(1)高分辨率方法，包括激波捕捉、前沿捕捉、数值均匀化和质点方法；(2)多尺度分析在这些问题和其他相关问题中的应用；(3)基于多尺度分析和高频渐近的线性发展方程的快速方法。所研究的工程和物理应用包括燃烧、流体动力学、晶体生长、斯特凡问题、微波散射和反应气体。这些创新的数值方法被用来模拟航空、石油开采、材料科学和环境科学领域的现实世界问题。在上述领域，以及在低可观测性、半导体器件建模和形状识别方面，已经过渡到工业和军事应用。这些方法适用于所要研究的现象具有状态变量的急剧变化或超出标准方法的解决能力的地方。

The principal investigators and their collaborators study the development, analysis, and applications of numerical methods for nonlinear partial differential equations. The three main topics addressed are: (1) high resolution methods, including shock capturing, front capturing, numerical homogenization, and particle methods, (2) multiscale analysis applied to these and other related problems, and (3) fast methods for linear evolution equations based on multiscale analysis and high frequency asymptotics. Engineering and physical applications studied include combustion, fluid dynamics, crystal growth, Stefan problems, microwave scattering, and reacting gases. These innovative numerical methods are used to simulate real world problems in the areas of aeronautics, oil recovery, materials science, and environmental science. Transition to industrial and military application is made in the above areas, as well as in low observables, semiconductor device modelling, and shape recognition. These methods are applicable wherever the phenomena to be studied have sharp changes of state variables or are beyond the resolving capabilities of standard methods.