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

主要研究人员和他们的合作者研究非线性偏微分方程的数值方法的发展，分析和应用。本文讨论了三个主要主题：(1)高分辨率方法，包括激波捕获、前捕获、数值均匀化和粒子方法；(2)应用于这些问题和其他相关问题的多尺度分析；(3)基于多尺度分析和高频渐近的线性演化方程的快速方法。研究的工程和物理应用包括燃烧、流体动力学、晶体生长、斯蒂芬问题、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等领域的现实问题。过渡到工业和军事应用是在上述领域，以及在低观测，半导体器件建模和形状识别。这些方法适用于任何状态变量变化剧烈或超出标准方法解析能力的现象。

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.