Modern numerical methods for problems in physics

本研究计划的主要主题是将科学计算、物理建模和应用数学分析中的经典方法结合起来，研究流体动力学和材料科学中出现的问题。在数学上，要解决的方程是高阶非线性偏微分方程，并且域通常非常复杂。底层的计算框架是采用“设计者”算法，即那些针对特定问题的算法。这些现代而强大的计算方法能够通过充分利用其底层分析结构来准确地捕获问题的解决方案。这与更标准的数值方法形成对比，后者已被开发出来，具有尽可能广泛的应用。

The primary theme in this research program is to bring together state-of-the-art methods in scientific computing, physical modeling, and classical methods in applied mathematical analysis for studying problems that arise in fluid dynamics and material science. Mathematically, the equations to be solved are high-order, nonlinear partial differential equations, and the domains are often extremely complex. The underlying computational framework is to employ, whenever indicated, ``designer'' algorithms, i.e. those that target specific problems. These modern and powerful computational methods are able to accurately capture the solution of a problem by fully exploiting its underlying analytical structure. This is in contrast to more standard numerical methods that have been developed to have the widest possible application.