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。这一建议的一个主要推力是发展积分方程方法的纳维-斯托克斯方程。这些方法有可能为传统的有限差分或有限单元提供令人兴奋的替代方案，并且它们具有显著的优点：复杂的物理边界易于合并，并且避免了与控制偏微分方程的直接离散相关的不良条件。迄今为止，由于明显的计算费用，积分方程方法被认为是不切实际的通用工具。我们打算通过采用快速算法（如快速多极方法）来改善这一费用。我们已经成功地开发了Stokes方程的方法，并且最近使用这些工具来研究粒子模拟和界面运动。一个新的研究兴趣领域将是研究周期性模式形态，一个物理例子是嵌段共聚物熔体中的相分离。同样，控制方程是高阶非线性偏微分方程（Cahn-Hilliard式）。

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.One main thrust of this proposal is to develop integral equation methods for the Navier-Stokesequations. These methods have the potential to offer an exciting alternative to conventional finite difference or finite elements, and they offer significant advantages:  complex physical boundaries are easy to incorporate and the ill-conditioning associated with direct discretization of the governing partial differential equations is avoided. To date, an integral equation approach has been deemed impractical as a general tool because of the apparent computational expense. We intend to ameliorate this expense by employing fast algorithms, such as the Fast Multipole Method. We have been successful in developing methods for the Stokes equations, and have recently used these tools to investigate particle simulations, and interfacial motion.A new area of research interest will be to investigate periodic pattern morphologies, with a physical example being phase separation in block copolymer melts. Again, the governing equations are high-order, nonlinear partial differential equations (Cahn-Hilliard like).