Modern numerical methods for problems in physics

该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在一起，研究流体力学和材料科学中出现的问题。从数学上讲，要求解的方程是高阶的、非线性的偏微分方程组，而且区域往往非常复杂。基本的计算框架是在任何需要的时候使用“设计者”算法，即那些针对具体问题的算法。这些现代而强大的计算方法能够通过充分利用问题的底层分析结构来准确地捕获问题的解决方案。这与已经发展为具有最广泛应用的更标准的数值方法形成了对比。这一建议的一个主要主旨是发展出求解Navier-Stokes方程的积分方程法。这些方法有可能提供一种令人兴奋的替代传统有限差分或有限元的方法，它们提供了显著的优势：易于合并复杂的物理边界，并且避免了与直接离散控制偏微分方程组相关的病态条件。到目前为止，积分方程法作为一种通用工具被认为是不切实际的，因为明显的计算费用。我们打算通过使用快速算法来改善这一开销，例如快速多极子方法。我们已经成功地开发了求解Stokes方程的方法，并最近使用这些工具来研究粒子模拟和界面运动。一个新的研究领域将是研究周期性图案形态，物理例子是嵌段共聚熔体中的相分离。同样，控制方程是高阶非线性偏微分方程组(Cahn-Hilliard Like)。

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).