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起来，以研究流体动力学和材料科学中出现的问题。在数学上，要求解的方程是高阶非线性偏微分方程，并且域通常非常复杂。基本的计算框架是采用，只要指出，“设计师”的算法，即那些针对具体问题。这些现代而强大的计算方法能够通过充分利用其潜在的分析结构来准确地捕获问题的解决方案。这是在对比更标准的数值方法，已经开发出具有最广泛的可能application.One的主要推力，这项建议是发展的Navier-Stokes方程的积分方程方法。这些方法有可能提供一个令人兴奋的替代传统的有限差分或有限元，他们提供了显着的优势：复杂的物理边界很容易纳入和病态相关的直接离散化的偏微分方程被避免。到目前为止，积分方程方法已被认为是不切实际的一般工具，因为明显的计算费用。我们打算通过采用快速算法（如快速多极子方法）来改善这一费用。我们已经成功地发展了Stokes方程的求解方法，并在最近利用这些工具研究了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，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（类似Cahn-Hilliard）。

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).