Modern numerical methods for problems in physics

点击翻译按钮获取中文摘要

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).