Modern numerical methods for problems in physics

物理问题的现代数值方法

• 批准号: 203326-2007
• 负责人: Kropinski, MaryCatherine
• 金额: $ 1.38万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=365352
• 关键词: Modern; numerical; methods; problems; physics

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).

该研究计划的主要主题是将科学计算、物理建模中的最先进方法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。无论何时，底层计算框架都是采用“设计者”算法，即针对特定问题的算法。这些现代且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捕获问题的解决方案。这与已开发出具有最广泛应用的更标准数值方法形成对比。该提案的一个主要推动力是开发纳维-斯托克斯方程的积分方程方法。这些方法有可能为传统的有限差分或有限元提供令人兴奋的替代方案，并且具有显着的优点：易于合并复杂的物理边界，并且避免了与控制偏微分方程直接离散相关的病态条件。迄今为止，由于明显的计算费用，积分方程方法被认为作为通用工具是不切实际的。我们打算通过采用快速算法（例如快速多极法）来减少这种费用。我们已经成功地开发了斯托克斯方程的方法，并且最近使用这些工具来研究颗粒模拟和界面运动。一个新的研究兴趣领域将是研究周期性图案形态，其中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）。