Superconvergent Discontinuous Galerkin methods for Partial Differential Equations

偏微分方程的超收敛间断伽辽金法

• 批准号: 1115331
• 负责人: Bernardo Cockburn
• 金额: $ 42万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-10-01 至 2015-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1115331&HistoricalAwards=false
• 关键词: Superconvergent; Discontinuous; Galerkin; methods; Partial

In this proposal, the investigator studies and develops a new, emerging generation of discontinuous Galerkin methods characterized by being easier to implement, by having enhanced stability and convergence properties, and by displaying an improved flexibility for handling arbitrarily-shaped domains. The investigator will focuses his effort in four particular problems. The first is in the area of fluid flow and consists in establishing a general theory of very competitive numerical methods for the incompressible Navier-Stokes equations. The second is in the area of continuum mechanics and consists in the study of optimally convergent methods for fourth-order problems in order to pave the way to the devising of numerical methods for non-linear shells. The third is in the area of non-linear conservation laws and consists in the introduction of new techniques geared towards overcoming the two main difficulties that have dragged down for more than a decade the development of efficient, high-order accurate methods for these useful equations. The last is in the area of techniques for handling curved boundaries and consists in replacing the traditional paradigm of meshing the domain with high accuracy by the new approach of using a very simple mesh of a box containing the domain and employing special approximation techniques near its border.The computer simulation of physical phenomena is a highly valued tool of practical interest in a wide variety of applications in Engineering and Physics. The investigator studies an emerging and promising technique of carrying out these simulations with highly accurate and more efficient algorithms for a wide range of problems of practical interest. They include many applications to Aerospace and Mechanics (incompressible fluid flow, subsonic and supersonic flow) as well as to Civil Engineering (solid structures).

在这项提案中，研究人员研究和开发了一个新的，新兴的一代不连续Galerkin方法，其特点是更容易实现，具有增强的稳定性和收敛性，并显示出更好的灵活性，处理任意形状的域。研究者将集中精力解决四个具体问题。第一个是在流体流动领域，包括建立一个非常有竞争力的不可压缩Navier-Stokes方程的数值方法的一般理论。第二个是在连续介质力学领域，包括研究四阶问题的最佳收敛方法，以便为非线性壳体的数值方法的设计铺平道路。第三个是在非线性守恒定律领域，包括引进新技术，以克服两个主要困难，这两个困难已经拖了十多年，为这些有用的方程开发高效，高阶精确的方法。最后是在处理弯曲边界的技术领域，包括用一种新的方法取代传统的高精度网格化域的范例，这种新方法使用一个非常简单的包含域的盒子网格，并在其边界附近采用特殊的近似技术。物理现象的计算机模拟在工程和物理学的各种应用中是一种具有实际意义的高度有价值的工具。研究人员研究了一种新兴的和有前途的技术，该技术可以用高度准确和更有效的算法来进行这些模拟，以解决各种实际问题。它们包括航空航天和机械（不可压缩流体流动，亚音速和超音速流动）以及土木工程（固体结构）的许多应用。