High Order Schemes for Hyperbolic and Convection-dominated Problems

双曲和对流主导问题的高阶方案

• 批准号: 1418750
• 负责人: Chi-Wang Shu
• 金额: $ 38.78万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1418750&HistoricalAwards=false
• 关键词: Order; Schemes; Hyperbolic; Convection; dominated

In this project, the PI will perform research in the algorithm design and analysis of high order numerical methods. These algorithms are used to solve scientific and engineering problems arising from diverse application fields such as aerospace engineering, semi-conductor device design, astrophysics, and biological problems. Even with today's fast computers, it is still essential to design efficient and reliable algorithms which can be used to obtain accurate solutions to these application problems. The broader impacts resulting from the proposed activity will be a suite of powerful computational tools, suitable for various applications mentioned above. These tools are expected to make positive contributions to computer simulations of the complicated solution structure in these applications. The algorithms to be investigated include the finite difference and finite volume weighted essentially non-oscillatory (WENO) schemes and discontinuous Galerkin finite element methods, for solving hyperbolic and other convection dominated partial differential equations (PDEs). While the emphasis of this project is on algorithm design and analysis, close attention will be paid to applications. Topics of proposed investigations will include the study on high order accurate bound-preserving algorithms and applications, an inverse Lax-Wendroff procedure for high order numerical boundary conditions for finite difference schemes on rectangular meshes when the physical boundary is not aligned with the meshes, WENO schemes with subcell resolution for nonconservative problems, Lagrangian type finite volume schemes for multi-material flows, energy-conserving discontinuous Galerkin methods for long time simulation of wave problems, efficient discontinuous Galerkin methods for front propagation problems with obstacles, superconvergence analysis of discontinuous Galerkin methods and its applications in adaptive computation, simple WENO limiters for discontinuous Galerkin methods in unstructured meshes for problems with strong shocks, multi-scale methods based on the discontinuous Galerkin framework, analysis and numerical solutions for traffic and pedestrian flow models, turbulence simulation in cosmology, and study on aggregation and coordinated movement in computational biology. Problems in applications will motivate the design of new algorithms or new features in existing algorithms; mathematics tools are used to analyze these algorithms to give guidelines for their applicability and limitations; practical considerations including parallel implementation issues are addressed to make the algorithms competitive in large scale calculations; and collaborations with engineers and other applied scientists enable the efficient application of these new algorithms or new features in existing algorithms.

在该项目中，PI将对高阶数值方法的算法设计和分析进行研究。 这些算法用于解决由航空航天工程，半导体设备设计，天体物理学和生物学问题等不同应用领域引起的科学和工程问题。 即使使用当今的快速计算机，设计有效可靠的算法仍然至关重要，这些算法可用于获得这些应用问题的准确解决方案。 拟议活动所产生的更广泛的影响将是一套强大的计算工具，适用于上述各种应用程序。 预计这些工具将为这些应用程序中复杂解决方案结构的计算机模拟做出积极贡献。要研究的算法包括有限的差异和有限体重加权基本上非振荡（WENO）方案以及不连续的Galerkin有限元方法，用于求解双曲线和其他对流主导的部分偏微分方程（PDES）。尽管该项目的重点是算法设计和分析，但将密切关注应用程序。 Topics of proposed investigations will include the study on high order accurate bound-preserving algorithms and applications, an inverse Lax-Wendroff procedure for high order numerical boundary conditions for finite difference schemes on rectangular meshes when the physical boundary is not aligned with the meshes, WENO schemes with subcell resolution for nonconservative problems, Lagrangian type finite volume schemes for multi-material长时间的不连续盖尔金方法，用于长期模拟波浪问题的流量，有效的不连续的盖尔金方法，用于障碍物的前传繁殖问题，对不连续的Galerkin方法的超授权分析及其在自适应计算中的应用，其在自适应计算中的应用，在不连续的Galerkin shess中的简单限制器，用于不连续的Galerkin shess中的问题，用于不连续的Galerke Meshes，用于远程销售方法，以构建不连续的梅塞斯，用于构建杂物，以实现不连续的梅什群岛。不连续的Galerkin框架，用于交通和行人流量模型的分析和数值解决方案，宇宙学中的湍流模拟以及计算生物学中的聚集和协调运动的研究。 应用中的问题将激发新算法或现有算法中的新功能的设计；数学工具用于分析这些算法，以提供有关其适用性和局限性的准则；解决了包括并行实施问题在内的实际考虑因素，以使算法在大规模计算中具有竞争力；以及与工程师和其他应用科学家的合作，可以在现有算法中有效地应用这些新算法或新功能。