Algorithm Development, Analysis, and Application of High Order Schemes

高阶方案的算法开发、分析与应用

基本信息

批准号：
1719410
负责人：
Chi-Wang Shu
金额：
$ 22.5万
依托单位：
Brown University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2021-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1719410&HistoricalAwards=false
关键词：
Algorithm Development Analysis Application Order

项目摘要

In this project the PI will perform research in algorithm design and analysis of high order accurate and efficient numerical methods for solving partial differential equations. 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 the PI plans to investigate 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 an inverse Lax-Wendroff procedure for high order numerical boundary conditions for finite difference schemes on Cartesian meshes solving problems in general geometry, Lagrangian type finite volume schemes for multi-material flows, a simple weighted essentially non-oscillatory limiter for discontinuous Galerkin methods with strong shocks, high order stable conservative methods on arbitrary point clouds, discontinuous Galerkin methods for weakly coupled hyperbolic multi-domain and network problems, efficient time-stepping techniques for discontinuous Galerkin schemes, high order accurate bound-preserving schemes and applications, bound-preserving high order discontinuous Galerkin schemes for radiative transfer equations, energy-conserving DG methods for Maxwell's equations in Drude metamaterials, efficient discontinuous Galerkin method for front propagation problems with obstacles, superconvergence analysis of discontinuous Galerkin methods and its applications, multi-scale methods based on the discontinuous Galerkin framework, and applications in areas including traffic and pedestrian flow models and aggregation, coordinated movement and cell proliferation 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将研究解决偏微分方程的高阶精确高效数值方法的算法设计和分析。这些算法用于解决航空航天工程、半导体器件设计、天体物理学和生物问题等不同应用领域产生的科学和工程问题。即使当今的计算机速度很快，设计高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。拟议活动产生的更广泛影响将是一套强大的计算工具，适用于上述各种应用。这些工具预计将为这些应用中复杂解结构的计算机模拟做出积极贡献。PI计划研究的算法包括有限差分和有限体积加权本质上非振荡（WENO）方案和不连续伽辽金有限元方法，用于求解双曲型和其他对流主导的偏微分方程 (PDE)。虽然该项目的重点是算法设计和分析，但也会密切关注应用。拟议研究的主题将包括研究笛卡尔网格有限差分格式的高阶数值边界条件的逆Lax-Wendroff过程，解决一般几何中的问题，多材料流的拉格朗日型有限体积格式，简单的加权本质上非-强冲击的不连续伽辽金方法的振荡限制器，任意点云上的高阶稳定保守方法，弱耦合双曲多域和网络问题的不连续伽辽金方法，高效不连续伽辽金格式的时间步进技术、高阶精确保界方案和应用、辐射传递方程的保界高阶不连续伽辽金格式、德鲁德超材料中麦克斯韦方程组的能量守恒DG方法、前向高效不连续伽辽金方法障碍传播问题、间断伽辽金方法的超收敛分析及其应用、基于间断伽辽金的多尺度方法框架，以及计算生物学中交通和行人流模型和聚合、协调运动和细胞增殖等领域的应用。应用中的问题将激发新算法的设计或现有算法的新功能；使用数学工具来分析这些算法，以为其适用性和局限性提供指导；解决了包括并行实现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；与工程师和其他应用科学家的合作可以有效地应用这些新算法或现有算法中的新功能。