Mathematical and Numerical Analysis of Asymptotically Compatible Discretization of Nonlocal Models

非局部模型渐近兼容离散化的数学和数值分析

• 批准号: 2012562
• 负责人: Qiang Du
• 金额: $ 30万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2023-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2012562&HistoricalAwards=false
• 关键词: Mathematical; Numerical; Analysis; Asymptotically; Compatible

The study of nonlocal models has attracted much attention in many science and engineering disciplines such as materials science, mechanics, biology, and social science, and they are therefore of interest to applied and computational mathematics. Nonlocal models differ from the more common local models because they account for the factors active on a range rather than only at a point at which they are considered. The project is aimed at advancing the mathematical and numerical analysis of robust and effective numerical methods for those nonlocal models with a finite range of interactions. The research will complement the ongoing development of effective simulation platforms for nonlocal modeling in various application domains. It will also contribute to the integrated interdisciplinary education and research training of students.An important class of robust numerical schemes for nonlocal models is provided by asymptotically compatible (AC) discretization schemes. The latter are designed to assure the convergence of approximate solutions, as numerical resolution gets refined, to correct physical solutions for problems with changing or even diminishing ranges of nonlocal interactions. The project will include a comprehensive study of AC schemes for nonlocal problems with heterogeneously distributed ranges of nonlocal interactions and/or having boundary/interfaces. Further investigations of AC schemes will be carried out for problems involving coupled local/nonlocal models and nonlinear problems motivated by important applications. The focus on robust discretization methods like the AC schemes is particularly relevant to reliable and efficient simulations of nonlocal models with application to complex physical systems involving multiscale, singular, and anomalous behaviors. An integrated analytical and computational approach will be used to develop both fundamental ideas and practical insight so that the research findings will not only enrich the mathematical theory of AC schemes but also offer guidance to their practical applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

非局部模型的研究在材料科学、力学、生物学和社会科学等许多科学和工程学科中引起了广泛的关注，因此它们也是应用数学和计算数学的研究热点。 非局部模型与更常见的局部模型不同，因为它们考虑了在一个范围内而不仅仅是在考虑它们的一个点上活跃的因素。 该项目的目的是推进数学和数值分析的强大和有效的数值方法，这些非局部模型与有限范围的相互作用。 该研究将补充正在进行的开发有效的仿真平台，在各种应用领域的非本地建模。它也将有助于综合跨学科的教育和研究培训的学生。一类重要的鲁棒数值计划的非局部模型提供了渐近相容（AC）离散计划。后者的目的是确保近似解的收敛性，因为数值分辨率得到细化，以纠正物理解决方案的变化，甚至缩小范围的非局部相互作用的问题。该项目将包括一个全面的研究AC计划的非局部问题与非局部相互作用的非均匀分布的范围和/或有边界/接口。AC计划将进行进一步的调查，涉及耦合的本地/非本地模型和非线性问题的重要应用。对AC方案等鲁棒离散化方法的关注与非局部模型的可靠和有效模拟特别相关，该模型应用于涉及多尺度，奇异和异常行为的复杂物理系统。 综合分析和计算的方法将被用来开发基本思想和实际的洞察力，使研究成果不仅丰富了AC方案的数学理论，而且还提供了指导，以他们的实际应用。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

项目成果

A Nonlocal Stokes System with Volume Constraints

具有体积约束的非局部斯托克斯系统

• DOI: 10.4208/nmtma.oa-2022-0002s
• 发表时间: 2022
• 期刊: Methods and Applications
• 作者: Qiang Du;Shi, Zuoqiang
• 通讯作者: Shi, Zuoqiang

On the Ternary Ohta–Kawasaki Free Energy and Its One-dimensional Global Minimizers

• DOI: 10.1007/s00332-022-09814-9
• 发表时间: 2021-11
• 期刊: Journal of Nonlinear Science
• 影响因子: 3
• 作者: Zirui Xu;Q. Du

Numerical Simulation of Singularity Propagation Modeled by Linear Convection Equations with Spatially Heterogeneous Nonlocal Interactions

• DOI: 10.1007/s10915-022-01915-7
• 发表时间: 2022-01
• 期刊: Journal of Scientific Computing
• 影响因子: 2.5
• 作者: X. Yu;Yan Xu;Q. Du

A space-time nonlocal traffic flow model: Relaxation representation and local limit

时空非局部交通流模型：松弛表示和局部极限

• DOI: 10.3934/dcds.2023054
• 发表时间: 2023
• 期刊: Discrete and Continuous Dynamical Systems
• 影响因子: 1.1
• 作者: Du, Qiang;Huang, Kuang;Scott, James;Shen, Wen
• 通讯作者: Shen, Wen

The Average Distance Problem with Perimeter-to-Area Ratio Penalization

周长与面积比惩罚的平均距离问题

• DOI: 10.1137/21m1422987
• 发表时间: 2022
• 期刊: SIAM Journal on Mathematical Analysis
• 影响因子: 2
• 作者: Du, Qiang;Lu, Xin Yang;Wang, Chong
• 通讯作者: Wang, Chong