Numerical Methods for Nonlocal Models with Applications to Multiscale and Nonlinear Systems

偏微分方程是现代科学的一个组成部分。然而，它们作为我们日益互联的世界以及金融危机、材料故障和疾病爆发等极端和异常事件的模型存在局限性。相比之下，其他非局部方程对于交通流、流行病、有缺陷的材料、人口和聚集动态、金融以及图像和数据分析模型很有用。该项目利用高效可靠的非局部模型数值方法开发系统的数学框架，并将其应用于现代科学和社会中几个重要的多尺度和非线性系统。学生将参与跨学科研究并接受培训。 该项目包括三个子项目。第一个涉及为无界域上制定的系统设计有效数值算法的重要问题。该项目将使用无反射完美匹配层的方法来求解非局部波动方程；将进行严格的数值研究，并提供对非局域波的离散和连续完美匹配层的系统理解。第二个子项目旨在开发非自伴非线性系统的渐近兼容有限元方案；该项目将为奇异扰动非局域对流主导扩散问题和半线性非局域问题开发新理论，这些问题对于参数化非局域模型的鲁棒模拟至关重要。第三个子项目旨在为一类非局部算子提供易于实现的保正性无网格有限差分离散化；该项目将解决稳定性问题，同时保持无网格方法的正性保持特性和其他优势。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。

Partial differential equations form an integral part of modern sciences. However, they have limitations as models for our increasingly connected world as well as for extreme and anomalous events such as financial crisis, material failures, and disease outbreaks. Other nonlocal equations in contrast are useful for models of traffic flow, epidemics, materials with defects, population and flocking dynamics, and finance, as well as for image and data analysis. This project develops systematic mathematical frameworks with efficient and reliable numerical methods for nonlocal models with applications to several important multiscale and nonlinear systems of importance in modern sciences and society. Students will be involved and trained in interdisciplinary research. The project comprises three sub-projects. The first is concerned with the important issue of designing efficient numerical algorithms for systems formulated on unbounded domains. The project will use an approach of a reflectionless perfectly matched layer for nonlocal wave equations; rigorous numerical studies will be conducted and a systematic understanding of the discrete and continuous perfectly matched layers for nonlocal waves will be provided. The second sub-project aims at developing asymptotically compatible finite element schemes for non-self-adjoint and nonlinear systems; the project will develop new theories for singular perturbed nonlocal convection-dominated diffusion problems and semi-linear nonlocal problems that are crucial for the robust simulation of parametrized nonlocal models. The third sub-project aims at providing easily implementable positivity-preserving meshfree finite difference discretization for a class of nonlocal operators; the project will tackle stability issues while maintaining the positivity-preserving properties and other advantages of meshfree methods.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.