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

University Of California-San Diego, La Jolla CA

Investigators

Abstract

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.