From peridynamics to stochastic jump process: Illustrations of nonlocal balance laws and nonlocal calculus framework

Abstract

Nonlocal effects are ubiquitous in nature. There is a need to go beyond the traditional setup to develop new mathematical insight and computational methods for their systematics study and understanding. In this paper, by working with peridynamic models of nonlocal mechanics and nonlocal diffusion models for stochastic jump processes, we systematically explore the mathematical description of nonlocal balance laws. We use simple examples to introduce the recently developed concepts of nonlocal vector calculus and nonlocal calculus of variations and to illustrate the connections to traditional local models and the key differences. We also present the notion of asymptotically compatible (AC) discretization schemes as robust numerical algorithms for simulating multiscale problems.