摘要
非局部效应在自然界随处可见,对它的系统描述和认识需要突破传统模式的局限来发展新的数学思想和计算方法.本文从非局部毗域动力学模型和有关随机跳跃过程的非局部扩散模型出发,探讨非局部的平衡关系及其系统的数学表述,并用简例来介绍最近发展的非局部向量微积分框架和非局部变分原理中的一些基本概念,从而说明非局部模型与传统局部模型的关联和不同.本文还进一步提出渐近兼容(AC)的数值离散格式,为多尺度问题的非局部模型提供稳健的计算模拟方法.
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.
出处
《中国科学：数学》
CSCD
北大核心
2015年第7期939-952,共14页
Scientia Sinica：Mathematica
关键词
毗域动力学
随机跳跃过程
Lévy飞行
非局部扩散
非局部微积分
非局部变分问题
渐近兼容离散格式
peridynamics, stochastic jump processes, Lévy flight, nonlocal diffusion, nonlocal vector calcu-lus, nonlocal variational problems, asymptotically compatible discretization
