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非局部和反常扩散模型的数值方法

张继伟   

  1. 武汉大学数学与统计学院, 计算科学湖北省重点实验室, 武汉 430072
  • 收稿日期:2021-08-06 出版日期:2021-09-15 发布日期:2021-09-17
  • 作者简介:张继伟,武汉大学数学与统计学院教授,博士生导师.2003和2006年在郑州大学获得学士和硕士学位,2009年在香港浸会大学获得博士学位.随后在南洋理工大学和纽约大学克朗所从事博士后研究,2014年5月到北京计算科学研究中心工作,2018年11月到武汉大学工作.主要研究领域包括偏微分方程和非局部模型的数值解法,以及神经科学的建模与计算.主要成果发表在SIAM系列,Mathematics of Computation,Journal of Computational Neuroscience,Plos Computational Biology等国际知名期刊上.
  • 基金资助:
    国家自然科学基金(12171376)和173重点基础研究项目(2020-JCJQ-ZD-029)资助.

张继伟. 非局部和反常扩散模型的数值方法[J]. 数值计算与计算机应用, 2021, 42(3): 183-214.

Zhang Jiwei. NUMERICAL METHODS FOR NONLOCAL AND ANOMALOUS DIFFUSION MODELS[J]. Journal on Numerica Methods and Computer Applications, 2021, 42(3): 183-214.

NUMERICAL METHODS FOR NONLOCAL AND ANOMALOUS DIFFUSION MODELS

Zhang Jiwei   

  1. School of Mathematics and Statistics, and Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China
  • Received:2021-08-06 Online:2021-09-15 Published:2021-09-17
由于非局部模型能够描述某些重要物理现象产生的奇性和间断机制,近些年来在很多领域受到广泛应用并对相关学科的发展产生了强有力的推动作用.反常扩散模型作为一个重要的非局部模型,常用于描述反常扩散等现象.非局部模型的非局部性和多尺度特征不仅推动了新的数学理论的发现,而且为现有的离散和局部连续模型及其联系提供了新的视角.尽管已经有很多成果,但无论是从数学方法和基础理论还是数值方法角度来看,多尺度非局部和反常扩散模型都有广阔的研究空间.进一步发展和完善基础数学理论和方法,在真实的解正则性条件下发展新的高效数值格式,尤其是具有稳定、收敛、满足渐近兼容的数值格式是一个研究重点.在过去的几年里,本文作者一直致力于非局部模型的数学理论和数值方法研究,在人工边界条件设计、非局部极值原理和渐近兼容的数值格式等方面,取得了一些有意义的研究成果.在反常扩散方程的数值分析方面,发展了Caputo导数的快速算法和离散分数阶类型的Grönwall不等式,并提出了误差卷积结构的思想来表示局部相容误差,为一类常用变步长数值格式的最优误差估计提供了一些基础分析框架.要完全解决非局部和反常扩散模型中的各种数学问题还有相当长的距离,需要进一步深入研究.希望本文能为推动多尺度非局部模型和反常扩散模型的基础理论和算法的深入发展起到抛砖引玉的作用.
As the nonlocal model can describe the singularity and discontinuity mechanism of some important physical phenomena, it has been widely used in many fields in recent years and has played a strong role in promoting the development of related disciplines. As an important nonlocal model, anomalous diffusion model is used to describe anomalous diffusion phenomena. The nonlocality and multi-scale characteristics of nonlocal models not only promote the discovery of new mathematical theories, but also provide a new perspective for the existing discrete and local continuous models and their connections. Although there have been many achievements, there is a broad space for the development of multi-scale nonlocal and anomalous diffusion models, whether from the perspective of mathematical methods, basic theories or numerical methods. It is a research focus to further develop and improve the basic mathematical theories and methods, and develop new efficient numerical schemes under the condition of real solution regularity, especially the numerical schemes with stability, convergence and asymptotic compatibility. In the past few years, the author of this paper has been committed to the research of mathematical theory and numerical methods of nonlocal models, and has made some meaningful research results in the design of artificial boundary conditions, nonlocal maximum principle and asymptotically compatible numerical schemes. In the numerical analysis of anomalous diffusion equations, the fast algorithm of Caputo derivative and the discrete fractional Grönwall inequality are developed, and the idea of error convolution structure is proposed to represent the local consistent error, which provides some basic analysis frameworks for the optimal error estimation of a class of widely-used variable-step-size numerical schemes. There is still a long way to completely solve various mathematical problems in nonlocal and anomalous diffusion models, which needs further study. It is hoped that this paper can play a role in promoting the in-depth development of basic theories and algorithms of multi-scale nonlocal and anomalous diffusion models.

MR(2010)主题分类: 

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