Communications in Mathematical Sciences

Volume 20 (2022)

Number 2

On uniform second-order nonlocal approximations to diffusion and subdiffusion equations with nonlocal effect parameter

Pages: 359 – 375

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n2.a3

Authors

Jerry Z. Yang (School of Mathematics and Statistics, and Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, China)

Xiaobo Yin (School of Mathematics and Statistics, and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, China)

Jiwei Zhang (School of Mathematics and Statistics, and Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, China)

Abstract

In this paper we focus on uniform convergence rates from nonlocal diffusion and subdiffusion solutions to the corresponding local limit with respect to a nonlocal effect parameter without extra assumptions on the regularity of nonlocal solutions, and present sufficient conditions to guarantee first- and second-order convergence rates, respectively. To do so, we first revisit the maximum principle for nonlocal models using the idea in [Luchko, J. Math. Anal. Appl., 2009], and present the uniqueness of the nonlocal solutions. After that, we extend the methodology developed in [Du, Zhang and Zheng, Commun. Math. Sci., 2019] to address the truncated errors on the volume constraints, and then combine the resulting errors from the boundary domain with the maximum principle to obtain the uniform convergence rates. Our analysis shows that the constant value continuation of the boundary conditions of local problems only leads to a first-order convergence rate. If one expects a second-order convergence rate, the information of first-order derivatives for local problems on the boundaries is required. One- and two-dimensional numerical examples are provided to validate our theoretical analysis.

Keywords

Caputo derivative, maximum principle, nonlocal model, local limit, asymptotically compatible

2010 Mathematics Subject Classification

45A05, 46N20, 65M60, 65R20, 82C21

Full Text (PDF format)

The research is supported by the National Natural Science Foundation of China (Nos. 11671165, 11771035, 12071362), the National Key Research and Development Program of China (No. 2020YFA0714200), 2020-JCJQZD-029, NSAF U1930402, the Natural Science Foundation of Hubei Province (No. 2019CFA007), Hubei Provincial Science and Technology Innovation Base (Platform) Special Project (No. 2020DFH002), and Xiangtan University 2018ICIP01.

The numerical calculations in this paper have been done on the supercomputing system in the Supercomputing Center of Wuhan University.

Received 21 July 2020

Accepted 1 July 2021

Published 28 January 2022