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A discontinuous Galerkin method for one-dimensional time-dependent nonlocal diffusion problems

Authors: Qiang Du, Lili Ju and Jianfang Lu
Journal: Math. Comp.
MSC (2010): Primary 65M60, 65R20, 45A05
Published electronically: April 12, 2018
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Abstract: Given that nonlocal diffusion (ND) problems allow their solutions to have spatial discontinuities, discontinuous Galerkin (DG) discretizations in space become natural choices when numerical approximations are considered. In this paper, we design and study a novel DG method for solving the one-dimensional time-dependent ND problem. The key idea of the method is the introduction of an auxiliary variable, analogous to the classic local discontinuous Galerkin (LDG) method but with some nonlocal extensions. Theoretical analysis shows that the proposed semi-discrete DG scheme is $ L^2$-stable, convergent, and in particular asymptotically compatible. This latter feature implies that the classical DG approximation of the local diffusion problem can be recovered in the zero horizon limit. We also present various numerical tests to demonstrate the effectiveness and robustness of our method.

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Additional Information

Qiang Du
Affiliation: Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027

Lili Ju
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

Jianfang Lu
Affiliation: Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing, 100193, People’s Republic of China

Keywords: Nonlocal diffusion, asymptotic compatibility, discontinuous Galerkin method, strong stability preserving, long time asymptotic behavior
Received by editor(s): July 15, 2017
Received by editor(s) in revised form: September 28, 2017
Published electronically: April 12, 2018
Additional Notes: The first author’s research was partially supported by US National Science Foundation grant DMS-1719699, US AFOSR MURI Center for Material Failure Prediction Through Peridynamics, and US Army Research Office MURI grant W911NF-15-1-0562.
The second author’s research is partially supported by US National Science Foundation grant DMS-1521965 and US Department of Energy grant DE-SC0016540.
The third author’s research is partially supported by Postdoctoral Science Foundation of China grant 2017M610749.
Article copyright: © Copyright 2018 American Mathematical Society

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