An accurate and asymptotically compatible collocation scheme for nonlocal diffusion problems

Zhang, Xiaoping; Wu, Jiming; Ju, Lili

doi:10.1016/j.apnum.2017.11.007

An accurate and asymptotically compatible collocation scheme for nonlocal diffusion problems

Journal Article · Tue Dec 05 00:00:00 UTC 2017 · Applied Numerical Mathematics

DOI:https://doi.org/10.1016/j.apnum.2017.11.007· OSTI ID:1612624

Zhang, Xiaoping ^[1]; Wu, Jiming ^[2]; Ju, Lili ^[3]

Wuhan Univ. (China); DOE/OSTI
Inst. of Applied Physics and Computational Mathematics (IAPCM), Beijing (China)
Univ. of South Carolina, Columbia, SC (United States); Beijing Computational Science Research Center (China)

Here, in this paper, we develop and analyze a collocation scheme for solving the linear nonlocal diffusion problem with general kernels. To approximate the nonlocal diffusion operator, we take a classic trapezoidal rule based on the linear interpolation as the starting point, and then carefully derive a new improved quadrature rule, which is not only more accurate but also could avoid the evaluations of singular integrals. We then use this rule to construct a collocation scheme for solving the nonlocal diffusion equations, that produces a symmetric positive definite stiffness matrix with Toeplitz structure. The proposed scheme is rigorously shown to be of second order accurate with respect to the mesh size for the nonlocal problem with fixed horizon, and in particular, it can achieve higher order accuracy for the commonly used kernels in the literature. Furthermore, we also prove that the scheme is asymptotically compatible, i.e., the approximate solution of the nonlocal diffusion problem converges to the exact solution of the corresponding local PDE problem when the horizon and the mesh size both go to zero. Finally, numerical experiments are presented to verify the theoretical results.

View Accepted Manuscript (DOE)

Research Organization:: Univ. of South Carolina, Columbia, SC (United States)

Sponsoring Organization:: National Science Foundation (NSF); USDOE; USDOE Office of Science (SC)

Grant/Contract Number:: SC0016540

OSTI ID:: 1612624

Alternate ID(s):: OSTI ID: 1693735

Journal Information:: Applied Numerical Mathematics, Journal Name: Applied Numerical Mathematics Vol. 133; ISSN 0168-9274

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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97 MATHEMATICS AND COMPUTING
Asymptotic compatibility
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An accurate and asymptotically compatible collocation scheme for nonlocal diffusion problems

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