Numerical Solution of the Nonlocal Diffusion Equation on the real line
Update: 2018-02-24 09:26:51      Author: yangjuan@csrc.ac.cn

This paper is concerned with the numeral simulation of a nonlocal diffusion equation defined on the whole real axis. A challenging problem is how to construct artificial boundary conditions (ABCs) for these problems due to the nonlocality of the interaction. With the application of the Laplace transform in the spacial direction, rather than the classical temporal direction, a DtN-like mapping is obtained to be taken as an exact ABC for the nonlocal diffusion problem. In the practical numerical implementation, the nonlocal diffusion equation is first discretized in space to lead to a discrete nonlocal system on the whole real axis. After that, an exact ABC for the discrete nonlocal diffusion system is achieved on artificial boundary gridpoints. This exact ABC allowed one to reformulate the nonlocal system on the whole real axis into a finite nonlocal one on the truncated computational domain. So far, this is a pioneering work in the designing of exact ABCs for the nonlocal problems. The numerical examples were presented to verify the effectiveness of the approach in two parts. First, the schemes converge to the nonlocal problem by fixing horizon size δ and taking hà0. Second, the numerical schemes are shown to converge to the correct local models when both δ and hà0. 

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Fig. 1The nonintegral kernel is used. One can see that our ABCs perfectly absorbed the heat flow which passes through the artificial boundary gridpoints, and do not generate any obvious approximation error

 

 

References:

[1]     Chunxiong Zheng, Jiashun Hu, Qiang Du and J. Zhang. SIAM J. Sci. Comput. 2017. A3067-A3088.

 


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