Volume 50, Issue 1
High-Order Local Artificial Boundary Conditions for the Fractional Diffusion Equation on One-Dimensional Unbounded Domain

Zhang Wei, Can Li, Xiaonan Wu & Jiwei Zhang

J. Math. Study, 50 (2017), pp. 28-53.

Published online: 2017-03

Preview Full PDF 880 4285
Export citation
  • Abstract

In this paper we consider the numerical solutions of the fractional diffusion equation on the unbounded spatial domain. With the application of Laplace transformation, we obtain one-way equations which absorb the wave touching on the artificial boundaries. By using Padé expansion to approximate the frequency in Laplace space and introducing auxiliary variables to reduce the order of the derivatives with respect to time t, we achieve a system of ODEs within the artificial boundaries. This system of ODEs, called high-order local absorbing boundary conditions (LABCs), reformulate the fractional diffusion problem on the unbounded domain to an initial-boundaryvalue (IBV) problem on a bounded computational domain. A fully discrete implicit difference scheme is constructed for the reduced problem. The stability and convergence rate are established for a finite difference scheme. Finally, numerical experiments are given to demonstrate the efficiency and accuracy of our approach.

  • Keywords

Fractional subdiffusion equation high-order absorbing boundary conditions Laplace transform Padé expansion artificial boundary methods

  • AMS Subject Headings

35R11, 65M06, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

jwzhang@csrc.ac.cn (Can Li)

xwu@hkbu.edu.hk (Xiaonan Wu)

jwzhang@csrc.ac.cn (Jiwei Zhang)

  • BibTex
  • RIS
  • TXT
@Article{JMS-50-28, author = {Wei , Zhang and Li , Can and Wu , Xiaonan and Zhang , Jiwei}, title = {High-Order Local Artificial Boundary Conditions for the Fractional Diffusion Equation on One-Dimensional Unbounded Domain}, journal = {Journal of Mathematical Study}, year = {2017}, volume = {50}, number = {1}, pages = {28--53}, abstract = {

In this paper we consider the numerical solutions of the fractional diffusion equation on the unbounded spatial domain. With the application of Laplace transformation, we obtain one-way equations which absorb the wave touching on the artificial boundaries. By using Padé expansion to approximate the frequency in Laplace space and introducing auxiliary variables to reduce the order of the derivatives with respect to time t, we achieve a system of ODEs within the artificial boundaries. This system of ODEs, called high-order local absorbing boundary conditions (LABCs), reformulate the fractional diffusion problem on the unbounded domain to an initial-boundaryvalue (IBV) problem on a bounded computational domain. A fully discrete implicit difference scheme is constructed for the reduced problem. The stability and convergence rate are established for a finite difference scheme. Finally, numerical experiments are given to demonstrate the efficiency and accuracy of our approach.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v50n1.17.03}, url = {http://global-sci.org/intro/article_detail/jms/978.html} }
TY - JOUR T1 - High-Order Local Artificial Boundary Conditions for the Fractional Diffusion Equation on One-Dimensional Unbounded Domain AU - Wei , Zhang AU - Li , Can AU - Wu , Xiaonan AU - Zhang , Jiwei JO - Journal of Mathematical Study VL - 1 SP - 28 EP - 53 PY - 2017 DA - 2017/03 SN - 50 DO - http://doi.org/10.4208/jms.v50n1.17.03 UR - https://global-sci.org/intro/article_detail/jms/978.html KW - Fractional subdiffusion equation KW - high-order absorbing boundary conditions KW - Laplace transform KW - Padé expansion KW - artificial boundary methods AB -

In this paper we consider the numerical solutions of the fractional diffusion equation on the unbounded spatial domain. With the application of Laplace transformation, we obtain one-way equations which absorb the wave touching on the artificial boundaries. By using Padé expansion to approximate the frequency in Laplace space and introducing auxiliary variables to reduce the order of the derivatives with respect to time t, we achieve a system of ODEs within the artificial boundaries. This system of ODEs, called high-order local absorbing boundary conditions (LABCs), reformulate the fractional diffusion problem on the unbounded domain to an initial-boundaryvalue (IBV) problem on a bounded computational domain. A fully discrete implicit difference scheme is constructed for the reduced problem. The stability and convergence rate are established for a finite difference scheme. Finally, numerical experiments are given to demonstrate the efficiency and accuracy of our approach.

Zhang Wei, Can Li, Xiaonan Wu & Jiwei Zhang. (2019). High-Order Local Artificial Boundary Conditions for the Fractional Diffusion Equation on One-Dimensional Unbounded Domain. Journal of Mathematical Study. 50 (1). 28-53. doi:10.4208/jms.v50n1.17.03
Copy to clipboard
The citation has been copied to your clipboard