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Efficient spatial second-/fourth-order finite difference ADI methods for multi-dimensional variable-order time-fractional diffusion equations

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Abstract

Variable-order time-fractional diffusion equations (VO-tFDEs), which can be used to model solute transport in heterogeneous porous media are considered. Concerning the well-posedness and regularity theory (cf., Zheng & Wang, Anal. Appl., 2020), two finite difference ADI and compact ADI schemes are respectively proposed for the two-dimensional VO-tFDE. We show that the two schemes are unconditionally stable and convergent with second and fourth orders in space with respect to corresponding discrete norms. Besides, efficiency and practical computation of the ADI schemes are also discussed. Furthermore, the ADI and compact ADI methods are extended to model three-dimensional VO-tFDE, and unconditional stability and convergence are also proved. Finally, several numerical examples are given to validate the theoretical analysis and show efficiency of the ADI methods.

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Acknowledgements

The authors would like to express their most sincere thanks to the referees for their very helpful comments and suggestions, which greatly improved the quality of this paper.

Funding

This work was supported in part by the National Natural Science Foundation of China (No. 11971482), by the Natural Science Foundation of Shandong Province (No. ZR2017MA006), and by the OUC Scientific Research Program for Young Talented Professionals.

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Authors and Affiliations

  1. School of Mathematical Sciences, Ocean University of China, Qingdao, Shandong, 266100, China

    Hongfei Fu

  2. College of Science, China University of Petroleum (East China), Qingdao, Shandong, 266580, China

    Chen Zhu, Xueting Liang & Bingyin Zhang

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  1. Hongfei Fu
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  2. Chen Zhu
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Correspondence to Hongfei Fu.

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Communicated by: Bangti Jin

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Fu, H., Zhu, C., Liang, X. et al. Efficient spatial second-/fourth-order finite difference ADI methods for multi-dimensional variable-order time-fractional diffusion equations. Adv Comput Math 47, 58 (2021). https://doi.org/10.1007/s10444-021-09881-8

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