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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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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DOI: https://doi.org/10.1007/s10444-021-09881-8
Keywords
- Variable-order time-fractional diffusion equations
- Finite difference method
- ADI method
- Compact ADI method
- Stability and convergence