A fast high-order compact difference method for the fractal mobile/immobile transport equation

ABSTRACT

In this paper, we present a high-order compact finite difference scheme for fractal mobile/immobile transport model with a Caputo fractional time derivative. The compact finite difference scheme is stable and convergent with convergence order $O({\tau }^{2-\alpha }+h^4)$ in $L_2$-norm. Furthermore, because of the non-local property of fractional differential operators, it is necessary to find a fast technique to reduce the computational cost. We develop a fast solution technique which is based on a fast Fourier transform. The computational work will reduce to $O(MN{\log }^{2}N)$ while the direct method requires the computational cost of $O(M{N}^2)$, where $N={\tau }^{-1}$ and τ is the size of time step, $M=h^{-1}$ and h is the size of space step. Moreover, numerical results are consistent with the theoretical analysis.

Keywords:

• Fractal mobile/immobile transport model
• compact finite difference method
• Topelitz
• fast Fourier transform
• numerical experiments

2010 AMS Subject Classifications:

• 65M06
• 65M12
• 65M15
• 26A33

Acknowledgments

We would like to acknowledge the assistance of volunteers in putting together this example manuscript and supplement.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the Postdoctoral Science Foundation of China [grant numbers BX20190187 and 2019M650152], by National Natural Science Foundation of China [grant numbers 11931003, 41974133, 11901489, 11971276]; National Science Foundation DMS-1216923; OSD/ARO MURI W911NF-15-1-0562; Shandong Provincial Natural Science Foundation, China ZR2011AM015; National Science and Technology Major Project of China 2011ZX05052, 2011ZX05011-004.