A Fast Averaged L1 Finite Difference Method for Time Fractional Mobile/Immobile Diffusion Equation with Weakly Singular Solution

Abstract

In this paper, the time fractional mobile/immobile diffusion equation with the weak singular solution at the initial time is studied. The averaged L1 finite difference scheme is established for the equation. The stability of the numerical scheme is analyzed by the Fourier analysis method. The convergence order of the scheme is

O({\tau }^2|ln\tau |+h^2)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(\tau ^2|\ln \tau |+h^2)$$\end{document}

, where

\tau

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tau $$\end{document}

and h are the sizes of the time and space steps, respectively. In addition, due to the historical dependence of the time fractional derivative, we establish a fast method based on the exponential-sum-approximation, effectively reducing computation and storage. Furthermore, we provide an error estimate of the fast algorithm. Finally, a numerical experiment verifies the effectiveness of the algorithm.

Keywords

Time fractional mobile/immobile diffusion equation / Averaged L1 scheme / Exponential-sum-approximation / Weak singularity

References

[1] FardiM, GhasemiM. A numerical solution strategy based on error analysis for time-fractional mobile/immobile transport model. Soft. Comput., 2021, 25(16): 11307-11331

[2] Huang, Y., Li, Q., Li, R., Zeng, F., Guo, L.: A unified fast memory-saving time-stepping method for fractional operators and its applications. Numer. Math.: Theory Methods Appl. 15 (3), 679–714 (2022)

[3] Huang, Y., Zeng, F., Guo, L.: Error estimate of the fast L1 method for time-fractional subdiffusion equations. Appl. Math. Lett. 133, 108288 (2022)

[4] JiangH, XuD, QiuW, ZhouJ. An ADI compact difference scheme for the two-dimensional semilinear time-fractional mobile-immobile equation. Comput. Appl. Math., 2020, 39: 1-17

[5] LiuZ, LiX, ZhangX. A fast high-order compact difference method for the fractal mobile/immobile transport equation. Int. J. Comput. Math., 2020, 97(9): 1860-1883

[6] MaryshevB, JoelsonM, LyubimovD, LyubimovaT, NéelM-C. Non Fickian flux for advection-dispersion with immobile periods. J. Phys. A: Math. Theor., 2009, 42(11) ArticleID: 115001

[7] NikanO, MachadoJT, GolbabaiA, NikazadT. Numerical approach for modeling fractal mobile/immobile transport model in porous and fractured media. Int. Commun. Heat Mass Transfer, 2020, 111, ArticleID: 104443

[8] Rusagara, I., Baleanu, D.: Numerical solution of a kind of fractional parabolic equations via two difference schemes. Abstract and Applied Analysis, 2013, 324–331 (2013)

[9] Schumer, R., Benson, D.A., Meerschaert, M.M., Baeumer, B.: Fractal mobile/immobile solute transport. Water Resour. Res. 39 (10) (2003). https://doi.org/10.1029/2003WR002141

[10] ShenJ, ZengF, StynesM. Second-order error analysis of the averaged L1 scheme L 1 ¯ \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline{\rm L1}$$\end{document} for time-fractional initial-value and subdiffusion problems. Sci. China Math., 2024, 67 7): 1641-1664

[11] YangZ, ZengF. A corrected L1 method for a time-fractional subdiffusion equation. J. Sci. Comput., 2023, 95(3): 85

[12] YinB, LiuY, LiH. A class of shifted high-order numerical methods for the fractional mobile/immobile transport equations. Appl. Math. Comput., 2020, 368 124799

[13] Yu, F., Chen, M.: Second-order error analysis for fractal mobile/immobile Allen-Cahn equation on graded meshes. J. Sci. Comput. 96(2), 49 (2023)

[14] ZhangH, JiangX, LiuF. Error analysis of nonlinear time fractional mobile/immobile advection-diffusion equation with weakly singular solutions. Fract. Calculus Appl. Anal., 2021, 24(1): 202-224

[15] Zhang, J., Fang, Z., Sun, H.: Exponential-sum-approximation technique for variable-order time-fractional diffusion equations. J. Appl. Math. Comput. 68(1), 323–347 (2022)

[16] ZhaoJ, FangZ, LiH, LiuY. Finite volume element method with the WSGD formula for nonlinear fractional mobile/immobile transport equations. Adv. Differ. Equ., 2020, 2020(1): 1-20

[17] ZhengZ, WangY. An averaged L1-type compact difference method for time-fractional mobile/immobile diffusion equations with weakly singular solutions. Appl. Math. Lett., 2022, 131 ArticleID: 108076

Funding

Natural Science Foundation of Shandong Province(ZR2022QA038)

Doctoral Research Foundation of Liaocheng University(318052155)

RIGHTS & PERMISSIONS

Shanghai University