Skip to main content
SpringerLink
Log in

Second-Order Error Analysis for Fractal Mobile/Immobile Allen–Cahn Equation on Graded Meshes

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

The fractal mobile/immobile model bridges between Fickian fluxes at early times and non-Gaussian behavior at late times. In this work, an averaged L1 scheme for solving the fractal mobile/immobile Allen–Cahn equation with a Caputo temporal derivative of order \(\alpha \in (0,1)\) is developed and analyzed on graded meshes. The unique solvability and discrete energy stability are established rigorously on arbitrary nonuniform time meshes. Based on the spectral norm inequality, the unconditional stability and the second-order convergence analysis under the weakly regularity assumption are investigated on graded meshes. Finally, several numerical examples are presented to illustrate the theoretical analysis. To the best of our knowledge, this is the first topic on the convergence analysis for the fractal mobile/immobile Allen–Cahn equation on graded meshes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Data availability

Not applicable.

References

  1. Akrivis, G., Chen, M.H., Yu, F., Zhou, Z.: The energy technique for the six-step BDF method. SIAM J. Numer. Anal. 59, 2449–2472 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cao, R.J., Chen, M.H., Ng, M.K., Wu, Y.J.: Fast and high-order accuracy numerical methods for time-dependent nonlocal problem in \({\mathbb{R} }^2\). J. Sci. Comput. 84, 8 (2020)

    Article  MATH  Google Scholar 

  3. Chen, M.H., Yu, F., Zhang, Q.D., Zhang, Z.M.: Variable step-size BDF3 method for Allen–Cahn equation. arXiv:2112.13613

  4. Doerries, T.J., Chechkin, A.V., Schumer, R., Metzler, R.: Rate equations, spatial moments, and concentration profiles for mobile-immobile models with power-law and mixed waiting time distributions. Phys. Rev. E 105, 014105 (2022)

    Article  MathSciNet  Google Scholar 

  5. Guan, Z., Wang, J.G., Liu, Y., Nie, Y.F.: Unconditionally optimal convergence of a linearized Galerkin FEM for the nonlinear time-fractional mobile/immobile transport equation. Appl. Numer. Math. 172, 133–156 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ji, B.Q., Liao, H.L., Gong, Y.Z., Zhang, L.M.: Adaptive second-order Crank-Nicolson time-stepping schemes for time-fractional molecular beam epitaxial growth models. SIAM J. Sci. Comput. 42, B738–B760 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jiang, H.F., Xu, D., Qiu, W.L., Zhou, J.: An ADI compact difference scheme for the two-dimensional semilinear time-fractional mobile-immobile equation. Comput. Appl. Math. 39, 1–17 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  8. Liao, H.L., Zhang, Z.M.: Analysis of adaptive BDF2 scheme for diffusion equations. Math. Comput. 90, 1207–1226 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  9. Liao, H.L., Tang, T., Zhou, T.: An energy stable and maximum bound preserving scheme with variable time steps for time fractional Allen–Cahn equation. SIAM J. Sci. Comput. 43, A3503–A3526 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  10. Liu, J., Fu, H.F.: An efficient QSC approximation of variable-order time-fractional mobile-immobile diffusion equations with variably diffusive coefficients. J. Sci. Comput. 93, 44 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  11. Maryshev, B., Joelson, M., Lyubimov, D., Lyubimova, T., Néel, M.: Non Fickian flux for advection-dispersion with immobile periods. J. Phys. A Math. Theor. 42, 115001 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. McLean, W., Mustapha, K.: A second-order accurate numerical method for a fractional wave equation. Numer. Math. 105, 481–510 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mclean, W., Thomée, V., Wahlbin, L.: Discretization with variable time steps of an evolution equation with a positive-type memory term. J. Comput. Appl. Math. 69, 49–69 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mustapha, K.: An L1 approximation for a fractional reaction-diffusion equation, a second-order error analysis over time-graded meshes. SIAM J. Numer. Anal. 58, 1319–1338 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mustapha, K., Knio, O.M., Le Maître, O.P.: A second-order accurate numerical scheme for a time-fractional Fokker–Planck equation. IMA J. Numer. Anal. https://doi.org/10.1093/imanum/drac031

  17. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  18. Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics, 2nd edn. Springer, Berlin (2007)

    Book  MATH  Google Scholar 

  19. Schumer, R., Benson, D.A., Meerschaert, M.M., Baeumer, B.: Fractal mobile/immobile solute transport. Water Resour. Res. 39, 1296–1307 (2003)

    Article  Google Scholar 

  20. Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  21. Shen, J.Y., Zeng, F.H., Stynes, M.: Second-order error analysis of the averaged L1 scheme \(\overline{\text{L}1}\) for time-fractional initial-value and subdiffusion problems. https://doi.org/10.13140/RG.2.2.36480.71683

  22. Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55, 1057–1079 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tang, T., Yu, H.J., Zhou, T.: On energy dissipation theory and numerical stability for time-fractional phase-field equations. SIAM J. Sci. Comput. 41, A3757–A3778 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Varah, J.M.: A lower bound for the smallest singular value of a matrix. Linear Algebra Appl. 11, 3–5 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yan, Y.Y., Egwu, B.A., Liang, Z.Q., Yan, Y.B.: Error estimates of a continuous Galerkin time stepping method for subdiffusion problem. J. Sci. Comput. 88, 68 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yin, B.L., Liu, Y., Li, H.: A class of shifted high-order numerical methods for the fractional mobile/immobile transport equations. Appl. Math. Comput. 368, 124799 (2020)

    MathSciNet  MATH  Google Scholar 

  27. Zhang, H., Jiang, X.Y., Liu, F.W.: Error analysis of nonlinear time fractional mobile/immobile advection-diffusion equation with weakly singular solutions. Fract. Calc. Appl. Anal. 24, 202–224 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zheng, X.C., Wang, H.: Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions. IMA J. Numer. Anal. 41, 1522–1545 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zheng, Z.Y., Wang, Y.M.: An averaged L1-type compact difference method for time-fractional mobile/immobile diffusion equations with weakly singular solutions. Appl. Math. Lett. 131, 108076 (2022)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This work was supported by the Science Fund for Distinguished Young Scholars of Gansu Province (Grant No. 23JRRA1020) and NSFC 11601206.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Fan Yu & Minghua Chen

Authors
  1. Fan Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Minghua Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Minghua Chen.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the Science Fund for Distinguished Young Scholars of Gansu Province (Grant No. 23JRRA1020) and NSFC 11601206.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yu, F., Chen, M. Second-Order Error Analysis for Fractal Mobile/Immobile Allen–Cahn Equation on Graded Meshes. J Sci Comput 96, 49 (2023). https://doi.org/10.1007/s10915-023-02276-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-023-02276-5

Keywords

Search

Navigation