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Communications in Analysis and Mechanics

2024, Volume 16, Issue 1: 53-70. doi: 10.3934/cam.2024003
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Research article

Pointwise-in-time $ \alpha $-robust error estimate of the ADI difference scheme for three-dimensional fractional subdiffusion equations with variable coefficients

  • College of Sciences, Hunan University of Technology, Zhuzhou, 412008, China
  • Received: 30 October 2023 Revised: 12 December 2023 Accepted: 02 January 2024 Published: 10 January 2024

  • 65M15, 65M06

  • In this paper, a fully-discrete alternating direction implicit (ADI) difference method is proposed for solving three-dimensional (3D) fractional subdiffusion equations with variable coefficients, whose solution presents a weak singularity at $ t = 0 $. The proposed method is established via the L1 scheme on graded mesh for the Caputo fractional derivative and central difference method for spatial derivative, and an ADI method is structured to change the 3D problem into three 1D problems. Using the modified Grönwall inequality we prove the stability and $ \alpha $-robust convergence. The results presented in numerical experiments are in accordance with the theoretical analysis.

    Citation: Wang Xiao, Xuehua Yang, Ziyi Zhou. Pointwise-in-time $ \alpha $-robust error estimate of the ADI difference scheme for three-dimensional fractional subdiffusion equations with variable coefficients[J]. Communications in Analysis and Mechanics, 2024, 16(1): 53-70. doi: 10.3934/cam.2024003

    Related Papers:

  • In this paper, a fully-discrete alternating direction implicit (ADI) difference method is proposed for solving three-dimensional (3D) fractional subdiffusion equations with variable coefficients, whose solution presents a weak singularity at $ t = 0 $. The proposed method is established via the L1 scheme on graded mesh for the Caputo fractional derivative and central difference method for spatial derivative, and an ADI method is structured to change the 3D problem into three 1D problems. Using the modified Grönwall inequality we prove the stability and $ \alpha $-robust convergence. The results presented in numerical experiments are in accordance with the theoretical analysis.


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    [1] A. Saadatmandi, M. Dehghan, M. R. Azizi, The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients, Commun. Nonlinear. Sci., 17 (2012), 4125–4136. https://doi.org/10.1016/j.cnsns.2012.03.003 doi: 10.1016/j.cnsns.2012.03.003
    [2] J. E. Restrepo, M. Ruzhansky, D. Suragan, Explicit solutions for linear variable-coefficient fractional differential equations with respect to functions, Appl. Math. Comput., 403 (2021), 126177. https://doi.org/10.1016/j.amc.2021.126177 doi: 10.1016/j.amc.2021.126177
    [3] M. Abdulhameed, M. M. Muhammad, A. Y. Gital, D. G. Yakubu, I. Khan, Effect of fractional derivatives on transient MHD flow and radiative heat transfer in a micro-parallel channel at high zeta potentials, Physica A, 519 (2019), 42–71. https://doi.org/10.1016/j.physa.2018.12.019 doi: 10.1016/j.physa.2018.12.019
    [4] C. D. Constantinescu, J. M. Ramirez, W. R. Zhu, An application of fractional differential equations to risk theory, Financ. Stoch., 23 (2019), 1001–1024. https://doi.org/10.1007/s00780-019-00400-8 doi: 10.1007/s00780-019-00400-8
    [5] S. E. Chidiac, M. Shafikhani, Electrical resistivity model for quantifying concrete chloride diffusion coefficient, Cement. Concrete. Comp., 113 (2020), 103707. https://doi.org/10.1016/j.cemconcomp.2020.103707 doi: 10.1016/j.cemconcomp.2020.103707
    [6] G. H. Gao, Z. Z. Sun, A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys., 230 (2011), 586–595. https://doi.org/10.1016/j.jcp.2010.10.007 doi: 10.1016/j.jcp.2010.10.007
    [7] X. H. Yang, H. X. Zhang, The uniform l1 long-time behavior of time discretization for time-fractional partial differential equations with nonsmooth data, Appl. Math. Lett., 124 (2022), 107644. https://doi.org/10.1016/j.aml.2021.107644 doi: 10.1016/j.aml.2021.107644
    [8] C. J. Li, H. X. Zhang, X. H. Yang, A high-precision Richardson extrapolation method for a class of elliptic Dirichlet boundary value calculation, Journal of Hunan University of Technology, 38 (2024), 91–97. doi:10.3969/j.issn.1673-9833.2024.01.013 doi: 10.3969/j.issn.1673-9833.2024.01.013
    [9] X. H. Yang, L. J. Wu, H, X, Zhang, A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity, Appl. Math. Comput., 457 (2023), 128192. https://doi.org/10.1016/j.amc.2023.128192 doi: 10.1016/j.amc.2023.128192
    [10] Q. Q. Tian, X. H. Yang, H. X. Zhang, D. Xu, An implicit robust numerical scheme with graded meshes for the modified Burgers model with nonlocal dynamic properties, Comput. Appl. Math., 42 (2023), 246. https://doi.org/10.1007/s40314-023-02373-z doi: 10.1007/s40314-023-02373-z
    [11] W. Wan, H. X. Zhang, X. X. Jiang, X. H. Yang, A high-order and efficient numerical technique for the nonlocal neutron diffusion equation representing neutron transport in a nuclear reactor, Ann. Nucl. Energy., 195 (2024), 110163. https://doi.org/10.1016/j.anucene.2023.110163 doi: 10.1016/j.anucene.2023.110163
    [12] H. X. Zhang, X. H. Yang, Q. Tang, D. Xu, A robust error analysis of the OSC method for a multi-term fourth-order sub-diffusion equation, Comput. Math. Appl, 109 (2022), 180–190. https://doi.org/10.1016/j.camwa.2022.01.007 doi: 10.1016/j.camwa.2022.01.007
    [13] X. H. Yang, H. X. Zhang, Q. Zhang, G. W. Yuan, Simple positivity-preserving nonlinear finite volume scheme for subdiffusion equations on general non-conforming distorted meshes, Nonlinear. Dynam., 108 (2022), 3859–3886. https://doi.org/10.1007/s11071-022-07399-2 doi: 10.1007/s11071-022-07399-2
    [14] J. W. Wang, H. X. Zhang, X. H. Yang, A predictor-corrector compact difference scheme for a class of nonlinear Burgers equations, Journal of Hunan University of Technology, 38 (2024), 98–104. doi:10.3969/j.issn.1673-9833.2024.01.014 doi: 10.3969/j.issn.1673-9833.2024.01.014
    [15] X. H. Yang, Z. M. Zhang, On conservative, positivity preserving, nonlinear FV scheme on distorted meshes for the multi-term nonlocal Nagumo-type equations, Appl. Math. Lett., 150 (2024), 108972. https://doi.org/10.1016/j.aml.2023.108972 doi: 10.1016/j.aml.2023.108972
    [16] Z. F. Tian, Y. B. Ge, A fourth-order compact ADI method for solving two-dimensional unsteady convection-diffusion problems, Appl. Math. Lett., 198 (2007), 268–286. https://doi.org/10.1016/j.cam.2005.12.005 doi: 10.1016/j.cam.2005.12.005
    [17] Z. B. Wang, D. K. Cen, Y. Mo, Sharp error estimate of a compact L1-ADI scheme for the two-dimensional time-fractional integro-differential equation with singular kernels, Appl. Numer. Math., 159 (2021), 190–203. https://doi.org/10.1016/j.apnum.2020.09.006 doi: 10.1016/j.apnum.2020.09.006
    [18] Z. B. Wang, Y. X. Liang, Y. Mo, A novel high order compact ADI scheme for two dimensional fractional integro-differential equations, Appl. Numer. Math., 167 (2021), 257–272. https://doi.org/10.1016/j.apnum.2021.05.008 doi: 10.1016/j.apnum.2021.05.008
    [19] Q. F. Zhang, C. J. Zhang, L. Wang, The compact and Crank-Nicolson ADI schemes for two-dimensional semilinear multidelay parabolic equations, J. Comput. Appl. Math., 306 (2016), 217–230. https://doi.org/10.1016/j.cam.2016.04.016 doi: 10.1016/j.cam.2016.04.016
    [20] Q. F. Zhang, X. M. Lin, K. J. Pan, Y. Z. Ren, Linearized ADI schemes for two-dimensional space-fractional nonlinear Ginzburg-Landau equation, Comput. Math. Appl., 80 (2020), 1201–1220. https://doi.org/10.1016/j.camwa.2020.05.027 doi: 10.1016/j.camwa.2020.05.027
    [21] Y. Wang, H. Chen, T. Sun, $\alpha$-Robust $H^1$-norm convergence analysis of ADI scheme for two-dimensional time-fractional diffusion equation, Appl. Numer. Math., 168 (2021), 75–83. https://doi.org/10.1016/j.apnum.2021.05.025 doi: 10.1016/j.apnum.2021.05.025
    [22] Y. Wang, B. Zhu, H. Chen, $\alpha$-Robust $H^1$-norm convergence analysis of L1FEM-ADI scheme for 2D/3D subdiffusion equation with initial singularity, Math. Method. Appl. Sci., 46 (2023), 16144–16155. https://doi.org/10.1002/mma.9442 doi: 10.1002/mma.9442
    [23] D. Cao, H. Chen, Pointwise-in-time error estimate of an ADI scheme for two-dimensional multi-term subdiffusion equation, J. Appl. Math. Comput., 69 (2023), 707–729. https://doi.org/10.1007/s12190-022-01759-2 doi: 10.1007/s12190-022-01759-2
    [24] K. Li, H. Chen, S. Xie, Error estimate of L1-ADI scheme for two-dimensional multi-term time fractional diffusion equation, Netw. Heterog. Media., 18 (2023), 1454–1470. https://doi.org/10.3934/nhm.2023064 doi: 10.3934/nhm.2023064
    [25] L. J. Qiao, D. Xu, Z. B. Wang, An ADI difference scheme based on fractional trapezoidal rule for fractional integro-differential equation with a weakly singular kernel, Appl. Math. Comput., 354 (2019), 103–114. https://doi.org/10.1016/j.amc.2019.02.022 doi: 10.1016/j.amc.2019.02.022
    [26] L. J. Qiao, J. Guo, W. L. Qiu, Fast BDF2 ADI methods for the multi-dimensional tempered fractional integrodifferential equation of parabolic type, Comput. Math. Appl., 123 (2022), 89–104. https://doi.org/10.1016/j.camwa.2022.08.014 doi: 10.1016/j.camwa.2022.08.014
    [27] L. J. Qiao, D. Xu, A fast ADI orthogonal spline collocation method with graded meshes for the two-dimensional fractional integro-differential equation, Adv. Comput. Math., 47 (2021), 64. https://doi.org/10.1007/s10444-021-09884-5 doi: 10.1007/s10444-021-09884-5
    [28] H. X. Zhang, Y. Liu, X. H. Yang, An efficient ADI difference scheme for the nonlocal evolution problem in three-dimensional space, J. Appl. Math. Comput., 69 (2023), 651–674. https://doi.org/10.1007/s12190-022-01760-9 doi: 10.1007/s12190-022-01760-9
    [29] Z. Y. Zhou, H. X. Zhang, X. H. Yang, $H^1$-norm error analysis of a robust ADI method on graded mesh for three-dimensional subdiffusion problems, Numer. Algorithms., (2023), 1–19. https://doi.org/10.1007/s11075-023-01676-w
    [30] Z. Y. Zhou, H. X. Zhang, X. H. Yang, J. Tang, An efficient ADI difference scheme for the nonlocal evolution equation with multi-term weakly singular kernels in three dimensions, Int. J. Comput. Math., 2023, 1–18. https://doi.org/10.1080/00207160.2023.2212307
    [31] E. Ngondiep, A two-level fourth-order approach for time-fractional convection-diffusion-reaction equation with variable coefficients, Commun. Nonlinear. Sci., 111 (2022), 106444. https://doi.org/10.1016/j.cnsns.2022.106444 doi: 10.1016/j.cnsns.2022.106444
    [32] Y. B. Wei, S. J. L$\ddot{u}$, H. Chen, Y. M. Zhao, F. L. Wang, Convergence analysis of the anisotropic FEM for 2D time fractional variable coefficient diffusion equations on graded meshes, Appl. Math. Lett., 111 (2021), 106604. https://doi.org/10.1016/j.aml.2020.106604 doi: 10.1016/j.aml.2020.106604
    [33] Y. H. Zeng, Z. J. Tan, Two-grid finite element methods for nonlinear time fractional variable coefficient diffusion equations, Appl. Math. Comput., 434 (2022), 127408. https://doi.org/10.1016/j.amc.2022.127408 doi: 10.1016/j.amc.2022.127408
    [34] L. Ma, H. F. Fu, B. Y. Zhang, S. S. Xie, L1-robust analysis of a fourth-order block-centered finite difference method for two-dimensional variable-coefficient time-fractional reaction-diffusion equations, Comput. Math. Appl., 148 (2023), 211–227. https://doi.org/10.1016/j.camwa.2023.08.020 doi: 10.1016/j.camwa.2023.08.020
    [35] Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533–1552. https://doi.org/10.1016/j.jcp.2007.02.001 doi: 10.1016/j.jcp.2007.02.001
    [36] Z. Z. Sun, X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), 193–209. https://doi.org/10.1016/j.apnum.2005.03.003 doi: 10.1016/j.apnum.2005.03.003
    [37] H. L. Liao, W. McLean, J. W. Zhang, A discrete Gronwall inequality with applications to numerical schemes for subdiffusion problems, SIAM. J. Numer. Anal., 57 (2019), 218–237. https://doi.org/10.1137/16m1175742 doi: 10.1137/16m1175742
    [38] C. B. Huang, M. Stynes, $\alpha$-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation, Numer. Algorithms., 87 (2021), 1749–1766. https://doi.org/10.1007/s11075-020-01036-y doi: 10.1007/s11075-020-01036-y
    [39] C. B. Huang, N. An, H. Chen, Optimal pointwise-in-time error analysis of a mixed finite element method for a multi-term time-fractional fourth-order equation, Comput. Math. Appl., 135 (2023), 149–156. https://doi.org/10.1016/j.camwa.2023.01.028 doi: 10.1016/j.camwa.2023.01.028
    [40] Y. N. Zhang, Z. Z. Sun, Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, J. Comput. Phys., 230 (2011), 8713–8728. https://doi.org/10.1016/j.jcp.2011.08.020 doi: 10.1016/j.jcp.2011.08.020
    [41] J, F. Zhou, X. M. Gu, Y. L. Zhao, H. Li, A fast compact difference scheme with unequal time-steps for the tempered time-fractional Black-Scholes model, Int. J. Comput. Math., 2303 (2023), 10592. https://doi.org/10.1080/00207160.2023.2254412 doi: 10.1080/00207160.2023.2254412
    [42] J. Y. Shen, Z. Z. Sun, R. Du, Fast finite difference schemes for time-fractional diffusion equations with a weak singularity at initial time, East Asian J. Appl. Math, 8 (2018), 834–858. https://doi.org/10.13140/RG.2.2.18369.81767 doi: 10.13140/RG.2.2.18369.81767
    [43] Y. L. Zhao, X. M. Gu, A. Ostermann, A preconditioning technique for an all-at-once system from volterra subdiffusion equations with graded time steps, J. Sci. Comput., 88 (2021), 11. https://doi.org/10.1007/s10915-021-01527-7 doi: 10.1007/s10915-021-01527-7
    [44] X. M. Gu, S. L. Wu, A parallel-in-time iterative algorithm for Volterra partial integro-differential problems with weakly singular kernel, J. Sci. Comput., 417 (2020), 109576. https://doi.org/10.1016/j.jcp.2020.109576 doi: 10.1016/j.jcp.2020.109576
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