In light of the advantages of the Caputo–Hadamard fractional derivative in characterizing ultra-slow diffusion phenomena, this paper proposes a second-order approximation scheme to approximate it. Then, for the Allen–Cahn equation with the Caputo–Hadamard fractional derivative in time, a numerical algorithm is designed. This algorithm employs the proposed second-order formula for time discretization. Considering the potential anisotropic behavior of the solution in space, the anisotropic nonconforming quasi-Wilson finite element method is utilized for spatial approximation. The error in the -norm and the superclose error in the -norm of this algorithm are analyzed. The global superconvergence in the -norm is demonstrated through interpolation postprocessing techniques. Numerical examples are given to verify the theoretical results and further investigate the influence of different time derivatives on the dynamic behavior of the solution.
Citation: Luhan Sun, Zhen Wang, Yabing Wei. A second–order approximation scheme for Caputo–Hadamard derivative and its application in fractional Allen–Cahn equation[J]. Communications in Analysis and Mechanics, 2025, 17(2): 630-661. doi: 10.3934/cam.2025025
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In light of the advantages of the Caputo–Hadamard fractional derivative in characterizing ultra-slow diffusion phenomena, this paper proposes a second-order approximation scheme to approximate it. Then, for the Allen–Cahn equation with the Caputo–Hadamard fractional derivative in time, a numerical algorithm is designed. This algorithm employs the proposed second-order formula for time discretization. Considering the potential anisotropic behavior of the solution in space, the anisotropic nonconforming quasi-Wilson finite element method is utilized for spatial approximation. The error in the -norm and the superclose error in the -norm of this algorithm are analyzed. The global superconvergence in the -norm is demonstrated through interpolation postprocessing techniques. Numerical examples are given to verify the theoretical results and further investigate the influence of different time derivatives on the dynamic behavior of the solution.
[1] | I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. |
[2] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. |
[3] | C. P. Li, M. Cai, Theory and Numerical Approximations of Fractional Integrals and Derivatives, SIAM, Philadelphia, 2019. |
[4] |
R. Garra, F. Mainardi, G. Spada, A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus, Chaos, Solitons Fractals, 102 (2017), 333–338. https://doi.org/10.1016/j.chaos.2017.03.032 doi: 10.1016/j.chaos.2017.03.032
![]() |
[5] |
E. Y. Fan, C. P. Li, Z. Q. Li, Numerical approaches to Caputo-Hadamard fractional derivatives with applications to long-term integration of fractional differential systems, Commun. Nonlinear Sci. Numer. Simul., 106 (2022), 106096. https://doi.org/10.1016/j.cnsns.2021.106096 doi: 10.1016/j.cnsns.2021.106096
![]() |
[6] |
M. Cai, G. E. Karniadakis, C. P. Li, Fractional SEIR model and data-driven predictions of COVID-19 dynamics of Omicron variant, Chaos, 32 (2022), 071101. https://doi.org/10.1063/5.0099450 doi: 10.1063/5.0099450
![]() |
[7] | B. Ahmad, A. Alsaedi, S. K. Ntouyas, J. Tariboon, Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, Springer, Switzerland, 2017. |
[8] |
D. D. Cao, C. P. Li, Analysis and computation for quenching solution to the time-space fractional Kawarada problem, Fract. Calc. Appl. Anal., 28 (2025), 559–606. https://doi.org/10.1007/s13540-025-00384-7 doi: 10.1007/s13540-025-00384-7
![]() |
[9] |
D. D. Cao, C. P. Li, Quenching phenomenon in the Caputo-Hadamard time-fractional Kawarada problem: Analysis and computation, Math. Comput. Simulat., 233 (2025), 21–38. https://doi.org/10.1016/j.matcom.2025.01.014 doi: 10.1016/j.matcom.2025.01.014
![]() |
[10] |
R. Zhu, Z. Wang, Z. Zhang, Global existence and convergence of the solution to the nonlinear -Caputo fractional diffusion equation, J. Nonlinear Sci., 35 (2025), 36. https://doi.org/10.1007/s00332-025-10129-8 doi: 10.1007/s00332-025-10129-8
![]() |
[11] |
Z. Wang, L. Sun, The Allen-Cahn equation with a time Caputo-Hadamard derivative: Mathematical and Numerical Analysis, Communications in Analysis and Mechanics, 15 (2023), 611–637. https://doi.org/10.3934/cam.2023031 doi: 10.3934/cam.2023031
![]() |
[12] |
M. Gohar, C. P. Li, Z. Q. Li, Finite difference methods for Caputo-Hadamard fractional differential equations, Mediterr. J. Math., 17 (2020), 194. https://doi.org/10.1007/s00009-020-01605-4 doi: 10.1007/s00009-020-01605-4
![]() |
[13] |
C. P. Li, Z. Wang, The local discontinuous Galerkin finite element methods for Caputo-type partial differential equations: Mathematical analysis, Appl. Numer. Math., 150 (2020), 587–606. https://doi.org/10.1016/j.apnum.2019.11.007 doi: 10.1016/j.apnum.2019.11.007
![]() |
[14] |
T. G. Zhao, C. P. Li, D. X. Li, Efficient spectral collocation method for fractional differential equation with Caputo-Hadamard derivative, Fract. Calc. Appl. Anal., 26 (2023), 2903–2927. https://doi.org/10.1007/s13540-023-00216-6 doi: 10.1007/s13540-023-00216-6
![]() |
[15] |
C. X. Ou, D. K. Cen, S. Vong, Z. B. Wang, Mathematical analysis and numerical methods for Caputo-Hadamard fractional diffusion-wave equations, Appl. Numer. Math., 177 (2022), 34–57. https://doi.org/10.1016/j.apnum.2022.02.017 doi: 10.1016/j.apnum.2022.02.017
![]() |
[16] |
C. X. Ou, D. K. Cen, Z. B. Wang, S. Vong, Fitted schemes for Caputo-Hadamard fractional differential equations, Numer. Algorithms, 97 (2024), 135–164. https://doi.org/10.1007/s11075-023-01696-6 doi: 10.1007/s11075-023-01696-6
![]() |
[17] |
Z. Wang, L1/LDG method for Caputo-Hadamard time fractional diffusion equation, Commun. Appl. Math. Comput., 7 (2025), 203–227. https://doi.org/10.1007/s42967-023-00257-x doi: 10.1007/s42967-023-00257-x
![]() |
[18] |
Z. Wang, A nonuniform L2-/LDG method for the Caputo-Hadamard time-fractional convection-diffusion equation, Adv. Studies: Euro-Tbilisi Math. J., 16 (2023), 89–115. https://doi.org/10.32513/asetmj/193220082328 doi: 10.32513/asetmj/193220082328
![]() |
[19] |
C. P. Li, Z. Q. Li, Stability and logarithmic decay of the solution to Hadamard-type fractional differential equation, J. Nonlinear Sci., 31 (2021), 31. https://doi.org/10.1007/s00332-021-09691-8 doi: 10.1007/s00332-021-09691-8
![]() |
[20] |
X. C. Zheng, Logarithmic transformation between (variable-order) Caputo and Caputo-Hadamard fractional problems and applications, Appl. Math. Lett., 121 (2021), 107366. https://doi.org/10.1016/j.aml.2021.107366 doi: 10.1016/j.aml.2021.107366
![]() |
[21] |
A. Alikhanov, C. Huang, A class of time-fractional diffusion equations with generalized fractional derivatives, J. Comput. Appl. Math., 414 (2022), 114424. https://doi.org/10.1016/j.cam.2022.114424 doi: 10.1016/j.cam.2022.114424
![]() |
[22] |
Z. Wang, L. H. Sun, A numerical approximation for the Caputo-Hadamard derivative and its application in time-fractional variable-coefficient diffusion equation, Discrete Contin. Dyn. Syst., Series S, 17 (2024), 2679–2705. https://doi.org/10.3934/dcdss.2024027 doi: 10.3934/dcdss.2024027
![]() |
[23] |
Z. Wang, L. H. Sun, Y. B. Wei, Superconvergence analysis of the nonconforming FEM for the Allen-Cahn equation with time Caputo-Hadamard derivative, Phys. D, 465 (2024), 134201. https://doi.org/10.1016/j.physd.2024.134201 doi: 10.1016/j.physd.2024.134201
![]() |
[24] |
K. Mustapha, An L1 approximation for a fractional reaction-diffusion equation, a second-order error analysis over time-graded meshes, SIAM J. Numer. Anal., 58 (2020), 1319–1338. https://doi.org/10.1137/19M1260475 doi: 10.1137/19M1260475
![]() |
[25] |
B. Ji, H. L. Liao, Y. Z. Gong, L. M. Zhang, Adaptive second-order Crank-Nicolson time-step schemes for time-fractional molecular beam epitaxial growth models, SIAM J. Sci. Comput., 42 (2020), B738–B760. https://doi.org/10.1137/19M1259675 doi: 10.1137/19M1259675
![]() |
[26] |
H. L. Liao, N. Liu, P. Lyu, Discrete gradient structure of a second-order variable-step method for nonlinear integro-differential models, SIAM J. Numer. Anal., 61 (2023), 2157-2181. https://doi.org/10.1137/22M1520050 doi: 10.1137/22M1520050
![]() |
[27] |
Y. Yan, B. A. Egwu, Z. Liang, Y. Yan, Error estimates of a continuous Galerkin time step method for subdiffusion problem, J. Sci. Comput., 88 (2021), 1–30. https://doi.org/10.1007/s10915-021-01587-9 doi: 10.1007/s10915-021-01587-9
![]() |
[28] |
F. Yu, M. H. Chen, Second-order error analysis for fractal mobile/immobile Allen-Cahn equation on graded meshes, J. Sci. Comput., 96 (2023), 49. https://doi.org/10.1007/s10915-023-02276-5 doi: 10.1007/s10915-023-02276-5
![]() |
[29] |
J. Y. Shen, F. H. Zeng, M. Stynes, Second-order error analysis of the averaged L1 scheme for time-fractional initial-value and subdiffusion problems, Sci. China Math., 67 (2024), 1641–1664. https://doi.org/10.1007/s11425-022-2078-4 doi: 10.1007/s11425-022-2078-4
![]() |
[30] |
S. M. Allen, J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1085–1095. https://doi.org/10.1016/0001-6160(79)90196-2 doi: 10.1016/0001-6160(79)90196-2
![]() |
[31] |
L. Golubović, A. Levandovsky, D. Moldovan, Interface dynamics and far-from-equilibrium phase transitions in multilayer epitaxial growth and erosion on crystal surfaces: Continuum theory insights, East Asian J. Appl. Math., 1 (2011), 297–371. https://doi.org/10.4208/eajam.040411.030611a doi: 10.4208/eajam.040411.030611a
![]() |
[32] |
D. Fan, L. Q. Chen, Computer simulation of grain growth using a continuum field model, Acta Mater., 45 (1997), 611–622. https://doi.org/10.1016/S1359-6454(96)00200-5 doi: 10.1016/S1359-6454(96)00200-5
![]() |
[33] |
H. Liu, A. J. Cheng, H. Wang, J. Zhao, Time-fractional Allen-Cahn and Cahn-Hilliard phase-field models and their numerical investigation, Comput. Math. Appl., 76 (2018), 1876–1892. https://doi.org/10.1016/j.camwa.2018.07.036 doi: 10.1016/j.camwa.2018.07.036
![]() |
[34] |
B. Ji, H. L. Liao, L. Zhang, Simple maximum principle preserving time-stepping methods for time-fractional Allen-Cahn equation, Adv. Comput. Math., 46 (2020), 37. https://doi.org/10.1007/s10444-020-09782-2 doi: 10.1007/s10444-020-09782-2
![]() |
[35] |
H. L. Liao, T. Tang, T. Zhou, An energy stable and maximum bound preserving scheme with variable time steps for time fractional Allen-Cahn equation, SIAM J. Sci. Comput., 43 (2021), A3503–A3526. https://doi.org/10.1137/20M1384105 doi: 10.1137/20M1384105
![]() |
[36] |
H. L. Liao, X. Zhu, J. Wang, The variable-step L1 time-stepping scheme preserving a compatible energy law for the time-fractional Allen-Cahn equation, Numer. Math. Theory Method Appl., 15 (2022), 1128–1146. https://doi.org/10.4208/nmtma.OA-2022-0011s doi: 10.4208/nmtma.OA-2022-0011s
![]() |
[37] |
H. L. Liao, X. Zhu, H. Sun, Asymptotically compatible energy and dissipation law of the nonuniform L2- scheme for time fractional Allen-Cahn model, J. Sci. Comput., 99 (2024), 46. https://doi.org/10.1007/s10915-024-02515-3 doi: 10.1007/s10915-024-02515-3
![]() |
[38] |
C. Huang, M. Stynes, A sharp -robust error bound for a time-fractional Allen-Cahn problem discretised by the Alikhanov L2- scheme and a standard FEM, J. Sci. Comput., 91 (2022), 43. https://doi.org/10.1007/s10915-022-01810-1 doi: 10.1007/s10915-022-01810-1
![]() |
[39] |
E. Y. Fan, C. P. Li, Diffusion in Allen-Cahn equation: Normal vs anomalous, Phys. D, 457 (2024), 133973. https://doi.org/10.1016/j.physd.2023.133973 doi: 10.1016/j.physd.2023.133973
![]() |
[40] |
F. Jarad, T. Abdeljawad, D. Baleanu, Caputo-type modification of the Hadamard fractional derivatives, Adv. Differ. Equ., 2012 (2012), 1–8. https://doi.org/10.1186/1687-1847-2012-142 doi: 10.1186/1687-1847-2012-142
![]() |
[41] |
Y. Q. Long, Y. Xu, Generalized conforming quadrilateral membrane element with vertex rigid rotational freedom, Comput. Struct., 52 (1994), 749–755. https://doi.org/10.1016/0045-7949(94)90356-5 doi: 10.1016/0045-7949(94)90356-5
![]() |
[42] |
J. S. Jiang, X. L. Cheng, A nonconforming element like Wilson's for second order problems, Math. Numer. Sinica, 14 (1992), 274–278. https://doi.org/10.12286/jssx.1992.3.274 doi: 10.12286/jssx.1992.3.274
![]() |
[43] |
M. Stynes, E. O'Riordan, J. Gracia, Error analysis of a finite difference method on graded mesh for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057–1079. https://doi.org/10.1137/16M1082329 doi: 10.1137/16M1082329
![]() |
[44] |
H. L. Liao, D. Li, J. Zhang, Sharp error estimate of nonuniform L1 formula for linear reaction-subdiffusion equations, SIAM J. Numer. Anal., 56 (2018), 1112–1133. https://doi.org/10.1137/17M1131829 doi: 10.1137/17M1131829
![]() |
[45] |
S. C. Chen, D. Y. Shi, Y. C. Zhao, Anisotropic interpolation and quasi-Wilson element for narrow quadrilateral meshes, IMA J. Numer. Anal., 24 (2004), 77–95. https://doi.org/10.1093/imanum/24.1.77 doi: 10.1093/imanum/24.1.77
![]() |
[46] |
D. Y. Shi, Y. M. Zhao, F. L. Wang, Quasi-Wilson nonconforming element approximation for nonlinear dual phase lagging heat conduction equations, Appl. Math. Comput., 243 (2014), 454–464. https://doi.org/10.1016/j.amc.2014.05.083 doi: 10.1016/j.amc.2014.05.083
![]() |
[47] |
S. C. Chen, D. Y. Shi, Accuracy analysis for quasi-Wilson element, Acta Math. Sci., 20 (2000), 44–48. https://doi.org/10.1016/S0252-9602(17)30730-0 doi: 10.1016/S0252-9602(17)30730-0
![]() |
[48] | V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, second ed., Springer, Berlin, 2006. |
[49] |
D. Y. Shi, P. L. Wang, Y. M. Zhao, Superconvergence analysis of anisotropic linear triangular finite element for nonlinear Schrdinger equation, Appl. Math. Lett., 38 (2014), 129–134. https://doi.org/10.1016/j.aml.2014.07.019 doi: 10.1016/j.aml.2014.07.019
![]() |
[50] | Q. Lin, J. F. Lin, Finite Element Methods: Accuracy and Improvement, Elsevier, 2006. |
[51] |
B. Zhou, X. Chen, D. Li, Nonuniform Alikhanov linearized Galerkin finite element methods for nonlinear time-fractional parabolic equations, J. Sci. Comput., 85 (2020), 39. https://doi.org/10.1007/s10915-020-01350-6 doi: 10.1007/s10915-020-01350-6
![]() |
[52] |
J. Choi, H. Lee, D. Jeong, J. Kim, An unconditionally gradient stable numerical method for solving the Allen-Cahn equation, Physica A, 388 (2009), 1791–1803. https://doi.org/10.1016/j.physa.2009.01.026 doi: 10.1016/j.physa.2009.01.026
![]() |
[53] |
A. A. Alikhanov, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys., 280 (2015), 424–438. https://doi.org/10.1016/j.jcp.2014.09.031 doi: 10.1016/j.jcp.2014.09.031
![]() |
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