A fast second-order block-centered finite difference method for the fractional Cattaneo equation

Xianqiang Zhang; Xianqiang Zhang

doi:10.3934/math.2025640

AIMS Mathematics

2025, Volume 10, Issue 6: 14196-14221. doi: 10.3934/math.2025640

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Research article

A fast second-order block-centered finite difference method for the fractional Cattaneo equation

Xianqiang Zhang ^,

School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China

Received: 27 March 2025 Revised: 01 June 2025 Accepted: 06 June 2025 Published: 19 June 2025
MSC : 65N06, 65N15

In this paper, a fast second-order block-centered finite difference method based on the $ \mathcal {FL} $2-$ 1_{\sigma} $ formula and a weighted approach was proposed for the time fractional Cattaneo equation. Using the special properties of the discrete coefficients and mathematical induction method, we derived the unconditional stability and convergence of the method rigorously. Second-order superconvergence in time and space for the velocity and pressure in discrete $ L^{\infty}(L^{2}) $-norms was established on non-uniform rectangular grids. Numerical examples were provided to verify our theoretical results.
- fractional Cattaneo equation,
- Caputo derivative,
- block-centered finite difference,
- fast algorithm,
- $ \mathcal {F}\mathcal {L}2-1_{\sigma} $ formula,
- superconvergence
Citation: Xianqiang Zhang. A fast second-order block-centered finite difference method for the fractional Cattaneo equation[J]. AIMS Mathematics, 2025, 10(6): 14196-14221. doi: 10.3934/math.2025640

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Abstract

In this paper, a fast second-order block-centered finite difference method based on the $ \mathcal {FL} $2-$ 1_{\sigma} $ formula and a weighted approach was proposed for the time fractional Cattaneo equation. Using the special properties of the discrete coefficients and mathematical induction method, we derived the unconditional stability and convergence of the method rigorously. Second-order superconvergence in time and space for the velocity and pressure in discrete $ L^{\infty}(L^{2}) $-norms was established on non-uniform rectangular grids. Numerical examples were provided to verify our theoretical results.

References

[1]	A. Compte, R. Metzler, The generalized Cattaneo equation for the description of anomalous transport processes, J. Phys. A: Math. Gen., 30 (1997), 7277–7289. https://doi.org/10.1088/0305-4470/30/21/006 doi: 10.1088/0305-4470/30/21/006
[2]	Y. Z. Povstenko, Fractional Cattaneo-type equations and generalized thermoelasticity, J. Therm. Stresses, 34 (2011), 97–114. https://doi.org/10.1080/01495739.2010.511931 doi: 10.1080/01495739.2010.511931
[3]	H. T. Qi, X. Y. Jiang, Solutions of the space-time fractional Cattaneo diffusion equation, Phys. A, 390 (2011), 1876–1883. https://doi.org/10.1016/j.physa.2011.02.010 doi: 10.1016/j.physa.2011.02.010
[4]	Z. N. Xue, G. Q. Cao, Y. J. Yu, J. L. Liu, Coupled thermoelastic fracture analysis of a cracked fiber reinforced composite hollow cylinder by fractional Cattaneo-Vernotte models, Theor. Appl. Fract. Mec., 121 (2022), 103538. https://doi.org/10.1016/j.tafmec.2022.103538 doi: 10.1016/j.tafmec.2022.103538
[5]	Y. J. Yu, Z. C. Deng, Fractional order theory of Cattaneo-type thermoelasticity using new fractional derivatives, Appl. Math. Model., 87 (2020), 731–751. https://doi.org/10.1016/j.apm.2020.06.023 doi: 10.1016/j.apm.2020.06.023
[6]	S. Vong, H. Pang, X. Q. Jin, A high-order difference scheme for the generalized Cattaneo equation, East Asian J. Appl. Math., 2 (2012), 170–184. https://doi.org/10.4208/eajam.110312.240412a doi: 10.4208/eajam.110312.240412a
[7]	X. Zhao, Z. Z. Sun, Compact Crank-Nicolson schemes for a class of fractional Cattaneo equation in inhomogeneous medium, J. Sci. Comput., 62 (2014), 747–771. https://doi.org/10.1007/s10915-014-9874-5 doi: 10.1007/s10915-014-9874-5
[8]	J. C. Ren, G. H. Gao, Efficient and stable numerical methods for the two-dimensional fractional Cattaneo equation, Numer. Algor., 69 (2015), 795–818. https://doi.org/10.1007/s11075-014-9926-9 doi: 10.1007/s11075-014-9926-9
[9]	L. L. Wei, Analysis of a new finite difference/local discontinuous Galerkin method for the fractional Cattaneo equation, Numer. Algor., 77 (2018), 675–690. https://doi.org/10.1007/s11075-017-0334-9 doi: 10.1007/s11075-017-0334-9
[10]	O. Nikan, Z. Avazzadeh, J. A. T. Machado, Numerical approach for modeling fractional heat conduction in porous medium with the generalized Cattaneo model, Appl. Math. Model., 100 (2021), 107–124. https://doi.org/10.1016/j.apm.2021.07.025 doi: 10.1016/j.apm.2021.07.025
[11]	A. Weiser, M. F. Wheeler, On convergence of block-centered finite-differences for elliptic problems, SIAM J. Numer. Anal., 25 (1988), 351–375. https://doi.org/10.1137/0725025 doi: 10.1137/0725025
[12]	H. Rui, H. Pan, A block-centered finite difference method for the Darcy-Forchheimer model, SIAM J. Numer. Anal., 5 (2012), 2612–2631. https://doi.org/10.1137/110858239 doi: 10.1137/110858239
[13]	W. Liu, J. T. Cui, J. Xin, A block-centered finite difference method for an unsteady asymptotic coupled model in fractured media aquifer system, J. Comput. Appl. Math., 337 (2018), 319–340. https://doi.org/10.1016/j.cam.2017.12.035 doi: 10.1016/j.cam.2017.12.035
[14]	Y. L. Jing, C. Li, Block-centered finite difference method for a tempered subdiffusion model with time-dependent coefficients, Comput. Math. Appl., 145 (2023) 202–223. https://doi.org/10.1016/j.camwa.2023.06.014
[15]	S. Y. Zhai, X. L. Feng, A block-centered finite-difference method for the time-fractional diffusion equation on nonuniform grids, Numer. Heat Tr. B-Fund., 69 (2016), 217–233. https://doi.org/10.1080/10407790.2015.1097101 doi: 10.1080/10407790.2015.1097101
[16]	Z. G. Liu, X. L. Li, A parallel CGS block-centered finite difference method for a nonlinear time-fractional parabolic equation, Comput. Meth. Appl. Mech. Eng., 308 (2016), 330–348. https://doi.org/10.1016/j.cma.2016.05.028 doi: 10.1016/j.cma.2016.05.028
[17]	X. L. Li, H. X. Rui, A high-order fully conservative block-centered finite difference method for the time-fractional advection-dispersion equation, Appl. Numer. Math., 124 (2018), 89–109. https://doi.org/ 10.1016/j.apnum.2017.10.004 doi: 10.1016/j.apnum.2017.10.004
[18]	X. L. Li, H. X. Rui, Stability and convergence based on the finite difference method for the nonlinear fractional cable equation on non-uniform staggered grids, Appl. Numer. Math., 152 (2020), 403–421. https://doi.org/10.1016/j.apnum.2019.11.013 doi: 10.1016/j.apnum.2019.11.013
[19]	X. L. Li, H. X. Rui, A block-centered finite difference method for the distributed-order time-fractional diffusion-wave equation, Appl. Numer. Math., 131 (2018), 123–139. https://doi.org/10.1016/j.apnum.2018.04.013 doi: 10.1016/j.apnum.2018.04.013
[20]	X. L. Li, H. X. Rui, Z. G. Liu, A block-centered finite difference method for fractional Cattaneo equation, Numer. Meth. Part. D. E., 34 (2017), 296–316. https://doi.org/10.1002/num.22198 doi: 10.1002/num.22198
[21]	Y. G. Yan, Z. Z. Sun, J. W. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations: a second-order scheme, Commun. Comput. Phys., 22 (2017), 1028–1048. https://doi.org/10.4208/cicp.OA-2017-0019 doi: 10.4208/cicp.OA-2017-0019
[22]	P. Lyu, Y. X. Liang, Z. B. Wang, A fast linearized finite difference method for the nonlinear multi-term time-fractional wave equation, Appl. Numer. Math., 151 (2020), 448–471. https://doi.org/10.1016/j.apnum.2019.11.012 doi: 10.1016/j.apnum.2019.11.012
[23]	X. X. Bai, J. Huang, H. X. Rui, S. Wang, Numerical simulation for 2D/3D time fractional Maxwell's system based on a fast second-order FDTD algorithm, J. Comput. Appl. Math., 416 (2022), 114590. https://doi.org/10.1016/j.cam.2022.114590 doi: 10.1016/j.cam.2022.114590
[24]	L. Ma, H. F. Fu, B. Y. Zhang, S. S. Xie, A fast compact block-centered finite difference method on graded meshes for time-fractional reaction-diffusion equations and its robust analysis, Numer. Math. Theory Me., 17 (2024), 429–462. https://doi.org/10.4208/nmtma.OA-2023-0108 doi: 10.4208/nmtma.OA-2023-0108
[25]	X. Zhao, X. L. Li, Z. Y. Li, Fast and efficient finite difference method for the distributed-order diffusion equation based on the staggered grids, Appl. Numer. Math., 174 (2022), 34–45. https://doi.org/10.1016/j.apnum.2022.01.006 doi: 10.1016/j.apnum.2022.01.006
[26]	S. D. Jiang, J. W. Zhang, Q. Zhang, Z. M. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys., 21 (2017), 650–678. https://doi.org/10.4208/cicp.OA-2016-0136 doi: 10.4208/cicp.OA-2016-0136
[27]	Z. Guan, J. G. Wang, Y. Liu, Y. F. Nie, Unconditionally optimal convergence of a linearized Galerkin FEM for the nonlinear time-fractional mobile/immobile transport equation, Appl. Numer. Math., 172 (2022), 133–156. https://doi.org/10.1016/j.apnum.2021.10.004 doi: 10.1016/j.apnum.2021.10.004
[28]	P. Lyu, S. Vong, A linearized second-order scheme for nonlinear time fractional Klein-Gordon type equations, Numer. Algor., 78 (2018), 485–511. https://doi.org/10.1007/s11075-017-0385-y doi: 10.1007/s11075-017-0385-y
[29]	J. G. Heywood, R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem part Ⅳ: Error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353–384. https://doi.org/10.1137/0727022 doi: 10.1137/0727022
[30]	L. J. Nong, Q. Yi, A. Chen, A fast second-order ADI finite difference scheme for the two-dimensional time-fractional Cattaneo equation with spatially variable coefficients, Fractal Fract., 8 (2024), 453. https://doi.org/10.3390/fractalfract8080453 doi: 10.3390/fractalfract8080453

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