Error estimate of L1-ADI scheme for two-dimensional multi-term time fractional diffusion equation. (English) Zbl 1534.65140
Summary: A two-dimensional multi-term time fractional diffusion equation \(D_t^{\alpha} u(x, y, t)- \Delta u(x, y, t) = f(x, y, t)\) is considered in this paper, where \(D_t^{\alpha}\) is the multi-term time Caputo fractional derivative. To solve the equation numerically, L1 discretisation to each fractional derivative is used on a graded temporal mesh, together with a standard finite difference method for the spatial derivatives on a uniform spatial mesh. We provide a rigorous stability and convergence analysis of a fully discrete L1-ADI scheme for solving the multi-term time fractional diffusion problem. Numerical results show that the error estimate is sharp.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35R11 | Fractional partial differential equations |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
Keywords:
multi-term fractional derivative; ADI scheme; L1 scheme; finite difference method; time fractionalReferences:
| [1] | 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 · Zbl 1509.65069 · doi:10.1007/s12190-022-01759-2 |
| [2] | H. Chen, D. Xu, Y. Peng, A second order BDF alternating direction implicit difference scheme for the two-dimensional fractional evolution equation, Appl. Math. Model., 41 (2017), 54-67. https://doi.org/10.1016/j.apm.2016.05.047 · Zbl 1443.65439 · doi:10.1016/j.apm.2016.05.047 |
| [3] | H. Chen, X. Hu, J. Ren, T. Sun, Y. Tang, L1 scheme on graded mesh for the linearized time fractional KdV equation with initial singularity, Int. J. Mod. Sim. Sci. Comp., 10 (2019), 1941006. https://doi.org/10.1142/S179396231941006X · doi:10.1142/S179396231941006X |
| [4] | P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O’Riordan, G. I. Shishkin, Robust computational techniques for boundary layers, Boca Raton: Chapman & Hall/CRC, 2000. · Zbl 0964.65083 |
| [5] | X. M. Gu, H. W. Sun, Y. Zhang, Y. L. Zhao, Fast implicit difference schemes for time-space fractional diffusion equations with the integral fractional Laplacian, Math. Methods Appl. Sci., 44 (2021), 441-463. https://doi.org/10.1002/mma.6746 · Zbl 1512.65303 · doi:10.1002/mma.6746 |
| [6] | T. Guo, O. Nikan, Z. Avazzadeh, W. Qiu, Efficient alternating direction implicit numerical approaches for multi-dimensional distributed-order fractional integro differential problems, Comput. Appl. Math., 41 (2022), 236. https://doi.org/10.1007/s40314-022-01934-y · Zbl 1513.35401 · doi:10.1007/s40314-022-01934-y |
| [7] | R. Hilfer, Y. Luchko, Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal., 12 (2009), 299-318. · Zbl 1182.26011 |
| [8] | C. Huang, H. Chen, N. An, \( \beta \)-robust superconvergent analysis of a finite element method for the distributed order time-fractional diffusion equation, J. Sci. Comput., 90 (2022), 44. https://doi.org/10.1007/s10915-021-01726-2 · Zbl 1480.65258 · doi:10.1007/s10915-021-01726-2 |
| [9] | C. Huang, X. Liu, X. Meng, M. Stynes, Error analysis of a finite difference method on graded meshes for a multiterm time-fractional initial-boundary value problem, Comput. Methods Appl. Math., 20 (2020), 815-825. https://doi.org/10.1515/cmam-2019-0042 · Zbl 1451.65146 · doi:10.1515/cmam-2019-0042 |
| [10] | C. Huang, M. Stynes, H. Chen, An \(\alpha \)-robust finite element method for a multi-term time-fractional diffusion problem, J. Comput. Appl. Math., 389 (2021), 113334. https://doi.org/10.1016/j.cam.2020.113334 · Zbl 1457.65113 · doi:10.1016/j.cam.2020.113334 |
| [11] | J. Huang, J. Zhang, S. Arshad, Y. Tang, A numerical method for two-dimensional multi-term time-space fractional nonlinear diffusion-wave equations, Appl. Numer. Math., 159 (2021), 159-173. https://doi.org/10.1016/j.apnum.2020.09.003 · Zbl 1459.65145 · doi:10.1016/j.apnum.2020.09.003 |
| [12] | M. Hussain, S. Haq, Weighted meshless spectral method for the solutions of multi-term time fractional advection-diffusion problems arising in heat and mass transfer, Int J Heat Mass Tran, 129 (2019), 1305-1316. |
| [13] | B. Li, T. Wang, X. Xie, Analysis of the L1 scheme for fractional wave equations with nonsmooth data, Comput. Math. Appl., 90 (2021), 1-12. https://doi.org/10.1016/j.camwa.2021.03.006 · Zbl 1524.65563 · doi:10.1016/j.camwa.2021.03.006 |
| [14] | L. Li, D. Xu, M. Luo, Alternating direction implicit Galerkin finite element method for the two-dimensional fractional diffusion-wave equation, J. Comput. Phys., 255 (2013), 471-485. https://doi.org/10.1016/j.jcp.2013.08.031 · Zbl 1349.65456 · doi:10.1016/j.jcp.2013.08.031 |
| [15] | Z. Li, Y. Liu, M. Yamamoto, Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients, Appl. Math. Comput., 257 (2015), 381-397. https://doi.org/10.1016/j.amc.2014.11.073 · Zbl 1338.35471 · doi:10.1016/j.amc.2014.11.073 |
| [16] | 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 · Zbl 1126.65121 · doi:10.1016/j.jcp.2007.02.001 |
| [17] | Y. Luchko, Initial-boundary problems for the generalized multi-term time-fractional diffusion equation, J. Math. Anal. Appl., 374 (2011), 538-548. https://doi.org/10.1016/j.jmaa.2010.08.048 · Zbl 1202.35339 · doi:10.1016/j.jmaa.2010.08.048 |
| [18] | D. W. Peaceman, H. H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math., 3 (1955), 28-41. · Zbl 0067.35801 |
| [19] | L. Qiao, J. Guo, W. 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 · Zbl 1524.65385 · doi:10.1016/j.camwa.2022.08.014 |
| [20] | L. Qiao, W. Qiu, D. Xu, Error analysis of fast L1 ADI finite difference/compact difference schemes for the fractional telegraph equation in three dimensions, Math. Comput. Simulation, 205 (2023), 205-231. https://doi.org/10.1016/j.matcom.2022.10.001 · Zbl 1540.65320 · doi:10.1016/j.matcom.2022.10.001 |
| [21] | L. Qiao, D. Xu, Compact alternating direction implicit scheme for integro-differential equations of parabolic type, J. Sci. Comput., 76 (2018), 565-582. https://doi.org/10.1007/s10915-017-0630-5 · Zbl 1445.65050 · doi:10.1007/s10915-017-0630-5 |
| [22] | S. Qin, F. Liu, I. Turner, V. Vegh, Q. Yu, Q. Yang, Multi-term time-fractional Bloch equations and application in magnetic resonance imaging, J. Comput. Appl. Math., 319 (2017), 308-319. https://doi.org/10.1016/j.cam.2017.01.018 · Zbl 1394.92075 · doi:10.1016/j.cam.2017.01.018 |
| [23] | W. Qiu, D. Xu, H. Chen, J. Guo, An alternating direction implicit Galerkin finite element method for the distributed-order time-fractional mobile-immobile equation in two dimensions, Comput. Math. Appl., 80 (2020), 3156-3172. https://doi.org/10.1016/j.camwa.2020.11.003 · Zbl 1524.65584 · doi:10.1016/j.camwa.2020.11.003 |
| [24] | 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/10.4208/eajam.010418.020718 · Zbl 1468.65110 · doi:10.4208/eajam.010418.020718 |
| [25] | E. Sousa, C. Li, A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville derivative, Appl. Numer. Math., 90 (2015), 22-37. https://doi.org/10.1016/j.apnum.2014.11.007 · Zbl 1326.65111 · doi:10.1016/j.apnum.2014.11.007 |
| [26] | V. Srivastava, K. N. Rai, A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues, Math. Comput. Modelling, 51 (2010), 616-624. https://doi.org/10.1016/j.mcm.2009.11.002 · Zbl 1190.35226 · doi:10.1016/j.mcm.2009.11.002 |
| [27] | M. Stynes, A survey of the L1 scheme in the discretisation of time-fractional problems, Numer. Math. Theory Methods Appl., 15 (2022), 1173-1192. · Zbl 1524.65398 |
| [28] | M. Stynes, E. O’Riordan, J. L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057-1079. https://doi.org/10.1137/16M1082329 · Zbl 1362.65089 · doi:10.1137/16M1082329 |
| [29] | 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 · Zbl 1486.65133 · doi:10.1016/j.apnum.2021.05.025 |
| [30] | Y. Yan, M. Khan, N. J. Ford, An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data, SIAM J. Numer. Anal., 56 (2018), 210-227. https://doi.org/10.1137/16M1094257 · Zbl 1381.65070 · doi:10.1137/16M1094257 |
| [31] | 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 · Zbl 1242.65174 · doi:10.1016/j.jcp.2011.08.020 |
| [32] | 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 · Zbl 1468.76055 · doi:10.1007/s10915-021-01527-7 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
