Efficient L1-ADI finite difference method for the two-dimensional nonlinear time-fractional diffusion equation. (English) Zbl 07834527
Summary: In this work, we propose an efficient finite difference method for the two-dimensional nonlinear time-fractional diffusion equation with weakly singular solutions. By using backward formula for the approximation of nonlinear term, and L1 scheme on uniform mesh for discretisation of temporal Caputo fractional derivative, a linear scheme is constructed and analysed. Stability and pointwise-in-time convergence of the fully discrete scheme are rigorously established. Numerical results are provided to confirm the sharpness of theoretical analysis.
MSC:
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
References:
| [1] | Al-Maskari, Mariam; Karaa, Samir, Numerical approximation of semilinear subdiffusion equations with nonsmooth initial data, SIAM J. Numer. Anal., 57, 3, 1524-1544, 2019 · Zbl 1422.65246 |
| [2] | Cao, Dewei; Chen, Hu, Pointwise-in-time error estimate of an ADI scheme for two-dimensional multi-term subdiffusion equation, J. Appl. Math. Comput., 69, 1, 707-729, 2023 · Zbl 1509.65069 |
| [3] | Chen, Hu; Stynes, Martin, Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem, J. Sci. Comput., 79, 1, 624-647, 2019 · Zbl 1419.65010 |
| [4] | Chen, Hu; Stynes, Martin, Blow-up of error estimates in time-fractional initial-boundary value problems, IMA J. Numer. Anal., 41, 2, 974-997, 2021 · Zbl 07528268 |
| [5] | Chen, Hu; Stynes, Martin, Using complete monotonicity to deduce local error estimates for discretisations of a multi-term time-fractional diffusion equation, Comput. Methods Appl. Math., 22, 1, 15-29, 2022 · Zbl 1484.65175 |
| [6] | Chen, Minghua; Deng, Weihua, Discretized fractional substantial calculus, ESAIM Math. Model. Numer. Anal., 49, 2, 373-394, 2015 · Zbl 1314.26007 |
| [7] | Farrell, P. A.; Hegarty, A. F.; Miller, J. J.H.; O’Riordan, E.; Shishkin, G. I., Robust Computational Techniques for Boundary Layers, Applied Mathematics (Boca Raton), vol. 16, 2000, Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, FL · Zbl 0964.65083 |
| [8] | Luis Gracia, José; O’Riordan, Eugene; Stynes, Martin, Convergence in positive time for a finite difference method applied to a fractional convection-diffusion problem, Comput. Methods Appl. Math., 18, 1, 33-42, 2018 · Zbl 1382.65275 |
| [9] | Jin, Bangti; Lazarov, Raytcho; Zhou, Zhi, Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview, Comput. Methods Appl. Mech. Eng., 346, 332-358, 2019 · Zbl 1440.65138 |
| [10] | Jin, Bangti; Li, Buyang; Zhou, Zhi, Numerical analysis of nonlinear subdiffusion equations, SIAM J. Numer. Anal., 56, 1, 1-23, 2018 · Zbl 1422.65228 |
| [11] | Khan, Zareen A.; Ahmad, Israr; Shah, Kamal, Applications of fixed point theory to investigate a system of fractional order differential equations, J. Funct. Spaces, Article 1399764 pp., 2021 · Zbl 1480.34104 |
| [12] | Korobenko, L.; Kamrujjaman, Md.; Braverman, E., Persistence and extinction in spatial models with a carrying capacity driven diffusion and harvesting, J. Math. Anal. Appl., 399, 1, 352-368, 2013 · Zbl 1393.35101 |
| [13] | Li, Dongfang; Liao, Hong-Lin; Sun, Weiwei; Wang, Jilu; Zhang, Jiwei, Analysis of L1-Galerkin FEMs for time-fractional nonlinear parabolic problems, Commun. Comput. Phys., 24, 1, 86-103, 2018 · Zbl 1488.65431 |
| [14] | Liao, Hong-lin; Li, Dongfang; Zhang, Jiwei, Sharp error estimate of the nonuniform L1 formula for linear reaction-subdiffusion equations, SIAM J. Numer. Anal., 56, 2, 1112-1133, 2018 · Zbl 1447.65026 |
| [15] | Liao, Hong-lin; McLean, William; Zhang, Jiwei, A discrete Grönwall inequality with applications to numerical schemes for subdiffusion problems, SIAM J. Numer. Anal., 57, 1, 218-237, 2019 · Zbl 1414.65008 |
| [16] | Lin, Yumin; Xu, Chuanju, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225, 2, 1533-1552, 2007 · Zbl 1126.65121 |
| [17] | Magin, R. L., Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng., 104, 1, 1-104, 2004 |
| [18] | Scalas, Enrico; Gorenflo, Rudolf; Mainardi, Francesco, Fractional calculus and continuous-time finance, Physica A, 284, 1-4, 376-384, 2000 |
| [19] | Stynes, Martin; O’Riordan, Eugene; Gracia, José Luis, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55, 2, 1057-1079, 2017 · Zbl 1362.65089 |
| [20] | Sun, Zhi-zhong; Ji, Cui-cui; Du, Ruilian, A new analytical technique of the L-type difference schemes for time fractional mixed sub-diffusion and diffusion-wave equations, Appl. Math. Lett., 102, Article 106115 pp., 2020 · Zbl 1524.35717 |
| [21] | Uchaikin, Vladimir V., Fractional derivatives for physicists and engineers, vol. II. Nonlinear physical science, (Applications, 2013, Higher Education Press: Higher Education Press Beijing, Springer, Heidelberg) · Zbl 1312.26002 |
| [22] | Wang, Yue; Chen, Hu, Pointwise error estimate of an alternating direction implicit difference scheme for two-dimensional time-fractional diffusion equation, Comput. Math. Appl., 99, 155-161, 2021 · Zbl 1524.65416 |
| [23] | Zeng, Fanhai; Li, Changpin; Liu, Fawang; Turner, Ian, Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy, SIAM J. Sci. Comput., 37, 1, A55-A78, 2015 · Zbl 1334.65162 |
| [24] | Zhang, Zongbiao; Li, Meng; Wang, Zhongchi, A linearized Crank-Nicolson Galerkin FEMs for the nonlinear fractional Ginzburg-Landau equation, Appl. Anal., 98, 15, 2648-2667, 2019 · Zbl 1432.65151 |
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