Abstract
In this paper, we propose an alternating direction implicit (ADI) difference method to solve the two-dimensional time-fractional reaction-subdiffusion equation with weakly singular solutions, where L1 scheme on graded mesh is used to capture the initial weak singularity. Global and local convergences of L1-ADI scheme on graded mesh are studied with the comparison principle. The temporal global convergence rate of the fully discrete L1-ADI scheme is \(O(N^{-\min \{ 2-\alpha ,2\alpha ,\alpha r\}})\) and the temporal local convergence rate can attain \(O(N^{-\min \{ 2-\alpha , 2\alpha \}})\) when \(r> 2-\alpha \), where N, \(\alpha \), and r are the time partition number, order of fractional derivative, and grading parameter, respectively. Numerical experiments are proposed to verify the sharpness of our theoretical analysis.
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References
Garrappa, R.: Numerical solution of fractional differential equations: a survey and a software tutorial. Mathematics 6(2), 16 (2018)
Jin, B., Lazarov, R., Zhou, Z.: Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview. Comput. Methods Appl. Mech. Engrg. 346, 332–358 (2019). https://doi.org/10.1016/j.cma.2018.12.011
Stynes, M.: A survey of the L1 scheme in the discretisation of time-fractional problems. Numer. Math. Theory Methods Appl. 15(4), 1173–1192 (2022)
Liao, H.-L., Li, D., Zhang, J.: Sharp error estimate of the nonuniform L1 formula for linear reaction-subdiffusion equations. SIAM J. Numer. Anal. 56(2), 1112–1133 (2018). https://doi.org/10.1137/17M1131829
Kopteva, N.: Error analysis for time-fractional semilinear parabolic equations using upper and lower solutions. SIAM J. Numer. Anal. 58(4), 2212–2234 (2020). https://doi.org/10.1137/20M1313015
Liao, H.-l, Zhao, Y., Teng, X.-h: A weighted ADI scheme for subdiffusion equations. J. Sci. Comput 69(3), 1144–1164 (2016). https://doi.org/10.1007/s10915-016-0230-9
Wang, H., Du, N.: Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations. J. Comput. Phys. 258, 305–318 (2014). https://doi.org/10.1016/j.jcp.2013.10.040
Wang, Y., Chen, H.: Pointwise error estimate of an alternating direction implicit difference scheme for two-dimensional time-fractional diffusion equation. Comput. Math. Appl. 99, 155–161 (2021). https://doi.org/10.1016/j.camwa.2021.08.012
Wu, L., Zhai, S.: A new high order ADI numerical difference formula for time-fractional convection-diffusion equation. Appl. Math. Comput. 387, 124564–10 (2020). https://doi.org/10.1016/j.amc.2019.124564
Zeng, F., Zhang, Z., Karniadakis, G.E.: Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations. J. Comput. Phys. 307, 15–33 (2016). https://doi.org/10.1016/j.jcp.2015.11.058
Zhai, S., Feng, X., He, Y.: An unconditionally stable compact ADI method for three-dimensional time-fractional convection-diffusion equation. J. Comput. Phys. 269, 138–155 (2014). https://doi.org/10.1016/j.jcp.2014.03.020
Hou, Y., Wen, C., Liu, Y., Li, H.: A two-grid ADI finite element approximation for a nonlinear distributed-order fractional sub-diffusion equation. Netw. Heterog. Media 18(2), 855–876 (2023). https://doi.org/10.3934/nhm.2023037
Qiu, W., Fairweather, G., Yang, X., Zhang, H.: ADI finite element Galerkin methods for two-dimensional tempered fractional integro-differential equations. Calcolo 60(3), 41–34 (2023). https://doi.org/10.1007/s10092-023-00533-5
Saffarian, M., Mohebbi, A.: A novel ADI Galerkin spectral element method for the solution of two-dimensional time fractional subdiffusion equation. Int. J. Comput. Math. 98(4), 845–867 (2021). https://doi.org/10.1080/00207160.2020.1792450
Wang, Y., Zhu, B., Chen, H.: \(\alpha \)-robust \(H^1\)-norm convergence analysis of L1FEM-ADI scheme for 2D/3D subdiffusion equation with initial singularity. Math. Methods Appl. Sci. 46(15), 16144–16155 (2023)
Chen, H., Lü, S., Chen, W.: Spectral methods for the time fractional diffusion-wave equation in a semi-infinite channel. Comput. Math. Appl. 71(9), 1818–1830 (2016). https://doi.org/10.1016/j.camwa.2016.02.024
Zeng, F., Liu, F., Li, C., Burrage, K., Turner, I., Anh, V.: A Crank-Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation. SIAM J. Numer. Anal. 52(6), 2599–2622 (2014). https://doi.org/10.1137/130934192
Kopteva, N., Meng, X.: Error analysis for a fractional-derivative parabolic problem on quasi-graded meshes using barrier functions. SIAM J. Numer. Anal. 58(2), 1217–1238 (2020). https://doi.org/10.1137/19M1300686
Acknowledgements
The research is supported in part by Natural Science Foundation of Shandong Province under Grant ZR2023MA077, Fundamental Research Funds for the Central Universities (No. 202264006), and the National Natural Science Foundation of China under Grant 11801026.
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Jiang, Y., Chen, H. Local convergence analysis of L1-ADI scheme for two-dimensional reaction-subdiffusion equation. J. Appl. Math. Comput. 70, 1953–1964 (2024). https://doi.org/10.1007/s12190-024-02037-z
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DOI: https://doi.org/10.1007/s12190-024-02037-z