An efficient high order numerical scheme for the time-fractional diffusion equation with uniform accuracy

Junying Cao; Qing Tan; Zhongqing Wang; Ziqiang Wang; Junying Cao; Qing Tan; Zhongqing Wang; Ziqiang Wang

doi:10.3934/math.2023818

AIMS Mathematics

2023, Volume 8, Issue 7: 16031-16061. doi: 10.3934/math.2023818

Previous Article Next Article

Research article

An efficient high order numerical scheme for the time-fractional diffusion equation with uniform accuracy

School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, 550025, China

Received: 22 March 2023 Revised: 24 April 2023 Accepted: 27 April 2023 Published: 05 May 2023
MSC : 65L12, 65M06, 65M12

The construction of efficient numerical schemes with uniform convergence order for time-fractional diffusion equations (TFDEs) is an important research problem. We are committed to study an efficient uniform accuracy scheme for TFDEs. Firstly, we use the piecewise quadratic interpolation to construct an efficient uniform accuracy scheme for the fractional derivative of time. And the local truncation error of the efficient scheme is also given. Secondly, the full discrete numerical scheme for TFDEs is given by combing the spatial center second order scheme and the above efficient time scheme. Thirdly, the efficient scheme's stability and error estimates are strictly theoretical analysis to obtain that the unconditionally stable scheme is $ 3-\beta $ convergence order with uniform accuracy in time. Finally, some numerical examples are applied to show that the proposed scheme is an efficient unconditionally stable scheme.
- efficient numerical scheme,
- stability and convergence analysis,
- optimal convergence order
Citation: Junying Cao, Qing Tan, Zhongqing Wang, Ziqiang Wang. An efficient high order numerical scheme for the time-fractional diffusion equation with uniform accuracy[J]. AIMS Mathematics, 2023, 8(7): 16031-16061. doi: 10.3934/math.2023818

Related Papers:

Abstract

The construction of efficient numerical schemes with uniform convergence order for time-fractional diffusion equations (TFDEs) is an important research problem. We are committed to study an efficient uniform accuracy scheme for TFDEs. Firstly, we use the piecewise quadratic interpolation to construct an efficient uniform accuracy scheme for the fractional derivative of time. And the local truncation error of the efficient scheme is also given. Secondly, the full discrete numerical scheme for TFDEs is given by combing the spatial center second order scheme and the above efficient time scheme. Thirdly, the efficient scheme's stability and error estimates are strictly theoretical analysis to obtain that the unconditionally stable scheme is $ 3-\beta $ convergence order with uniform accuracy in time. Finally, some numerical examples are applied to show that the proposed scheme is an efficient unconditionally stable scheme.

References

[1]	Y. Wang, L. Ren, A high-order L2-compact difference method for Caputo-type time-fractional sub-diffusion equations with variable coefficients, Appl. Math. Comput., 342 (2019), 71–93. https://doi.org/10.1016/j.amc.2018.09.007 doi: 10.1016/j.amc.2018.09.007
[2]	H. Zhang, J. Jia, X. Jiang, An optimal error estimate for the two-dimensional nonlinear time fractional advection-diffusion equation with smooth and non-smooth solutions, Comput. Math. Appl., 79 (2020), 2819–2831. https://doi.org/10.1016/j.camwa.2019.12.013 doi: 10.1016/j.camwa.2019.12.013
[3]	Y. H. Youssri, A. G. Atta, Petrov-Galerkin Lucas polynomials procedure for the time-fractional diffusion equation, Contemp. Math., 4 (2023), 230–248. https://doi.org/10.37256/cm.4220232420 doi: 10.37256/cm.4220232420
[4]	T. Eftekhari, S. Hosseini, A new and efficient approach for solving linear and nonlinear time-fractional diffusion equations of distributed order, Comput. Appl. Math., 41 (2022), 281. https://doi.org/10.1007/s40314-022-01981-5 doi: 10.1007/s40314-022-01981-5
[5]	Y. Wang, H. Chen, Pointwise error estimate of an alternating direction implicit difference scheme for two-dimensional time-fractional diffusion equation, Comput. Math. Appl., 99 (2021), 155–161. https://doi.org/10.1016/j.camwa.2021.08.012 doi: 10.1016/j.camwa.2021.08.012
[6]	Z. Liu, A. Cheng, X. Li, A novel finite difference discrete scheme for the time fractional diffusion-wave equation, Appl. Numer. Math., 134 (2018), 17–30. https://doi.org/10.1016/j.apnum.2018.07.001 doi: 10.1016/j.apnum.2018.07.001
[7]	A. A. Alikhanov, C. Huang, A high-order L2 type difference scheme for the time-fractional diffusion equation, Appl. Math. Comput., 411 (2021), 126545. https://doi.org/10.1016/j.amc.2021.126545 doi: 10.1016/j.amc.2021.126545
[8]	M. Cui, Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation, Numer. Algor., 62 (2013), 383–409. https://doi.org/10.1007/s11075-012-9589-3 doi: 10.1007/s11075-012-9589-3
[9]	C. Lv, C. Xu, Error analysis of a high order method for time-fractional diffusion equations, SIAM J. Sci. Comput., 38 (2016), A2699–A2722. https://doi.org/10.1137/15M102664X doi: 10.1137/15M102664X
[10]	S. Kumar, D. Zeidan, An efficient Mittag-Leffler kernel approach for time-fractional advection-reaction-diffusion equation, Appl. Numer. Math., 170 (2021), 190–207. https://doi.org/10.1016/j.apnum.2021.07.025 doi: 10.1016/j.apnum.2021.07.025
[11]	J. Shen, C. Sheng, An efficient space-time method for time fractional diffusion equation, J. Sci. Comput., 81 (2019), 1088–1110. https://doi.org/10.1007/s10915-019-01052-8 doi: 10.1007/s10915-019-01052-8
[12]	M. A. Abdelkawy, A. M. Lopes, M. A. Zaky, Shifted fractional Jacobi spectral algorithm for solving distributed order time-fractional reaction-diffusion equations, Comput. Appl. Math., 38 (2019), 81. https://doi.org/10.1007/s40314-019-0845-1 doi: 10.1007/s40314-019-0845-1
[13]	M. A. Zaky, A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations, Comput. Appl. Math., 37 (2018), 3525–3538. https://doi.org/10.1007/s40314-017-0530-1 doi: 10.1007/s40314-017-0530-1
[14]	J. Zhang, Z. Fang, H. Sun, Exponential-sum-approximation technique for variable-order time-fractional diffusion equations, J. Appl. Math. Comput., 68 (2022), 323–347. https://doi.org/10.1007/s12190-021-01528-7 doi: 10.1007/s12190-021-01528-7
[15]	P. Lyu, S. Vong, A fast linearized numerical method for nonlinear time-fractional diffusion equations, Numer. Algor., 87 (2021), 381–408. https://doi.org/10.1007/s11075-020-00971-0 doi: 10.1007/s11075-020-00971-0
[16]	S. Jiang, J. Zhang, Q. Zhang, Z. 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
[17]	P. Roul, V. Rohil, A high-order numerical scheme based on graded mesh and its analysis for the two-dimensional time-fractional convection-diffusion equation, Comput. Math. Appl., 126 (2022), 1–13. https://doi.org/10.1016/j.camwa.2022.09.006 doi: 10.1016/j.camwa.2022.09.006
[18]	S. Martin, O. Eugene, L. Jose, 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 doi: 10.1137/16M1082329
[19]	N. Kedia, A. Alikhanov, V. Singh, Stable numerical schemes for time-fractional diffusion equation with generalized memory kernel, Appl. Numer. Math., 172 (2022), 546–565. https://doi.org/10.1016/j.apnum.2021.11.006 doi: 10.1016/j.apnum.2021.11.006
[20]	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
[21]	N. Kopteva, Error analysis of an $L2$-type method on graded meshes for a fractional-order parabolic problem, Math. Comp., 90 (2021), 19–40. https://doi.org/10.1090/mcom/3552 doi: 10.1090/mcom/3552
[22]	A. G. Atta, Y. H. Youssri, Advanced shifted first-kind Chebyshev collocation approach for solving the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel, Comp. Appl. Math., 41 (2022), 381. https://doi.org/10.1007/s40314-022-02096-7 doi: 10.1007/s40314-022-02096-7
[23]	J. Huang, D. Yang, L. O. Jay, Efficient methods for nonlinear time fractional diffusion-wave equations and their fast implementations, Numer. Algor., 85 (2020), 375–397. https://doi.org/10.1007/s11075-019-00817-4 doi: 10.1007/s11075-019-00817-4
[24]	J. Cao, C. Xu, A high order schema for the numercial solution of the fractional ordinary differential equations, J. Comput. Phys., 238 (2013), 154–168. https://doi.org/10.1016/j.jcp.2012.12.013 doi: 10.1016/j.jcp.2012.12.013
[25]	J. Cao, Z. Cai, Numerical analysis of a high-order scheme for nonlinear fractional differential equations with uniform accuracy, Numer. Math. Theor. Meth. Appl., 14 (2021), 71–112. https://doi.org/10.4208/nmtma.OA-2020-0039 doi: 10.4208/nmtma.OA-2020-0039
[26]	L. Feng, P. Zhuang, F. Liu, Y. T. Gu, Finite element method for space-time fractional diffusion equation, Numer. Algor., 72 (2016), 749–767. https://doi.org/10.1007/s11075-015-0065-8 doi: 10.1007/s11075-015-0065-8
[27]	H. Zhang, X. Jiang, F. Zeng, An $H^{1}$ convergence of the spectral method for the time-fractional non-linear diffusion equations, Adv. Comput. Math., 47 (2021), 63. https://doi.org/10.1007/s10444-021-09892-5 doi: 10.1007/s10444-021-09892-5
[28]	A. S. V. R. Kanth, N. Garg, An unconditionally stable algorithm for multiterm time fractional advection-diffusion equation with variable coefficients and convergence analysis, Numer. Methods Partial Differ. Equ., 37 (2021), 1928–1945. https://doi.org/10.1002/num.22629 doi: 10.1002/num.22629
[29]	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
[30]	D. A. Murio, Implicit finite difference approximation for time fractional diffusion equations, Comput. Math. Appl., 56 (2008), 1138–1145. https://doi.org/10.1016/j.camwa.2008.02.015 doi: 10.1016/j.camwa.2008.02.015
[31]	R. Gorenflo, E. A. Abdel-Rehim, Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion, J. Comput. Appl. Math., 205 (2007), 871–881. https://doi.org/10.1016/j.cam.2005.12.043 doi: 10.1016/j.cam.2005.12.043
[32]	R. L. Burden, J. D. Faires, A. M. Burden, Numerical analysis, Cengage Learning, 2015.
[33]	Y. H. Youssri, A. G. Atta, Spectral collocation approach via normalized shifted Jacobi polynomials for the nonlinear Lane-Emden equation with fractal-fractional derivative, Fractal Fract., 7 (2023), 133. https://doi.org/10.3390/fractalfract7020133 doi: 10.3390/fractalfract7020133

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)