Stability and convergence analysis for a uniform temporal high accuracy of the time-fractional diffusion equation with 1D and 2D spatial compact finite difference method

Junying Cao; Zhongqing Wang; Ziqiang Wang; Junying Cao; Zhongqing Wang; Ziqiang Wang

doi:10.3934/math.2024715

AIMS Mathematics

2024, Volume 9, Issue 6: 14697-14730. doi: 10.3934/math.2024715

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Research article

Stability and convergence analysis for a uniform temporal high accuracy of the time-fractional diffusion equation with 1D and 2D spatial compact finite difference method

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

Received: 22 February 2024 Revised: 26 March 2024 Accepted: 29 March 2024 Published: 23 April 2024
MSC : 65L12, 65M06, 65M12

The 1D and 2D spatial compact finite difference schemes (CFDSs) for time-fractional diffusion equations (TFDEs) were presented in this article with uniform temporal convergence order. Based on the idea of the modified block-by-block method, the CFDSs with uniform temporal convergence order for TFDEs were given by combining the fourth-order CFDSs in space and the high order scheme in time. The stability analysis and convergence order of CFDSs with uniform convergence order in time for TFDEs strictly proved that the provided uniform accuracy time scheme is $ (3-\alpha) $ temporal order and spatial fourth-order, respectively. Ultimately, the astringency of 1D and 2D spatial CFDSs was verified by some numerical examples.
- compact finite difference method,
- stability analysis,
- time-fractional diffusion equation,
- convergence analysis,
- optimal convergence order
Citation: Junying Cao, Zhongqing Wang, Ziqiang Wang. Stability and convergence analysis for a uniform temporal high accuracy of the time-fractional diffusion equation with 1D and 2D spatial compact finite difference method[J]. AIMS Mathematics, 2024, 9(6): 14697-14730. doi: 10.3934/math.2024715

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Abstract

The 1D and 2D spatial compact finite difference schemes (CFDSs) for time-fractional diffusion equations (TFDEs) were presented in this article with uniform temporal convergence order. Based on the idea of the modified block-by-block method, the CFDSs with uniform temporal convergence order for TFDEs were given by combining the fourth-order CFDSs in space and the high order scheme in time. The stability analysis and convergence order of CFDSs with uniform convergence order in time for TFDEs strictly proved that the provided uniform accuracy time scheme is $ (3-\alpha) $ temporal order and spatial fourth-order, respectively. Ultimately, the astringency of 1D and 2D spatial CFDSs was verified by some numerical examples.

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