| Citation: |
Ruimin Gao, Congping Lin, Boya Zhou. TRANSFORMED L1-ADI AND COMPACT ADI DIFFERENCE SCHEMES FOR TWO-DIMENSIONAL TIME-FRACTIONAL SUB-DIFFUSION EQUATIONS[J]. Journal of Applied Analysis & Computation, 2026, 16(3): 1490-1516. doi: 10.11948/20240560
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TRANSFORMED L1-ADI AND COMPACT ADI DIFFERENCE SCHEMES FOR TWO-DIMENSIONAL TIME-FRACTIONAL SUB-DIFFUSION EQUATIONS
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Abstract
This paper proposes two novel alternating direction implicit (ADI) difference schemes for solving two-dimensional time-fractional sub-diffusion equations, taking into account the weak initial singularity of the solutions. To accurately capture the rapid evolution near the initial time, the Caputo time-fractional derivative is discretized using a transformed L1 method. Furthermore, central and compact finite difference methods are employed in spatial discretization to enhance computational efficiency. By using a discrete Grönwall inequality, the error estimates in $ H^1 $-norm of the two schemes are obtained. Numerical examples are presented to illustrate the effectiveness of the proposed schemes.
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References
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Ruimin Gao, Congping Lin, Boya Zhou. TRANSFORMED L1-ADI AND COMPACT ADI DIFFERENCE SCHEMES FOR TWO-DIMENSIONAL TIME-FRACTIONAL SUB-DIFFUSION EQUATIONS[J]. Journal of Applied Analysis & Computation, 2026, 16(3): 1490-1516. doi: 10.11948/20240560
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Figure 1.
Time contract(s) of the L1 method (L1), grided mesh L1 method (GL1), TL1 method (TL1) and our method (ADI-TL1) with
$ \alpha=0.3 $ $ \alpha=0.5 $ $ \alpha=0.8 $