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Volume 18, Issue 1
Unconditional Convergence of Linearized TL1 Difference Methods for a Time-Fractional Coupled Nonlinear Schrödinger System

Min Li, Dongfang Li, Ju Ming & A. S. Hendy

DOI: 10.4208/nmtma.OA-2024-0095

Numer. Math. Theor. Meth. Appl., 18 (2025), pp. 1-30.

Published online: 2025-04

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  • Abstract

This paper presents a transformed L1 (TL1) finite difference method for the time-fractional coupled nonlinear Schrödinger system. Unconditionally optimal $L^2$ error estimates of the fully discrete scheme are obtained. The convergence results indicate that the method has an order of $2$ in the spatial direction and an order of $2 − α$ in the temporal direction. The error estimates hold without any spatial-temporal stepsize restriction. Such convergence results are obtained by applying a novel discrete fractional Grönwall inequality and the corresponding Sobolev embedding theorems. Numerical experiments for both two-dimensional and three-dimensional models are carried out to confirm our theoretical findings.

  • Keywords

Time-fractional coupled nonlinear Schrödinger system, transformed L1 schemes, unconditionally optimal error estimate, linearly implicit schemes.

  • AMS Subject Headings

65L04, 65L06, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-18-1, author = {Li , MinLi , DongfangMing , Ju and Hendy , A. S.}, title = {Unconditional Convergence of Linearized TL1 Difference Methods for a Time-Fractional Coupled Nonlinear Schrödinger System}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2025}, volume = {18}, number = {1}, pages = {1--30}, abstract = {

This paper presents a transformed L1 (TL1) finite difference method for the time-fractional coupled nonlinear Schrödinger system. Unconditionally optimal $L^2$ error estimates of the fully discrete scheme are obtained. The convergence results indicate that the method has an order of $2$ in the spatial direction and an order of $2 − α$ in the temporal direction. The error estimates hold without any spatial-temporal stepsize restriction. Such convergence results are obtained by applying a novel discrete fractional Grönwall inequality and the corresponding Sobolev embedding theorems. Numerical experiments for both two-dimensional and three-dimensional models are carried out to confirm our theoretical findings.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2024-0095}, url = {http://global-sci.org/intro/article_detail/nmtma/23940.html} }
TY - JOUR T1 - Unconditional Convergence of Linearized TL1 Difference Methods for a Time-Fractional Coupled Nonlinear Schrödinger System AU - Li , Min AU - Li , Dongfang AU - Ming , Ju AU - Hendy , A. S. JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 1 EP - 30 PY - 2025 DA - 2025/04 SN - 18 DO - http://doi.org/10.4208/nmtma.OA-2024-0095 UR - https://global-sci.org/intro/article_detail/nmtma/23940.html KW - Time-fractional coupled nonlinear Schrödinger system, transformed L1 schemes, unconditionally optimal error estimate, linearly implicit schemes. AB -

This paper presents a transformed L1 (TL1) finite difference method for the time-fractional coupled nonlinear Schrödinger system. Unconditionally optimal $L^2$ error estimates of the fully discrete scheme are obtained. The convergence results indicate that the method has an order of $2$ in the spatial direction and an order of $2 − α$ in the temporal direction. The error estimates hold without any spatial-temporal stepsize restriction. Such convergence results are obtained by applying a novel discrete fractional Grönwall inequality and the corresponding Sobolev embedding theorems. Numerical experiments for both two-dimensional and three-dimensional models are carried out to confirm our theoretical findings.

Li , MinLi , DongfangMing , Ju and Hendy , A. S.. (2025). Unconditional Convergence of Linearized TL1 Difference Methods for a Time-Fractional Coupled Nonlinear Schrödinger System. Numerical Mathematics: Theory, Methods and Applications. 18 (1). 1-30. doi:10.4208/nmtma.OA-2024-0095
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