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Volume 15, Issue 4
The Variable-Step L1 Scheme Preserving a Compatible Energy Law for Time-Fractional Allen-Cahn Equation

Hong-Lin Liao, Xiaohan Zhu & Jindi Wang

Numer. Math. Theor. Meth. Appl., 15 (2022), pp. 1128-1146.

Published online: 2022-10

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  • Abstract

In this work, we revisit the adaptive L1 time-stepping scheme for solving the time-fractional Allen-Cahn equation in the Caputo’s form. The L1 implicit scheme is shown to preserve a variational energy dissipation law on arbitrary nonuniform time meshes by using the recent discrete analysis tools, i.e., the discrete orthogonal convolution kernels and discrete complementary convolution kernels. Then the discrete embedding techniques and the fractional Grönwall inequality are applied to establish an $L^2$ norm error estimate on nonuniform time meshes. An adaptive time-stepping strategy according to the dynamical feature of the system is presented to capture the multi-scale behaviors and to improve the computational performance.

  • AMS Subject Headings

35Q99, 65M06, 65M12, 74A50

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COPYRIGHT: © Global Science Press

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@Article{NMTMA-15-1128, author = {Liao , Hong-LinZhu , Xiaohan and Wang , Jindi}, title = {The Variable-Step L1 Scheme Preserving a Compatible Energy Law for Time-Fractional Allen-Cahn Equation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2022}, volume = {15}, number = {4}, pages = {1128--1146}, abstract = {

In this work, we revisit the adaptive L1 time-stepping scheme for solving the time-fractional Allen-Cahn equation in the Caputo’s form. The L1 implicit scheme is shown to preserve a variational energy dissipation law on arbitrary nonuniform time meshes by using the recent discrete analysis tools, i.e., the discrete orthogonal convolution kernels and discrete complementary convolution kernels. Then the discrete embedding techniques and the fractional Grönwall inequality are applied to establish an $L^2$ norm error estimate on nonuniform time meshes. An adaptive time-stepping strategy according to the dynamical feature of the system is presented to capture the multi-scale behaviors and to improve the computational performance.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2022-0011s}, url = {http://global-sci.org/intro/article_detail/nmtma/21096.html} }
TY - JOUR T1 - The Variable-Step L1 Scheme Preserving a Compatible Energy Law for Time-Fractional Allen-Cahn Equation AU - Liao , Hong-Lin AU - Zhu , Xiaohan AU - Wang , Jindi JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 1128 EP - 1146 PY - 2022 DA - 2022/10 SN - 15 DO - http://doi.org/10.4208/nmtma.OA-2022-0011s UR - https://global-sci.org/intro/article_detail/nmtma/21096.html KW - Time-fractional Allen-Cahn equation, adaptive L1 scheme, variational energy dissipation law, orthogonal convolution kernels, complementary convolution kernels. AB -

In this work, we revisit the adaptive L1 time-stepping scheme for solving the time-fractional Allen-Cahn equation in the Caputo’s form. The L1 implicit scheme is shown to preserve a variational energy dissipation law on arbitrary nonuniform time meshes by using the recent discrete analysis tools, i.e., the discrete orthogonal convolution kernels and discrete complementary convolution kernels. Then the discrete embedding techniques and the fractional Grönwall inequality are applied to establish an $L^2$ norm error estimate on nonuniform time meshes. An adaptive time-stepping strategy according to the dynamical feature of the system is presented to capture the multi-scale behaviors and to improve the computational performance.

Hong-Lin Liao, Xiaohan Zhu & Jindi Wang. (2022). The Variable-Step L1 Scheme Preserving a Compatible Energy Law for Time-Fractional Allen-Cahn Equation. Numerical Mathematics: Theory, Methods and Applications. 15 (4). 1128-1146. doi:10.4208/nmtma.OA-2022-0011s
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