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Volume 16, Issue 5
A Linear Doubly Stabilized Crank-Nicolson Scheme for the Allen–Cahn Equation with a General Mobility

Dianming Hou, Lili Ju & Zhonghua Qiao

Adv. Appl. Math. Mech., 16 (2024), pp. 1009-1038.

Published online: 2024-07

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  • Abstract

In this paper, a linear second order numerical scheme is developed and investigated for the Allen–Cahn equation with a general positive mobility. In particular, our fully discrete scheme is mainly constructed based on the Crank-Nicolson formula for temporal discretization and the central finite difference method for spatial approximation, and two extra stabilizing terms are also introduced for the purpose of improving numerical stability. The proposed scheme is shown to unconditionally preserve the maximum bound principle (MBP) under mild restrictions on the stabilization parameters, which is of practical importance for achieving good accuracy and stability simultaneously. With the help of uniform boundedness of the numerical solutions due to MBP, we then successfully derive $H^1$-norm and $L^∞$-norm error estimates for the Allen–Cahn equation with a constant and a variable mobility, respectively. Moreover, the energy stability of the proposed scheme is also obtained in the sense that the discrete free energy is uniformly bounded by the one at the initial time plus a constant. Finally, some numerical experiments are carried out to verify the theoretical results and illustrate the performance of the proposed scheme with a time adaptive strategy.

  • AMS Subject Headings

65M06, 65M15, 41A05, 41A25

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COPYRIGHT: © Global Science Press

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@Article{AAMM-16-1009, author = {Hou , DianmingJu , Lili and Qiao , Zhonghua}, title = {A Linear Doubly Stabilized Crank-Nicolson Scheme for the Allen–Cahn Equation with a General Mobility}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2024}, volume = {16}, number = {5}, pages = {1009--1038}, abstract = {

In this paper, a linear second order numerical scheme is developed and investigated for the Allen–Cahn equation with a general positive mobility. In particular, our fully discrete scheme is mainly constructed based on the Crank-Nicolson formula for temporal discretization and the central finite difference method for spatial approximation, and two extra stabilizing terms are also introduced for the purpose of improving numerical stability. The proposed scheme is shown to unconditionally preserve the maximum bound principle (MBP) under mild restrictions on the stabilization parameters, which is of practical importance for achieving good accuracy and stability simultaneously. With the help of uniform boundedness of the numerical solutions due to MBP, we then successfully derive $H^1$-norm and $L^∞$-norm error estimates for the Allen–Cahn equation with a constant and a variable mobility, respectively. Moreover, the energy stability of the proposed scheme is also obtained in the sense that the discrete free energy is uniformly bounded by the one at the initial time plus a constant. Finally, some numerical experiments are carried out to verify the theoretical results and illustrate the performance of the proposed scheme with a time adaptive strategy.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2023-0067}, url = {http://global-sci.org/intro/article_detail/aamm/23284.html} }
TY - JOUR T1 - A Linear Doubly Stabilized Crank-Nicolson Scheme for the Allen–Cahn Equation with a General Mobility AU - Hou , Dianming AU - Ju , Lili AU - Qiao , Zhonghua JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1009 EP - 1038 PY - 2024 DA - 2024/07 SN - 16 DO - http://doi.org/10.4208/aamm.OA-2023-0067 UR - https://global-sci.org/intro/article_detail/aamm/23284.html KW - Allen–Cahn equation, general mobility, linear scheme, Crank-Nicolson. AB -

In this paper, a linear second order numerical scheme is developed and investigated for the Allen–Cahn equation with a general positive mobility. In particular, our fully discrete scheme is mainly constructed based on the Crank-Nicolson formula for temporal discretization and the central finite difference method for spatial approximation, and two extra stabilizing terms are also introduced for the purpose of improving numerical stability. The proposed scheme is shown to unconditionally preserve the maximum bound principle (MBP) under mild restrictions on the stabilization parameters, which is of practical importance for achieving good accuracy and stability simultaneously. With the help of uniform boundedness of the numerical solutions due to MBP, we then successfully derive $H^1$-norm and $L^∞$-norm error estimates for the Allen–Cahn equation with a constant and a variable mobility, respectively. Moreover, the energy stability of the proposed scheme is also obtained in the sense that the discrete free energy is uniformly bounded by the one at the initial time plus a constant. Finally, some numerical experiments are carried out to verify the theoretical results and illustrate the performance of the proposed scheme with a time adaptive strategy.

Dianming Hou, Lili Ju & Zhonghua Qiao. (2024). A Linear Doubly Stabilized Crank-Nicolson Scheme for the Allen–Cahn Equation with a General Mobility. Advances in Applied Mathematics and Mechanics. 16 (5). 1009-1038. doi:10.4208/aamm.OA-2023-0067
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