Efficient Variable Steps BDF2 Scheme for the Two-Dimensional Space Fractional Cahn-Hilliard Model

doi:10.1007/s42967-023-00350-1

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (4): 1489-1515.doi: 10.1007/s42967-023-00350-1

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Efficient Variable Steps BDF2 Scheme for the Two-Dimensional Space Fractional Cahn-Hilliard Model

Xuan Zhao, Zhongqin Xue

School of Mathematics, Southeast University, Nanjing, 210096, Jiangsu, China

收稿日期:2023-07-26 修回日期:2023-10-29 接受日期:2023-11-03 出版日期:2024-01-22 发布日期:2024-01-22
通讯作者: Xuan Zhao,E-mail:xuanzhao11@seu.edu.cn E-mail:xuanzhao11@seu.edu.cn
作者简介:Zhongqin Xue, E-mail:zqxue@seu.edu.cn
基金资助:
The authors appreciate the anonymous reviewers for their valuable comments and suggestions which improved the quality of this paper. The authors would like to acknowledge support by the National Natural Science Foundation of China (Nos. 11701081, 11861060), the State Key Program of National Natural Science Foundation of China (No. 61833005), and ZhiShan Youth Scholar Program of SEU.

Efficient Variable Steps BDF2 Scheme for the Two-Dimensional Space Fractional Cahn-Hilliard Model

Xuan Zhao, Zhongqin Xue

School of Mathematics, Southeast University, Nanjing, 210096, Jiangsu, China

Received:2023-07-26 Revised:2023-10-29 Accepted:2023-11-03 Online:2024-01-22 Published:2024-01-22
Supported by:
The authors appreciate the anonymous reviewers for their valuable comments and suggestions which improved the quality of this paper. The authors would like to acknowledge support by the National Natural Science Foundation of China (Nos. 11701081, 11861060), the State Key Program of National Natural Science Foundation of China (No. 61833005), and ZhiShan Youth Scholar Program of SEU.

摘要/Abstract

摘要： An implicit variable-step BDF2 scheme is established for solving the space fractional Cahn-Hilliard equation derived from a gradient flow in the negative order Sobolev space H^-α, α∈ (0, 1). The Fourier pseudo-spectral method is applied for the spatial approximation. The space fractional Cahn-Hilliard model poses significant challenges in theoretical analysis for variable time-stepping algorithms compared to the classical model, primarily due to the introduction of the fractional Laplacian. This issue is settled by developing a general discrete Hölder inequality involving the discretization of the fractional Laplacian. Subsequently, the unique solvability and the modified energy dissipation law are theoretically guaranteed. We further rigorously provided the convergence of the fully discrete scheme by utilizing the newly proved discrete Young-type convolution inequality to deal with the nonlinear term. Numerical examples with various interface widths and mobility are conducted to show the accuracy and the energy decay for different orders of the fractional Laplacian. In particular, we demonstrate that the adaptive time-stepping strategy, compared with the uniform time steps, captures the multiple time scale evolutions of the solution in simulations.

关键词: Space fractional Cahn-Hilliard equation, Variable-step BDF2, Modified discrete energy, Convergence, Adaptive time-stepping

Abstract: An implicit variable-step BDF2 scheme is established for solving the space fractional Cahn-Hilliard equation derived from a gradient flow in the negative order Sobolev space H^-α, α∈ (0, 1). The Fourier pseudo-spectral method is applied for the spatial approximation. The space fractional Cahn-Hilliard model poses significant challenges in theoretical analysis for variable time-stepping algorithms compared to the classical model, primarily due to the introduction of the fractional Laplacian. This issue is settled by developing a general discrete Hölder inequality involving the discretization of the fractional Laplacian. Subsequently, the unique solvability and the modified energy dissipation law are theoretically guaranteed. We further rigorously provided the convergence of the fully discrete scheme by utilizing the newly proved discrete Young-type convolution inequality to deal with the nonlinear term. Numerical examples with various interface widths and mobility are conducted to show the accuracy and the energy decay for different orders of the fractional Laplacian. In particular, we demonstrate that the adaptive time-stepping strategy, compared with the uniform time steps, captures the multiple time scale evolutions of the solution in simulations.

Key words: Space fractional Cahn-Hilliard equation, Variable-step BDF2, Modified discrete energy, Convergence, Adaptive time-stepping

中图分类号:

Xuan Zhao, Zhongqin Xue. Efficient Variable Steps BDF2 Scheme for the Two-Dimensional Space Fractional Cahn-Hilliard Model[J]. Communications on Applied Mathematics and Computation, 2025, 7(4): 1489-1515.

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Efficient Variable Steps BDF2 Scheme for the Two-Dimensional Space Fractional Cahn-Hilliard Model

Efficient Variable Steps BDF2 Scheme for the Two-Dimensional Space Fractional Cahn-Hilliard Model

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