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Numerical approximation of the fractional Cahn-Hilliard equation by operator splitting method

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Abstract

In this paper, we consider a fast explicit operator splitting method for a fractional Cahn-Hilliard equation with spatial derivative \((-{\varDelta })^{\frac {\alpha }{2}}\)(α ∈ (1,2]), where the choice α = 2 corresponds to the classical Cahn-Hilliard equation. The original problem is split into linear and nonlinear subproblems. For the linear part, the pseudo-spectral method is adopted, and thus an ordinary differential equation is obtained. For the nonlinear part, a second-order SSP-RK method together with the pseudo-spectral method is used. The stability and convergence of the proposed method in L2-norm are studied. We also carry out a comparative study of two classical definitions for fractional Laplacian \((-{\varDelta })^{\frac {\alpha }{2}}\), and numerical results obtained using computational simulation of the fractional Cahn-Hilliard equation for a variety of choices of fractional order α are presented. It is observed that the fractional order α controls the sharpness of the interface, which is typically diffusive in integer-order phase-field models.

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Acknowledgments

The authors thank the editor and referees for their constructive comments and useful suggestions which improved greatly the quality of our paper.

Funding

This work is in part supported by the NSF of China (nos. 11701196 and 11701197), the Promotion Program for Young and Middle-Aged Teacher in Science and Technology Research of Huaqiao University (no. ZQN-YX502), and the Fundamental Research Funds for the Central Universities (No. ZQN-702).

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  1. Fujian Province University Key Laboratory of Computation Science, Huaqiao University, Quanzhou, 362021, People’s Republic of China

    Shuying Zhai, Longyuan Wu, Jingying Wang & Zhifeng Weng

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  1. Shuying Zhai
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Correspondence to Zhifeng Weng.

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Zhai, S., Wu, L., Wang, J. et al. Numerical approximation of the fractional Cahn-Hilliard equation by operator splitting method. Numer Algor 84, 1155–1178 (2020). https://doi.org/10.1007/s11075-019-00795-7

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