research-article

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

Authors:

Zhifeng WengAuthors Info & Claims

Numerical Algorithms, Volume 84, Issue 3

Pages 1155 - 1178

https://doi.org/10.1007/s11075-019-00795-7

Published: 01 July 2020 Publication History

Abstract

In this paper, we consider a fast explicit operator splitting method for a fractional Cahn-Hilliard equation with spatial derivative

{(- Δ)}^{\frac{α}{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

{(- Δ)}^{\frac{α}{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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Cited By

Huang XLi DSun HZhang F(2022)Preconditioners with Symmetrized Techniques for Space Fractional Cahn-Hilliard EquationsJournal of Scientific Computing10.1007/s10915-022-01900-092:2Online publication date: 28-Jun-2022
https://dl.acm.org/doi/10.1007/s10915-022-01900-0
Zhai SWeng ZFeng XHe Y(2021)Stability and Error Estimate of the Operator Splitting Method for the Phase Field Crystal EquationJournal of Scientific Computing10.1007/s10915-020-01386-886:1Online publication date: 1-Jan-2021
https://dl.acm.org/doi/10.1007/s10915-020-01386-8
Zhao YLi MOstermann AGu X(2021)An efficient second-order energy stable BDF scheme for the space fractional Cahn–Hilliard equationBIT10.1007/s10543-021-00843-661:3(1061-1092)Online publication date: 1-Sep-2021
https://dl.acm.org/doi/10.1007/s10543-021-00843-6

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

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Published In

cover image Numerical Algorithms

Numerical Algorithms Volume 84, Issue 3

Jul 2020

408 pages

ISSN:1017-1398

Issue’s Table of Contents

© Springer Science+Business Media, LLC, part of Springer Nature 2019.

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 July 2020

Accepted: 08 August 2019

Received: 08 January 2019

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Cited By

Huang XLi DSun HZhang F(2022)Preconditioners with Symmetrized Techniques for Space Fractional Cahn-Hilliard EquationsJournal of Scientific Computing10.1007/s10915-022-01900-092:2Online publication date: 28-Jun-2022
https://dl.acm.org/doi/10.1007/s10915-022-01900-0
Zhai SWeng ZFeng XHe Y(2021)Stability and Error Estimate of the Operator Splitting Method for the Phase Field Crystal EquationJournal of Scientific Computing10.1007/s10915-020-01386-886:1Online publication date: 1-Jan-2021
https://dl.acm.org/doi/10.1007/s10915-020-01386-8
Zhao YLi MOstermann AGu X(2021)An efficient second-order energy stable BDF scheme for the space fractional Cahn–Hilliard equationBIT10.1007/s10543-021-00843-661:3(1061-1092)Online publication date: 1-Sep-2021
https://dl.acm.org/doi/10.1007/s10543-021-00843-6

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