Numerical approximation of the fractional Cahn-Hilliard equation by operator splitting method
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Numerical approximation of the fractional Cahn-Hilliard equation by operator splitting method Shuying Zhai1 · Longyuan Wu1 · Jingying Wang1 · Zhifeng Weng1 Received: 8 January 2019 / Accepted: 8 August 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this paper, we consider a fast explicit operator splitting method for a fractional α Cahn-Hilliard equation with spatial derivative (−Δ) 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 α Laplacian (−Δ) 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. Keywords Fractional-in-space Cahn-Hilliard equation · Operator splitting method · Pseudo-spectral method · SSP-RK method · Stability and convergence Mathematics Subject Classification (2010) 35R11 · 65M70 · 65M06 · 65M12
Zhifeng Weng
[email protected] Shuying Zhai [email protected] Longyuan Wu [email protected] Jingying Wang [email protected] 1
Fujian Province University Key Laboratory of Computation Science, Huaqiao University, Quanzhou 362021, People’s Republic of China
Numerical Algorithms
1 Introduction The Cahn-Hillard (CH) equation was proposed in 1958 to describe the complicated phase separation and coarsening phenomena in a solid [1, 2]. Due to its connection with many physically motivated problems such as phase separation and pattern formation, this equation has been used extensively in materials science [3], image processing [4], and so on. Since the CH equation is stiff, fourth-order, nonlinear, and possesses multiple time and space scales, it is difficult to solve accurately. Thus, research on the CH equation has drawn extensive attention of many researchers, and many numerical methods for this equation have been studied, such as finite difference methods [5–7], finite element methods [8–10], spectral methods [11–13], and finite volume methods [14]. A large number of experiments show that nonhomogeneities of the medium lead to long-range fluxes and non-Gaussian heavy-tailed densities; these motions no longer obey the classical Fick’s law and thus cannot be modeled properly by integer-order diffusion equations. This phenomenon is called anomalous diffusion. Recent studies show that fractional diffusion equations provide an adequate and accurate description of these anomalous diffusion transport processes. With the applicat
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