A Fourier spectral method for fractional-in-space Cahn–Hilliard equation

Weng, Zhifeng; Zhai, Shuying; Feng, Xinlong

doi:10.1016/j.apm.2016.10.035

Cited by 67 publications

(31 citation statements)

References 28 publications

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“…To apply the Fourier spectral method on the 3CNFSE, we follow the procedure given by Weng et al [17], and we transform ψ(x, t) into the discrete Fourier spacê…”

Section: The Fourier Spectral Methodsmentioning

confidence: 99%

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Exponential time differencing schemes for the 3-coupled nonlinear fractional Schrödinger equation

Liang

Bhatt

2018

Adv Differ Equ

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Two modified exponential time differencing schemes based on the Fourier spectral method are developed to solve the 3-coupled nonlinear fractional Schrödinger equation. We compare the stability of the schemes by plotting their stability regions. The local truncation errors of the time integrators are proved to be fourth-order. Numerical experiments illustrating the solution to the equations with various parameters and the mass conservation results of the numerical methods are carried out.

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“…To apply the Fourier spectral method on the 3CNFSE, we follow the procedure given by Weng et al [17], and we transform ψ(x, t) into the discrete Fourier spacê…”

Section: The Fourier Spectral Methodsmentioning

confidence: 99%

“…We assume that a fixed point u 0 satisfying cu 0 + F(u 0 ) = 0 exists, and u is the perturbation of u 0 . We denote λ = F (u 0 ), and then we lin- earize equation (17) to obtain…”

Section: Stability Regionsmentioning

confidence: 99%

Exponential time differencing schemes for the 3-coupled nonlinear fractional Schrödinger equation

Liang

Bhatt

2018

Adv Differ Equ

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“…However, compared to the large amount of studies in numerical methods for the AC equation and the CH equation [20][21][22][23][24][25][26], limited algorithms for the low-dimensional VCH equation has been investigated [27,28]. Until recently, Shin et al [29] established a nonlinearly stabilized scheme for the three dimensional VCH equation.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the operator splitting scheme for the Cahn‐Hilliard equation with a viscosity term

Weng

Zhai

Feng

2019

Numerical Methods Partial

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In this paper, we consider a second-order fast explicit operator splitting method for the viscous Cahn-Hilliard equation, which includes a viscosity term Δu t ( ∈ (0, 1)) described the influences of internal micro-forces. The choice = 0 corresponds to the classical Cahn-Hilliard equation whilst the choice = 1 recovers the nonlocal Allen-Cahn equation. The fundamental idea of our method is to split the original problem into linear and nonlinear parts. The linear subproblem is numerically solved using a pseudo-spectral method, and thus an ordinary differential equation is obtained. The nonlinear one is solved via TVD-RK method. The stability and convergence are discussed in L 2 -norm. Numerical experiments are performed to validate the accuracy and efficiency of the proposed method. Besides, a detailed comparison is made for the dynamics and the coarsening process of the metastable pattern for various values of . Moreover, energy degradation and mass conservation are also verified. KEYWORDSfast explicit operator splitting method, pseudo-spectral method, stability and convergence, TVD-RK method, viscous Cahn-Hilliard equation

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“…Over the past 30 years, Fourier spectral method has been actively applied to solve integer partial differential equations. [40] An unconditionally energy stable Fourier spectral scheme for the fractional equation with periodic or Neumann boundary conditions was developed in [41]. Recently, Zhang and Jiang [42] gave a Crank-Nicolson Fourier spectral method to solve the space fractional nonlinear Schrödinger equations.…”

Section: Introductionmentioning

confidence: 99%

A Fourier spectral method for the nonlinear coupled space fractional Klein‐Gordon‐Schrödinger equations

et al. 2019

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In this paper, we give an efficient numerical method for the nonlinear coupled space fractional Klein-Gordon-Schrödinger (NCSFKGS) equations, based on the Crank-Nicolson method, the central difference method and the Fourier spectral method. As far as we know, no one has studied the Equations (5)-(6) in our paper, these equations are different from those in [39] which only considers space fractional Schrödinger equation while the Klein-Gordon equation is classical, here, we consider the two equations which are both space fractional. In this paper, the Crank-Nicolson method and the central difference method are used to discretize the space fractional Schrödinger equation and the space fractional Klein-Gordon equation in time direction, respectively. The Fourier spectral method is used to discretize the NCSFKGS equations in space direction. This numerical method conserves the mass and energy in the discrete level. The convergence of the numerical method is of second order accuracy in time and spectral accuracy in space. Rigorous proofs are developed here for the conservation laws and the convergence of the numerical method. Numerical experiments are presented to confirm the theoretical results and they proved that our numerical method is very efficient (it only takes a few seconds to get the numerical solutions). K E Y W O R D Sconvergence analysis, fast Fourier transform, Fourier spectral method, mass and energy conservation laws, nonlinear coupled space fractional Klein-Gordon-Schrödinger equations

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A Fourier spectral method for fractional-in-space Cahn–Hilliard equation

Cited by 67 publications

References 28 publications

Exponential time differencing schemes for the 3-coupled nonlinear fractional Schrödinger equation

Exponential time differencing schemes for the 3-coupled nonlinear fractional Schrödinger equation

Analysis of the operator splitting scheme for the Cahn‐Hilliard equation with a viscosity term

A Fourier spectral method for the nonlinear coupled space fractional Klein‐Gordon‐Schrödinger equations

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