A spatial fourth-order maximum principle preserving operator splitting scheme for the multi-dimensional fractional Allen-Cahn equation

He, Dongdong; Pan, Kejia; Hu, Hongling

doi:10.1016/j.apnum.2019.12.018

Cited by 36 publications

(13 citation statements)

References 58 publications

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“…Several numerical techniques have been recently developed for space and time non-local versions of equation (1.3), most of them based on finite differences or spectral methods [28,21,35,22,27,7]. Also, numerical methods have been studied for nonlocal versions of related phase separation models, like the Cahn-Hilliard equation [5,6].…”

Section: Introductionmentioning

confidence: 99%

Numerical approximations for a fully fractional Allen–Cahn equation

Acosta

Bersetche

2021

ESAIM: M2AN

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A finite element scheme for an entirely fractional Allen-Cahn equation is introduced and analyzed. In the proposed nonlocal model, the Caputo fractional in-time derivative and the fractional Laplacian replace the standard local operators. Piecewise linear finite elements and convolution quadratures are the basic tools involved in the presented numerical method. Error analysis and implementation issues are addressed together with the needed results of regularity for the continuous model. Also, the asymptotic behavior of solutions, for a vanishing diffusion parameter, is analyzed within the framework of the Γ-convergence theory.On the other hand, even though (1.1) makes perfect sense for n = 1, lateral versions are necessary for dealing with the time variable. In such a case, the so-called 2010 Mathematics Subject Classification. 65R20, 65M60, 35R11.

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Section: Introductionmentioning

confidence: 99%

Numerical approximations for a fully fractional Allen–Cahn equation

Acosta

Bersetche

2021

ESAIM: M2AN

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“…Under the assumption of the homogenous boundary condition (15), the second-order difference scheme for space fractional derivatives ( 14) is given below…”

Section: A Three-level Linearized Implicit Difference Schemementioning

confidence: 99%

A conservative difference scheme with optimal pointwise error estimates for two‐dimensional space fractional nonlinear Schrödinger equations

Jin

et al. 2021

Numerical Methods Partial

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In this paper, a linearized semi-implicit finite difference scheme is proposed for solving the two-dimensional (2D) space fractional nonlinear Schrödinger equation (SFNSE). The scheme has the property of mass and energy conservation at the discrete level, with an unconditional stability and a second-order accuracy for both time and spatial variables. The main contribution of this paper is an optimal pointwise error estimate for the 2D SFNSE, which is rigorously established for the first time. Moreover, a novel technique is proposed for dealing with the nonlinear term in the equation, which plays an essential role in the error estimation. Finally, the numerical results confirm well with the theoretical findings.

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“…For the computational problems, we consider the interesting part of higher dimensional and nonlinear PDEs as nonlinear and fractional convection-diffusion equations, see [13,14]. The nonlinear PDEs are applied in nonlinear flow problems, for example, to simulate traffic-flow problems, see [15] and flow problems that are related to Navier-Stokes equations, see [16], which can be modeled with the Burgers' equation.…”

Section: Introductionmentioning

confidence: 99%

“…• α = β = 2: viscous Burgers' equation, see [2,26], • α = β = 2 and ν = 0: Diffusion equation, see [2,26], and • α, β ∈ (1, 2): fractional diffusion equation, see [14,27].…”

Section: Introductionmentioning

confidence: 99%

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Serial and Parallel Iterative Splitting Methods: Algorithms and Applications to Fractional Convection-Diffusion Equations

2020

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The benefits and properties of iterative splitting methods, which are based on serial versions, have been studied in recent years, this work, we extend the iterative splitting methods to novel classes of parallel versions to solve nonlinear fractional convection-diffusion equations. For such interesting partial differential examples with higher dimensional, fractional, and nonlinear terms, we could apply the parallel iterative splitting methods, which allow for accelerating the solver methods and reduce the computational time. Here, we could apply the benefits of the higher accuracy of the iterative splitting methods. We present a novel parallel iterative splitting method, which is based on the multi-splitting methods, The flexibilisation with multisplitting methods allows for decomposing large scale operator equations. In combination with iterative splitting methods, which use characteristics of waveform-relaxation (WR) methods, we could embed the relaxation behavior and deal better with the nonlinearities of the operators. We consider the convergence results of the parallel iterative splitting methods, reformulating the underlying methods with a summation of the individual convergence results of the WR methods. We discuss the numerical convergence of the serial and parallel iterative splitting methods with respect to the synchronous and asynchronous treatments. Furthermore, we present different numerical applications of fluid and phase field problems in order to validate the benefit of the parallel versions.

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A spatial fourth-order maximum principle preserving operator splitting scheme for the multi-dimensional fractional Allen-Cahn equation

Cited by 36 publications

References 58 publications

Numerical approximations for a fully fractional Allen–Cahn equation

Numerical approximations for a fully fractional Allen–Cahn equation

A conservative difference scheme with optimal pointwise error estimates for two‐dimensional space fractional nonlinear Schrödinger equations

Serial and Parallel Iterative Splitting Methods: Algorithms and Applications to Fractional Convection-Diffusion Equations

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