Finite element simulation and efficient algorithm for fractional Cahn–Hilliard equation

Wang, Feng; Chen, Huanzhen; Wang, Hong

doi:10.1016/j.cam.2019.01.037

Cited by 28 publications

(19 citation statements)

References 25 publications

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“…The aim for this section is to reduce the storage and calculation to O(M) and O(M log M), respectively. For this, we shall combine the stabilized bi-conjugate gradient algorithm (SBiCG) with the Toeplitz structure of the coefficient matrices to construct the fast stabilized bi-conjugate gradient algorithm (FSBiCG) [28]. This needs the following three steps:…”

Section: C)mentioning

confidence: 99%

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Fourth-order finite difference scheme and efficient algorithm for nonlinear fractional Schrödinger equations

Chang

Chen

2020

Adv Differ Equ

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To improve the computing efficiency, a fourth-order difference scheme is proposed and a fast algorithm is designed to simulate the nonlinear fractional Schrödinger (FNLS) equation oriented from the fractional quantum mechanics. The numerical analysis and experiments conducted in this article show that the proposed difference scheme has the optimal second-order and fourth-order convergence rates in time and space respectively, reduces its computation cost to O(M log M), and recognizes accurately its physical feature of FNLS such as the mass balance.

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Section: C)mentioning

confidence: 99%

“…It has been shown in [28] that the decomposition of circulant matrix C could be carried out within a computational cost of O(M log M).…”

Section: C)mentioning

confidence: 99%

Fourth-order finite difference scheme and efficient algorithm for nonlinear fractional Schrödinger equations

Chang

Chen

2020

Adv Differ Equ

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“…Numerical methods for time-fractional IBVPs with constant or time-independent diffusion parameter have received a huge amount of attention over the last decade. For such problems, several numerical methods have been proposed and analyzed, such as finite difference method [7, 19-21, 27, 36, 38], finite element method [6,32,39,41,43,44,48], discontinuous Galerkin (DG) methods [3, 4, 9-11, 31, 34], spectral method [23], and finite volume method [15,46], etc. The time-fractional IBVPs (1a)-(1b) with time-space dependent diffusivity is indeed very interesting and also practically important, and the numerical solutions of this problems were considered by a few authors only.…”

Section: Introductionmentioning

confidence: 99%

Superconvergence of a finite element method for the time-fractional diffusion equation with a time-space dependent diffusivity

2020

Adv Differ Equ

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In this work, a time-fractional diffusion problem with a time-space dependent diffusivity is considered. The solution of such a problem has a weak singularity at the initial time $t=0$ t = 0 . Based on the L1 scheme in time on a graded mesh and the conforming finite element method in space on a uniform mesh, the fully discrete L1 conforming finite element method (L1 FEM) of a time-fractional diffusion problem is investigated. The error analysis is based on a nonstandard discrete Gronwall inequality. The final superconvergence result shows that an optimal grading of the temporal mesh should be selected as $r\geq (2-\alpha )/\alpha $ r ≥ ( 2 − α ) / α . Numerical results confirm that our analysis is sharp.

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“…This makes the numerical methods for solving these problems be necessary and significant. In [9][10][11][12][13][14][15][16][17], numerical methods for some nonlinear second-order/fourth-order equations are developed. As well as, various numerical methods have been developed for the linear cases of (1.1)-(1.4).…”

Section: Introductionmentioning

confidence: 99%

Compact difference scheme for two‐dimensional fourth‐order nonlinear hyperbolic equation

Yang

Chen

2020

Numerical Methods Partial

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High-order compact finite difference method for solving the two-dimensional fourth-order nonlinear hyperbolic equation is considered in this article. In order to design an implicit compact finite difference scheme, the fourth-order equation is written as a system of two second-order equations by introducing the second-order spatial derivative as a new variable. The second-order spatial derivatives are approximated by the compact finite difference operators to obtain a fourth-order convergence. As well as, the second-order time derivative is approximated by the central difference method. Then, existence and uniqueness of numerical solution is given. The stability and convergence of the compact finite difference scheme are proved by the energy method. Numerical results are provided to verify the accuracy and efficiency of this scheme.

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Finite element simulation and efficient algorithm for fractional Cahn–Hilliard equation

Cited by 28 publications

References 25 publications

Fourth-order finite difference scheme and efficient algorithm for nonlinear fractional Schrödinger equations

Fourth-order finite difference scheme and efficient algorithm for nonlinear fractional Schrödinger equations

Superconvergence of a finite element method for the time-fractional diffusion equation with a time-space dependent diffusivity

Compact difference scheme for two‐dimensional fourth‐order nonlinear hyperbolic equation

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