An efficient second order stabilized scheme for the two dimensional time fractional Allen-Cahn equation

Jia, Junqing; Zhang, Hui; Xu, Huanying; Jiang, Xiaoyun

doi:10.1016/j.apnum.2021.02.016

Cited by 14 publications

(4 citation statements)

References 33 publications

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“…Next, we study the stability of scheme (18) using the numerical flux (16). The case of choosing numerical flux (17) is almost the same, so is omitted here. Firstly, we state a discrete fractional Gronwall inequality and a property of the nonuniform L1 scheme.…”

Section: Lemma 3 ([27]mentioning

confidence: 99%

“…In [16], Hou et al constructed a first-order scheme and a (2 − α)thorder scheme for the time-fractional Allen-Cahn equation. In [17], Jiang et al considered the Legendre spectral method for the time-fractional Allen-Cahn equation. In a series of works [18][19][20], Liao et al proposed several efficient finite difference schemes to solve the time-fractional phase-field type models.…”

Section: Introductionmentioning

confidence: 99%

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Local Discontinuous Galerkin Method Coupled with Nonuniform Time Discretizations for Solving the Time-Fractional Allen-Cahn Equation

2022

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This paper aims to numerically study the time-fractional Allen-Cahn equation, where the time-fractional derivative is in the sense of Caputo with order α∈(0,1). Considering the weak singularity of the solution u(x,t) at the starting time, i.e., its first and/or second derivatives with respect to time blowing-up as t→0+ albeit the function itself being right continuous at t=0, two well-known difference formulas, including the nonuniform L1 formula and the nonuniform L2-1σ formula, which are used to approximate the Caputo time-fractional derivative, respectively, and the local discontinuous Galerkin (LDG) method is applied to discretize the spatial derivative. With the help of discrete fractional Gronwall-type inequalities, the stability and optimal error estimates of the fully discrete numerical schemes are demonstrated. Numerical experiments are presented to validate the theoretical results.

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Section: Lemma 3 ([27]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Local Discontinuous Galerkin Method Coupled with Nonuniform Time Discretizations for Solving the Time-Fractional Allen-Cahn Equation

2022

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“…In the discrete case, it is hence necessary to develop numerical schemes that preserve the energy decay and maximum-principle for model (3). Historically, there has been enormous numerical exploration for tFAC equations or phase field models or more generally, the nonlinear subdiffustion models, where some energy stable schemes were developed (see, e.g., [16,15,14,25,24,19,5,21,29,44,33,23,9]). However, the studies of preservation on discrete energy dissipation, or energy decay and maximum-principle are still limited.…”

Section: Introductionmentioning

confidence: 99%

Preface: Celebrating the 70th Anniversary of School of Mechanical Science and Engineering, Huazhong University of Science and Technology

Huang

Yin

2022

Sci. China Technol. Sci.

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The shifted fractional trapezoidal rule (SFTR) with a special shift is adopted to construct a finite difference scheme for the time-fractional Allen-Cahn (tFAC) equation. Some essential key properties of the weights of SFTR are explored for the first time. Based on these properties, we rigorously demonstrate the discrete energy decay property and maximum-principle preservation for the scheme. Numerical investigations show that the scheme can resolve the intrinsic initial singularity of such nonlinear fractional equations as tFAC equation on uniform meshes without any correction. Comparison with the classic fractional BDF2 and L2-1 σ method further validates the superiority of SFTR in solving the tFAC equation. Experiments concerning both discrete energy decay and discrete maximum-principle also verify the correctness of the theoretical results.

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“…Meanwhile, the unique solvability and energy stability of the numerical schemes were proved. The numerical methods for time-fractional Allen-Cahn equations were also studied in [3,18,22]. The nonlocal Allen-Cahn equation is similar to the space-fractional Allen-Cahn equation.…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis of a linear second-order finite difference scheme for space-fractional Allen–Cahn equations

et al. 2022

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In this paper, we construct a new linear second-order finite difference scheme with two parameters for space-fractional Allen–Cahn equations. We first prove that the discrete maximum principle holds under reasonable constraints on time step size and coefficient of stabilized term. Secondly, we analyze the maximum-norm error. Thirdly, we can see that the proposed scheme is unconditionally energy-stable by defining the modified energy and selecting the appropriate parameters. Finally, two numerical examples are presented to verify the theoretical results.

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An efficient second order stabilized scheme for the two dimensional time fractional Allen-Cahn equation

Cited by 14 publications

References 33 publications

Local Discontinuous Galerkin Method Coupled with Nonuniform Time Discretizations for Solving the Time-Fractional Allen-Cahn Equation

Local Discontinuous Galerkin Method Coupled with Nonuniform Time Discretizations for Solving the Time-Fractional Allen-Cahn Equation

Preface: Celebrating the 70th Anniversary of School of Mechanical Science and Engineering, Huazhong University of Science and Technology

Numerical analysis of a linear second-order finite difference scheme for space-fractional Allen–Cahn equations

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