Time-fractional Allen-Cahn Equation Research Articles

Long-term dynamics of a stabilized time-space discretization scheme for 2D time-fractional Allen–Cahn equation with double well potential

This paper deals with the long-term discrete dynamics of 2D time-fractional Allen–Cahn equation (TFACE) with double well potential. A stabilized time-space discretization scheme is proposed to solve the equation. Under no constraint of the stepsizes, the proposed scheme is proved to be maximum-principle-preserving, energy-stability-preserving and variational-energy-dissipation-preserving in the discrete sense. Moreover, an error estimate of the numerical scheme is also derived. In the end, a series of numerical experiments are carried out to illustrate the obtained theoretical results.

Time-fractional Allen–Cahn equations versus powers of the mean curvature

We show by a formal asymptotic expansion that level sets of solutions of a time-fractional Allen–Cahn equation evolve by a geometric flow whose normal velocity is a positive power of the mean curvature.This connection is quite intriguing, since the original equation is nonlocal and the evolution of its solutions depends on all previous states, but the associated geometric flow is of purely local type, with no memory effect involved.

Extension of Cubic B-Spline for Solving the Time-Fractional Allen–Cahn Equation in the Context of Mathematical Physics

A B-spline is defined by the degree and quantity of knots, and it is observed to provide a higher level of flexibility in curve and surface layout. The extended cubic B-spline (ExCBS) functions with new approximation for second derivative and finite difference technique are incorporated in this study to solve the time-fractional Allen–Cahn equation (TFACE). Initially, Caputo’s formula is used to discretize the time-fractional derivative, while a new ExCBS is used for the spatial derivative’s discretization. Convergence analysis is carried out and the stability of the proposed method is also analyzed. The scheme’s applicability and feasibility are demonstrated through numerical analysis.

An efficient technique based on barycentric interpolation collocation method for the time fractional Allen-Cahn equation

An efficient technique based on barycentric interpolation collocation method for the time fractional Allen-Cahn equation

A High-Order Discrete Energy Decay and Maximum-Principle Preserving Scheme for Time Fractional Allen–Cahn Equation

A High-Order Discrete Energy Decay and Maximum-Principle Preserving Scheme for Time Fractional Allen–Cahn Equation

A Decreasing Upper Bound of the Energy for Time-Fractional Phase-Field Equations

In this article, we study the energy dissipation property of time-fractional Allen-Cahn equation. We propose a decreasing upper bound of energy that decreases with respect to time and coincides with the original energy at $t = 0$ and as $t$ tends to $\infty$. This upper bound can also be viewed as a nonlocal-in-time modified energy, the summation of the original energy and an accumulation term due to the memory effect of time fractional derivative. In particular, this indicates that the original energy indeed decays w.r.t. time in a small neighborhood at $t=0$. We illustrate the theory mainly with the time-fractional Allen-Cahn equation, but it could be applied to other time-fractional phase-field models such as the Cahn-Hilliard equation. On the discrete level, the first-order L1 and second-order L2 schemes for time-fractional Allen-Cahn equation have similar decreasing modified energies, so that the stability can be established. Some numerical results are provided to illustrate the behavior of this modified energy and to verify our theoretical results.

Exact special solutions of space–time fractional Cahn–Allen equation by beta and M-truncated derivatives

In this paper, we consider the nonlinear space–time fractional form of Cahn–Allen equation (FCAE) with beta and M-truncated derivatives. Cahn–Allen equation (CAE) is commonly used in many problems of physics and engineering, such as, solidification problems, phase separation in iron alloys and others. We apply the improved [Formula: see text]-expansion method (ITEM). We obtain four types of traveling wave solutions, including, trigonometric, hyperbolic, rational and exponential function solutions. We demonstrate some of the extracted solutions using definitions of the beta (BD) and M-truncated derivatives (MTD) to understand their dynamical behavior. We observe the fractional effects of the aforementioned derivatives on the related physical phenomena up to possible extent.

A linearly stabilized convolution quadrature method for the time-fractional Allen–Cahn equation

This paper proposes a linearly stabilized convolution quadrature method for solving the time-fractional Allen–Cahn equation. The stability condition is explicitly given such that the method is unconditionally stable for any time step size. The space is discretized by the central difference method. We prove that the fully discrete scheme preserves the discrete maximum principle, the discrete energy is bounded, and the modified discrete energy decays monotonically. Numerical simulations support the theoretical analysis.

Superconvergence analysis of nonconforming finite element method for two-dimensional time-fractional Allen–Cahn equation

In this paper, we investigate the numerical solution of the time-fractional Allen–Cahn equation with variable diffusion coefficients. A fully discrete approximation is presented by the L2-1σ scheme on graded meshes and spatial high precision nonconforming finite element method (FEM). The nonlinear term is treated with the Newton linearization method. Based on the modified discrete fractional Grönwall inequality, we rigorously prove that the proposed scheme can achieve optimal convergence accuracy in both time and space directions. Furthermore, the H1-norm global superconvergence result is obtained through the interpolation postprocessing technique. Finally, numerical experiments confirm the correctness of our theoretical analysis.

Strong convergence rates for the approximation of a stochastic time-fractional Allen–Cahn equation

The paper is concerned with the strong approximation of a stochastic time-fractional Allen-Cahn equation driven by an additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time; namely, a Caputo fractional derivative of order α∈(0,1), and a Riemann–Liouville fractional integral operator of order γ∈[0,1] applied to a Gaussian noise. We approximate the model by a standard piecewise linear finite element method (FEM) in space and the classical Grünwald–Letnikov method in time (for both time-fractional operators), and the noise by the L2-projection. Spatially semidiscrete and fully discrete schemes are analyzed and strong convergence rates are obtained by exploiting the temporal Hölder continuity property of the solution. Numerical experiments are presented to illustrate the theoretical results.

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

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.

The Variable-Step L1 Scheme Preserving a Compatible Energy Law for Time-Fractional Allen-Cahn Equation

In this work, we revisit the adaptive L1 time-stepping scheme for solving the time-fractional Allen-Cahn equation in the Caputo's form. The L1 implicit scheme is shown to preserve a variational energy dissipation law on arbitrary nonuniform time meshes by using the recent discrete analysis tools, i.e., the discrete orthogonal convolution kernels and discrete complementary convolution kernels. Then the discrete embedding techniques and the fractional Gr\"onwall inequality were applied to establish an $L^2$ norm error estimate on nonuniform time meshes. An adaptive time-stepping strategy according to the dynamical feature of the system is presented to capture the multi-scale behaviors and to improve the computational performance.

Energy stable L2 schemes for time-fractional phase-field equations

In this work, we establish the energy stability of high-order L2-type schemes for time-fractional phase-field equations. We propose a reformulation of the discrete L2 operator, show the monotonicity of some associated coefficients, and then obtain two positive definiteness properties for the L2 operator. Based on these two properties, we show that the energy is bounded by the initial energy for the L2 scalar auxiliary variable schemes of time-fractional gradient flows. Furthermore, a fractional energy law can be established for the L2 implicit-explicit scheme of the time-fractional Allen–Cahn equation. Several numerical experiments are provided to verify the stability as well as the convergence.

Highly efficient schemes for time-fractional Allen-Cahn equation using extended SAV approach

In this paper, we propose and analyze high-order efficient schemes for the time-fractional Allen-Cahn equation. The proposed schemes are based on the L1 discretization for the time-fractional derivative and the extended scalar auxiliary variable (SAV) approach developed very recently to deal with the nonlinear terms in the equation. The main contributions of the paper consist of (1) constructing first- and higher order unconditionally stable schemes for different mesh types, and proving the unconditional stability of the constructed schemes for the uniform mesh; (2) carrying out numerical experiments to verify the efficiency of the schemes and to investigate the coarsening dynamics governed by the time-fractional Allen-Cahn equation. In particular, the influence of the fractional order on the coarsening behavior is carefully examined. Our numerical evidence shows that the proposed schemes are more robust than the existing methods, and their efficiency is less restricted to particular forms of the nonlinear potentials.

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

In this paper, we give a stabilized second order scheme for the time fractional Allen-Cahn equation. The scheme uses the fractional backward difference formula (FBDF) for the time fractional derivative and the Legendre spectral method for the space approximation. The nonlinear terms are treated implicitly with a second order stabilized term. Based on the fractional Grönwall inequality, we strictly prove that the proposed scheme converges to second order accuracy in time and spectral accuracy in space. To save computation time and storage, a fast evaluation is developed. Finally, we give some numerical examples to show the configurations of phase field evolution and verify the effectiveness of the proposed methods.

A modified exp-function method for fractional partial differential equations

This paper proposes a novel exponential rational function method, a modification of the well-known exp-function method, to find exact solutions of the time fractional Cahn-Allen equation and the time fractional Phi-4 equation. The solution procedure is reduced to solve a system of algebraic equations, which is then solved by Wu?s method. The results show that the present method is effective, and can be applied to other fractional differential equations.

An Energy Stable and Maximum Bound Preserving Scheme with Variable Time Steps for Time Fractional Allen--Cahn Equation

In this work, we propose a Crank-Nicolson-type scheme with variable steps for the time fractional Allen-Cahn equation. The proposed scheme is shown to be unconditionally stable (in a variational energy sense), and is maximum bound preserving. Interestingly, the discrete energy stability result obtained in this paper can recover the classical energy dissipation law when the fractional order $\alpha \rightarrow 1.$ That is, our scheme can asymptotically preserve the energy dissipation law in the $\alpha \rightarrow 1$ limit. This seems to be the first work on variable time-stepping scheme that can preserve both the energy stability and the maximum bound principle. Our Crank-Nicolson scheme is build upon a reformulated problem associated with the Riemann-Liouville derivative. As a by product, we build up a reversible transformation between the L1-type formula of the Riemann-Liouville derivative and a new L1-type formula of the Caputo derivative, with the help of a class of discrete orthogonal convolution kernels. This is the first time such a \textit{discrete} transformation is established between two discrete fractional derivatives. We finally present several numerical examples with an adaptive time-stepping strategy to show the effectiveness of the proposed scheme.

Time-Fractional Allen–Cahn Equations: Analysis and Numerical Methods

In this work, we consider a time-fractional Allen–Cahn equation, where the conventional first order time derivative is replaced by a Caputo fractional derivative with order $$\alpha \in (0,1)$$ . First, the well-posedness and (limited) smoothing property are studied, by using the maximal $$L^p$$ regularity of fractional evolution equations and the fractional Grönwall’s inequality. We also show the maximum principle like their conventional local-in-time counterpart, that is, the time-fractional equation preserves the property that the solution only takes value between the wells of the double-well potential when the initial data does the same. Second, after discretizing the fractional derivative by backward Euler convolution quadrature, we develop several unconditionally solvable and stable time stepping schemes, such as a convex splitting scheme, a weighted convex splitting scheme and a linear weighted stabilized scheme. Meanwhile, we study the discrete energy dissipation property (in a weighted average sense), which is important for gradient flow type models, for the two weighted schemes. In addition, we prove the fractional energy dissipation law for the gradient flow associated with a convex free energy. Finally, using a discrete version of fractional Grönwall’s inequality and maximal $$\ell ^p$$ regularity, we prove that the convergence rates of those time-stepping schemes are $$O(\tau ^\alpha )$$ without any extra regularity assumption on the solution. We also present extensive numerical results to support our theoretical findings and to offer new insight on the time-fractional Allen–Cahn dynamics.

A high-efficiency second-order numerical scheme for time-fractional phase field models by using extended SAV method

In this paper, a second-order numerical scheme for the time-fractional phase field models is proposed. In this scheme, the fractional backward difference formula is used to approximate the time-fractional derivative and the extended scalar auxiliary variable method is used to deal with the nonlinear terms. The energy dissipation property for the numerical scheme is proved. Our discussion includes the time-fractional Allen–Cahn equation, the time-fractional Cahn–Hilliard equation, and the time-fractional molecular beam epitaxy model. In the numerical implementation, a fast method based on a globally uniform approximation of the trapezoidal rule for the integral on the real line is adopted to decrease the memory requirement and computational cost. Finally, some numerical examples are given to confirm the effectiveness of the proposed methods.

Optimal H1 spatial convergence of a fully discrete finite element method for the time-fractional Allen-Cahn equation

A time-fractional Allen-Cahn problem is considered, where the spatial domain Ω is a bounded subset of $\mathbb {R}^{d}$ for some d ∈{1,2,3}. New bounds on certain derivatives of the solution are derived. These are used in the analysis of a numerical method (L1 discretization of the temporal fractional derivative on a graded mesh, with a standard finite element discretization of the spatial diffusion term, and Newton linearization of the nonlinear driving term), showing that the computed solution achieves the optimal rate of convergence in the Sobolev H1(Ω) norm. (Previous papers considered only convergence in L2(Ω).) Numerical results confirm our theoretical findings.